SOURCES IN RECREATIONAL
MATHEMATICS
AN ANNOTATED BIBLIOGRAPHY
EIGHTH PRELIMINARY EDITION
DAVID SINGMASTER
Contact via the Puzzle Museum
http://puzzlemuseum.com
Last
updated on 19 March 2004.
This is a copy of the current version
from my source files. I had intended to
reorganise the material before producing a Word version, but have decided to
produce this version for G4G6 and to renumber it as the Eighth Preliminary
Edition.
If I have perchance omitted anything
more or less proper or necessary, I beg indulgence, since there is no one who
is blameless and utterly provident in all things. [Fibonacci, translated by Grimm.])
DIACRITICAL MARKS AND NOTATION
SOME OTHER RECURRING REFERENCES
1. BIOGRAPHICAL
MATERIAL -- in chronological order
2. GENERAL
PUZZLE COLLECTIONS AND SURVEYS
3. GENERAL
HISTORICAL AND BIBLIOGRAPHICAL MATERIAL
3.A. GENERAL
HISTORICAL MATERIAL
4.A. GENERAL
THEORY AND NIM‑LIKE GAMES
4.B.1. TIC‑TAC‑TOE =
NOUGHTS AND CROSSES
4.B.5. OVID'S GAME AND NINE MEN'S MORRIS
4.B.12. RITHMOMACHIA
= THE PHILOSOPHERS' GAME
5.A.2. THREE DIMENSIONAL VERSIONS
5.B.1. LOWERING FROM TOWER PROBLEM
5.B.2. CROSSING A BRIDGE WITH A TORCH
5.C. FALSE
COINS WITH A BALANCE
5.C.1RANKING COINS WITH A BALANCE
5.D.2. RULER WITH MINIMAL NUMBER OF MARKS
5.D.3 FALSE COINS WITH A WEIGHING SCALE
5.D.4. TIMING WITH HOURGLASSES
5.E.2. MEMORY WHEELS =
CHAIN CODES
5.F.1. KNIGHT'S TOURS AND PATHS
5.F.2. OTHER HAMILTONIAN CIRCUITS
5.F.3. KNIGHT'S TOURS IN HIGHER DIMENSIONS
5.F.4. OTHER CIRCUITS IN AND ON A CUBE
5.G.1. GAS, WATER AND ELECTRICITY
5.H. COLOURED
SQUARES AND CUBES, ETC.
5.H.1. INSTANT INSANITY =
THE TANTALIZER
5.I. LATIN
SQUARES AND EULER SQUARES
5.I.2. COLOURING CHESSBOARD WITH NO REPEATS
IN A LINE
5.J.3. TILING A SQUARE OF SIDE 70 WITH
SQUARES OF SIDES 1, 2, ..., 24
5.K.1. DERANGED BOXES OF A,
B AND A & B
5.K.2. OTHER LOGIC PUZZLES BASED ON
DERANGEMENTS
5.M. SIX
PEOPLE AT A PARTY -- RAMSEY THEORY
5.N. JEEP OR
EXPLORER'S PROBLEM
5.O. TAIT'S
COUNTER PUZZLE: BBBBWWWW TO
WBWBWBWB
5.P. GENERAL
MOVING PIECE PUZZLES
5.Q. NUMBER
OF REGIONS DETERMINED BY N LINES OR PLANES
5.Q.1. NUMBER
OF INTERSECTIONS DETERMINED BY N LINES
5.R.2. FROGS AND TOADS: BBB_WWW
TO WWW_BBB
5.R.3. FORE AND AFT -- 3
BY 3 SQUARES MEETING AT A CORNER
5.R.4. REVERSING FROGS AND TOADS: _12...n
TO _n...21 , ETC.
5.S. CHAIN
CUTTING AND REJOINING
5.S.1. USING CHAIN LINKS TO PAY FOR A ROOM
5.W. MAKING
THREE PIECES OF TOAST
5.X. COUNTING
FIGURES IN A PATTERN
5.X.2. COUNTING RECTANGLES OR SQUARES
5.Y. NUMBER
OF ROUTES IN A LATTICE
5.Z. CHESSBOARD
PLACING PROBLEMS
5.AB. FOLDING
A STRIP OF STAMPS
5.AC. PROPERTIES OF THE SEVEN BAR DIGITAL DISPLAY
5.AD. STACKING
A DECK TO PRODUCE A SPECIAL EFFECT
5.AG. RUBIK'S
CUBE AND SIMILAR PUZZLES
6.F.1. OTHER CHESSBOARD DISSECTIONS
6.F.2. COVERING DELETED CHESSBOARD WITH
DOMINOES
6.F.3. DISSECTING A CROSS INTO Zs
AND Ls
6.F.4. QUADRISECT AN L‑TROMINO, ETC.
6.F.5. OTHER DISSECTIONS INTO POLYOMINOES
6.G.2. DISSECTION OF 63 INTO 33, 43 AND 53, ETC.
6.G.3. DISSECTION OF A DIE INTO NINE 1 x 1 x 3
6.G.4. USE OF OTHER POLYHEDRAL PIECES
6.I. SYLVESTER'S
PROBLEM OF COLLINEAR POINTS
6.J. FOUR BUGS
AND OTHER PURSUIT PROBLEMS
6.K. DUDENEY'S
SQUARE TO TRIANGLE DISSECTION
6.N. DISSECTION
OF A 1 x 1 x 2 BLOCK TO A CUBE
6.O. PASSING
A CUBE THROUGH AN EQUAL OR SMALLER CUBE. PRINCE RUPERT'S PROBLEM
6.P.1. PARADOXICAL DISSECTIONS OF THE
CHESSBOARD BASED ON FIBONACCI NUMBERS 286
6.Q. KNOTTING
A STRIP TO MAKE A REGULAR PENTAGON
6.R.1. EVERY TRIANGLE IS ISOSCELES
6.R.2. A RIGHT ANGLE IS OBTUSE
6.R.3. LINES APPROACHING BUT NOT MEETING
6.T. NO THREE
IN A LINE PROBLEM
6.U.2. PACKING BRICKS IN BOXES
6.V. SILHOUETTE
AND VIEWING PUZZLES
6.W.2. SIX PIECE BURR =
CHINESE CROSS
6.W.3. THREE PIECE BURR WITH IDENTICAL
PIECES
6.W.4. DIAGONAL SIX PIECE BURR =
TRICK STAR
6.W.5. SIX PIECE BURR WITH IDENTICAL
PIECES
6.X. ROTATING
RINGS OF POLYHEDRA
6.Z. LANGLEY'S
ADVENTITIOUS ANGLES
6.AD.1. LARGEST PARCEL ONE CAN POST
6.AE. 6" HOLE THROUGH SPHERE LEAVES CONSTANT VOLUME
6.AF. WHAT
COLOUR WAS THE BEAR?
6.AJ.2. TRIBAR AND IMPOSSIBLE STAIRCASE
6.AK. POLYGONAL
PATH COVERING N x N LATTICE OF POINTS,
QUEEN'S TOURS, ETC.
6.AN. VOLUME OF
THE INTERSECTION OF TWO CYLINDERS
6.AO.1. PLACE FOUR POINTS EQUIDISTANTLY =
MAKE FOUR TRIANGLES WITH SIX MATCHSTICKS
6.AO.2. PLACE AN EVEN NUMBER ON EACH LINE
6.AP .
DISSECTIONS OF A TETRAHEDRON
6.AQ. DISSECTIONS
OF A CROSS, T OR H
6.AR. QUADRISECTED
SQUARE PUZZLE
6.AS. DISSECTION
OF SQUARES INTO A SQUARE
6.AS.1. TWENTY 1, 2, Ö5
TRIANGLES MAKE A SQUARE OR FIVE EQUAL SQUARES TO A SQUARE
6.AS.1.a. GREEK CROSS TO A SQUARE
6.AS.1.b. OTHER GREEK CROSS DISSECTIONS
6.AS.2. TWO (ADJACENT) SQUARES TO A SQUARE
6.AS.2.a. TWO EQUAL SQUARES TO A SQUARE
6.AS.3. THREE EQUAL SQUARES TO A SQUARE
6.AS.3.a. THREE EQUAL 'SQUARES' TO A HEXAGON
6.AS.4. EIGHT EQUAL SQUARES TO A SQUARE
6.AS.5. RECTANGLE TO A SQUARE OR OTHER
RECTANGLE
6.AT. POLYHEDRA
AND TESSELLATIONS
6.AT.2 STAR AND STELLATED POLYHEDRA
6.AT.5. REGULAR‑FACED POLYHEDRA
6.AT.6.a. TESSELLATING WITH CONGRUENT FIGURES
6.AU. THREE
RABBITS, DEAD DOGS AND TRICK MULES
MODERN VERSIONS OF THE THREE RABBITS PUZZLE
6.AV. CUTTING
UP IN FEWEST CUTS
6.AW. DIVISION
INTO CONGRUENT PIECES
6.AW.3. DIVIDING A SQUARE INTO CONGRUENT
PARTS
6.AW.4. DIVIDING AN L-TROMINO INTO CONGRUENT PARTS
6.AY. DISSECT 3A x 2B
TO MAKE 2A x 3B, ETC.
6.BA. CUTTING
A CARD SO ONE CAN PASS THROUGH IT
6.BB. DOUBLING
AN AREA WITHOUT CHANGING ITS HEIGHT OR WIDTH
6.BD. BRIDGE A MOAT WITH PLANKS
6.BE. REVERSE A
TRIANGULAR ARRAY OF TEN CIRCLES
6.BF.2. SLIDING SPEAR =
LEANING REED
6.BF.3. WELL BETWEEN TWO TOWERS
6.BF.5. TRAVELLING ON SIDES OF A RIGHT
TRIANGLE.
6.BG. QUADRISECT
A PAPER SQUARE WITH ONE CUT
6.BI. VENN DIAGRAMS FOR N SETS
6.BL. TAN-1
⅓ + TAN-1 ½ = TAN-1 1, ETC.
6.BM. DISSECT
CIRCLE INTO TWO HOLLOW OVALS
6.BN. ROUND PEG
IN SQUARE HOLE OR VICE VERSA
6.BP. EARLY MATCHSTICK
PUZZLES
6.BQ. COVERING A
DISC WITH DISCS
6.BR. WHAT IS A
GENERAL TRIANGLE?
6.BS. FORM SIX
COINS INTO A HEXAGON
6.BT. PLACING
OBJECTS IN CONTACT
6.BW. DISTANCES
TO CORNERS OF A SQUARE
7. ARITHMETIC &
NUMBER‑THEORETIC RECREATIONS
7.B. JOSEPHUS
OR SURVIVOR PROBLEM
7.E. MONKEY
AND COCONUTS PROBLEMS
7.E.1. VERSIONS WITH ALL GETTING THE SAME
7.F. ILLEGAL
OPERATIONS GIVING CORRECT RESULT
7.G.1. HALF + THIRD + NINTH, ETC.
7.H. DIVISION
AND SHARING PROBLEMS -- CISTERN PROBLEMS
7.H.1. WITH GROWTH -- NEWTON'S CATTLE
PROBLEM
7.H.3. SHARING UNEQUAL RESOURCES --
PROBLEM OF THE PANDECTS
7.H.4. EACH DOUBLES OTHERS' MONEY TO MAKE
ALL EQUAL, ETC.
7.H.5. SHARING COST OF STAIRS, ETC.
7.H.7. DIGGING PART OF A WELL.
7.I.1. LARGEST NUMBER USING FOUR ONES, ETC.
7.L.1. 1 + 7 + 49 + ... &
ST. IVES
7.L.2.b. HORSESHOE NAILS PROBLEM
7.L.2.c. USE OF 1, 2, 4, ... AS WEIGHTS, ETC.
7.L.3. 1 + 3 + 9 + ... AND OTHER SYSTEMS OF
WEIGHTS
7.M. BINARY
SYSTEM AND BINARY RECREATIONS
7.M.2.a. TOWER OF HANOI WITH MORE PEGS
7.M.4.b. OTHER DIVINATIONS USING BINARY OR
TERNARY
7.M.5. LOONY
LOOP =
GORDIAN KNOT
7.N.3. ANTI‑MAGIC SQUARES AND
TRIANGLES
7.P.1. HUNDRED FOWLS AND OTHER LINEAR
PROBLEMS
7.P.2. CHINESE REMAINDER THEOREM
7.P.3. ARCHIMEDES' CATTLE PROBLEM
7.P.5 . SELLING DIFFERENT AMOUNTS 'AT SAME
PRICES' YIELDING THE SAME
7.P.6. CONJUNCTION OF PLANETS, ETC.
7.Q.1. REARRANGEMENT ON A CROSS
7.Q.2. REARRANGE A CROSS OF SIX TO MAKE TWO
LINES OF FOUR, ETC.
7.R. "IF
I HAD ONE FROM YOU, I'D HAVE TWICE YOU"
7.R.1. MEN FIND A PURSE AND 'BLOOM OF
THYMARIDAS'
7.R.2. "IF I HAD 1/3
OF YOUR MONEY, I COULD BUY THE HORSE"
7.R.4. "IF I SOLD YOUR EGGS AT MY
PRICE, I'D GET ...."
7.S. DILUTION
AND MIXING PROBLEMS
7.S.1. DISHONEST BUTLER DRINKING SOME AND
REPLACING WITH WATER
7.S.2. WATER IN WINE VERSUS WINE IN WATER
7.V. XY = YX AND ITERATED EXPONENTIALS
7.X. HOW OLD
IS ANN? AND OTHER AGE PROBLEMS
7.Y. COMBINING
AMOUNTS AND PRICES INCOHERENTLY
7.Y.1. REVERSAL OF AVERAGES PARADOX
7.Z. MISSING
DOLLAR AND OTHER ERRONEOUS ACCOUNTING
7.AC. CRYPTARITHMS,
ALPHAMETICS AND SKELETON ARITHMETIC
7.AC.1. CRYPTARITHMS: SEND + MORE
= MONEY, ETC.
7.AC.2. SKELETON ARITHMETIC: SOLITARY SEVEN, ETC.
7.AC.3.a INSERTION OF SIGNS TO MAKE 100,
ETC.
7.AC.6. OTHER PAN‑DIGITAL AND SIMILAR
PROBLEMS
7.AC.7. SELF-DESCRIPTIVE NUMBERS, PANGRAMS,
ETC.
7.AD. SELLING, BUYING AND SELLING
SAME ITEM
7.AE. USE OF
COUNTERFEIT BILL OR FORGED CHEQUE
7.AH. MULTIPLYING
BY REVERSING
7.AH.1. OTHER REVERSAL PROBLEMS
7.AI.
IMPOSSIBLE EXCHANGE RATES
7.AJ.1. MULTIPLYING BY APPENDING DIGITS
7.AN. THREE
ODDS MAKE AN EVEN, ETC.
7.AO. DIVINATION
OF A PERMUTATION
7.AP. KNOWING SUM
VS KNOWING PRODUCT
7.AQ. NUMBERS IN
ALPHABETIC ORDER
7.AU. NUMBER OF
CUTS TO MAKE N PIECES
7.AV. HOW LONG
TO STRIKE TWELVE?
7.AZ.
DIVINATION OF A PAIR OF CARDS FROM ITS ROWS
7.BA. CYCLE OF
NUMBERS WITH EACH CLOSER TO TEN THAN THE PREVIOUS
7.BB. ITERATED
FUNCTIONS OF INTEGERS
7.BC. UNUSUAL DIFFICULTY
IN GIVING CHANGE
8.D. ATTEMPTS
TO MODIFY BOY‑GIRL RATIO
8.G. PROBABILITY
THAT THREE LENGTHS FORM A TRIANGLE
8.H.2. BERTRAND'S CHORD PARADOX
8.J. CLOCK
PATIENCE OR SOLITAIRE
9.A. ALL
CRETANS ARE LIARS, ETC.
9.B. SMITH
-- JONES -- ROBINSON PROBLEM
GENERAL STUDIES OF KINSHIP RELATIONS
9.E.1. THAT MAN'S FATHER IS MY FATHER'S SON,
ETC.
9.E.2. IDENTICAL SIBLINGS WHO ARE NOT TWINS
10.A. OVERTAKING
AND MEETING PROBLEMS
10.A.3. TIMES FROM MEETING TO FINISH GIVEN
10.A.6. DOUBLE CROSSING PROBLEMS
10.C. LEWIS
CARROLL'S MONKEY PROBLEM
10.D.1 MIRROR REVERSAL PARADOX
10.E.1. ARISTOTLE'S WHEEL PARADOX
10.E.2. ONE WHEEL ROLLING AROUND ANOTHER
10.E.4. RAILWAY WHEELS PARADOX
10.H. SNAIL
CLIMBING OUT OF WELL
10.I. LIMITED
MEANS OF TRANSPORT -- TWO MEN AND A BIKE, ETC.
10.K. PROBLEM
OF THE DATE LINE
10.L. FALLING
DOWN A HOLE THROUGH THE EARTH
10.N. SHIP'S
LADDER IN RISING TIDE
10.O. ERRONEOUS
AVERAGING OF VELOCITIES
10.U. SHORTEST
ROUTE VIA A WALL, ETC.
10.V. PICK UP
PUZZLES = PLUCK IT
10.X. HOW FAR
DOES A PHONOGRAPH NEEDLE TRAVEL?
10.Y. DOUBLE
CONE ROLLING UPHILL
10.AB. BICYCLE
TRACK PROBLEMS.
11.B. TWO
PEOPLE JOINED BY ROPES AT WRISTS
11.C. TWO BALLS
ON STRING THROUGH LEATHER HOLE AND STRAP
= CHERRIES PUZZLE
11.H. REMOVING
WAISTCOAT WITHOUT REMOVING COAT
11.H.1. REMOVING LOOP FROM ARM
11.I. HEART
AND BALL PUZZLE AND OTHER LOOP PUZZLES
11.K.1. RING AND SPRING PUZZLE
11.K.2. STRING AND SPRING PUZZLE
11.K.3. MAGIC CHAIN =
TUMBLE RINGS
11.K.6. INTERLOCKED NAILS, HOOKS, HORNS,
ETC.
11.L. JACOB'S
LADDER AND OTHER HINGING DEVICES
11.N. THREE
KNIVES MAKE A SUPPORT
11.Q. TURNING AN
INNER TUBE INSIDE OUT
Recreational mathematics is as old as
mathematics itself. Recreational
problems already occur in the oldest extant sources -- the Rhind Papyrus and
Old Babylonian tablets. The Rhind
Papyrus has an example of a purely recreational problem -- Problem 79 is like
the "As I was going to St. Ives" nursery rhyme. The Babylonians give fairly standard
practical problems with a recreational context -- a man knows the area plus the
difference of the length and width of his field, a measurement which no
surveyor would ever make! There is even
some prehistoric mathematics which could not have been practical -- numerous
'carved stone balls' have been found in eastern Scotland, dating from the
Neolithic period and they include rounded forms of all the regular polyhedra
and some less regular ones. Since these
early times, recreations have been a feature of mathematics, both as pure
recreations and as pedagogic tools. In
this work, I use recreational in a fairly broad sense, but I tend to omit the
more straightforward problems and concentrate on those which 'stimulate the
curiosity' (as Montucla says).
In addition, recreational mathematics
is certainly as diffuse as mathematics.
Every main culture and many minor ones have contributed to the
history. A glance at the Common
References below, or at almost any topic in the text, will reveal the diversity
of sources which are relevant to this study.
Much information arises from material outside the purview of the
ordinary historian of mathematics -- e.g.
patents; articles in newspapers,
popular magazines and minor journals;
instruction leaflets; actual
artifacts and even oral tradition.
Consequently, it is very difficult to
determine the history of any recreational topic and the history given in
popular books is often extremely dubious or even simply fanciful. For example, Nim, Tangrams, and Magic
Squares are often traced back to China of about 2000 BC. The oldest
known reference to Nim is in America in 1903.
Tangrams appear in China and Europe at essentially the same time, about
1800, though there are related puzzles in 18C Japan and in the Hellenistic
world. Magic Squares seem to be
genuinely a Chinese invention, but go back to perhaps a few centuries BC and
are not clearly described until about 80AD.
Because of the lack of a history of the field, results are frequently
rediscovered.
When I began this bibliography in
1982, I had the the idea of producing a book (or books) of the original
sources, translated into English, so people could read the original
material. This bibliography began as
the table of contents of such a book. I
thought that this would be an easy project, but it has become increasingly
apparent that the history of most recreations is hardly known. I have recently realised that mathematical
recreations are really the folklore of mathematics and that the historical
problems are similar to those of folklore.
One might even say that mathematical recreations are the urban myths or
the jokes or the campfire stories of mathematics. Consequently I decided that an annotated bibliography was the
first necessity to make the history clearer.
This bibliography alone has grown into a book, something like Dickson's History
of the Theory of Numbers. Like that
work, the present work divides the subject into a number of topics and treats
them chronologically.
I have printed six preliminary
editions of this work, with slightly varying titles. The first version of 4 Jul 1986 had 224 topics and was spaced out
so entries would not be spread over two pages and to give room for page
numbers. This stretched the text from
110pp to 129pp and was printed for the Strens Memorial Conference at the Univ.
of Calgary in Jul/Aug 1986. I no longer
worry about page breaks. The following
editions had: 250 topics on 152 pages; 290 topics on 192 pages; 307 topics on 223 pages; 357 topics on 311 pages and 392 topics on 456 pages. The seventh edition was never printed, but
was a continually changing computer file.
It had about 419 topics (as of 20 Oct 95) and 587 pages, as of 20 Oct
1995. I then carried out the conversion
to proportional spacing and this reduced the total length from 587 to 488
pages, a reduction of 16.87% which is conveniently estimated as 1/6. This reduction was fairly consistent
throughout the conversion process.
This eighth edition is being prepared
for the Gathering for Gardner 6 in March 2004.
The text is 818 pages as of 18 Mar 2004. There are about 457 topics as of 18 Mar 2004.
A fuller description of this project
in 1984-1985 is given in my article Some early sources in recreational
mathematics, in: C. Hay et al., eds.; Mathematics from Manuscript to
Print; Oxford Univ. Press, 1988, pp. 195‑208. A more recent description is in my article: Recreational
mathematics; in: Encyclopedia of the History and Philosophy of the
Mathematical Sciences; ed. by I. Grattan-Guinness; Routledge & Kegan
Paul, 1993; pp. 1568-1575.
Below I compare this work with Dickson
and similar works and discuss the coverage of this work.
As already mentioned, the work which
the present most resembles is Dickson's History of the Theory of Numbers.
The history of science can
be made entirely impartial, and perhaps that is what it should be, by merely
recording who did what, and leaving all "evaluations" to those who
like them. To my knowledge there is
only one history of a scientific subject (Dickson's, of the Theory of Numbers)
which has been written in this coldblooded, scientific way. The complete success of that unique example
-- admitted by all who ever have occasion to use such a history in their work
-- seems to indicate that historians who draw morals should have their own
morals drawn.
E. T. Bell. The Search for Truth. George Allen & Unwin, London, 1935,
p. 131.
Dickson attempted to be exhaustive and
certainly is pretty much so. Since his time,
many older sources have been published, but their number-theoretic content is
limited and most of Dickson's topics do not go back that far, so it remains the
authoritative work in its field.
The best previous book covering the
history of recreational mathematics is the second edition of Wilhelm Ahrens's Mathematische
Unterhaltungen und Spiele in two volumes.
Although it is a book on recreations, it includes extensive histories of
most of the topics covered, far more than in any other recreational book. He also gives a good index and a
bibliography of 762 items, often with some bibliographical notes. I will indicate the appropriate pages at the
beginning of any topic that Ahrens covers.
This has been out of print for many years but Teubner has some plans to
reissue it.
Another similar book is the 4th
edition of J. Tropfke's Geschichte der Elementarmathematik, revised by
Vogel, Reich and Gericke. This is quite
exhaustive, but is concerned with older problems and sources. It presents the material on a topic as a
history with references to the sources, but it doesn't detail what is in each
of the sources. Sadly, only one volume,
on arithmetic and algebra, appeared before Vogel's death. A second volume, on geometry, is being
prepared. For any topic covered in
Tropfke, it should be consulted for further references to early material which
I have not seen, particularly material not available in any western
language. I cite the appropriate pages
of Tropfke at the beginning of any topic covered by Tropfke.
Another book in the field is W. L.
Schaaf's Bibliography of Recreational Mathematics, in four volumes. This is a quite exhaustive bibliography of
recent articles, but it is not chronological, is without annotation and is
somewhat less classified than the present work. Nonetheless it is a valuable guide to recent material.
Collecting books on magic has been
popular for many years and quite notable collections and bibliographies have
been made. Magic overlaps recreational
mathematics, particularly in older books, and I have now added references to
items listed in the bibliographies of Christopher, Clarke & Blind, Hall,
Heyl, Toole Stott and Volkmann & Tummers -- details of these works are
given in the list of Common References below.
There is a notable collection of Harry Price at Senate House, University
of London, and a catalogue was printed in 1929 & 1935 -- see HPL in Common
References.
Another related bibliography is
Santi's Bibliografia della Enigmistica, which is primarily about word
puzzles, riddles, etc., but has some overlap with recreational mathematics --
again see the entry in the list of Common References. I have not finished working through this.
Other relevant bibliographies are
listed in Section 3.B.
In selecting topics, I tend to avoid
classical number theory and classical geometry. These are both pretty well known. Dickson's History of the Theory of Numbers and Leveque's
and Guy's Reviews in Number Theory cover number theory quite well. I also tend to avoid simple exercises, e.g.
in the rule of three, in 'aha' or 'heap' problems, in the Pythagorean theorem
(though I have now included 6.BF) or in two linear equations in two unknowns,
though these often have fanciful settings which are intended to make them
amusing and some of these are included -- see
7.R, 7.X, 7.AX.
I also leave out most divination (or 'think of a number') techniques
(but a little is covered in 7.M.4.b) and most arithmetic fallacies. I also leave out Conway's approach to
mathematical games -- this is extensively covered by Winning Ways and
Frankel's Bibliography.
The classification of topics is still
ad‑hoc and will eventually get rationalised -- but it is hard to sort
things until you know what they are! At
present I have only grouped them under the general headings: Biography,
General, History & Bibliography, Games, Combinatorics, Geometry,
Arithmetic, Probability, Logic, Physics, Topology. Even the order of these should be amended. The General section should be subsumed under
the History & Bibliography.
Geometry and Arithmetic need to be subdivided.
I have recently realised that some
general topics are spread over several sections in different parts. E.g. fallacies are covered in 6.P, 6.R,
6.AD, 6.AW.1, 6.AY, 7.F, 7.Y, 7.Z, 7.AD, 7.AI, 7.AL, 7.AN, most of 8, 10.D,
10.E, 10.O. Perhaps I will produce an
index to such topics. I try to make
appropriate cross-references.
Some topics are so extensive that I
include introductory or classifactory material at the beginning. I often give a notation for the problems
being considered. I give brief
explanations of those problems which are not well known or are not described in
the notation or the early references.
There may be a section index. I
have started to include references to comprehensive surveys of a given topic --
these are sometimes given at the beginning.
Recreational problems are repeated so
often that it is impossible to include all their occurrences. I try to be exhaustive with early material,
but once a problem passes into mathematical and general circulation, I only
include references which show new aspects of the problem or show how the
problem is transmitted in time and/or space.
However, the point at which I start leaving out items may vary with time
and generally slowly increases as I learn more about a topic. I include numerous variants and developments
on problems, especially when the actual origin is obscure.
When I began, I made minimal
annotations, often nothing at all. In rereading
sections, particularly when adding more material, I have often added
annotations, but I have not done this for all the early entries yet.
Recently added topics often may exist
in standard sources that I have not reread recently, so the references for such
topics often have gaps -- I constantly discover that Loyd or Dudeney or Ahrens
or Lucas or Fibonacci has covered such a topic but I have forgotten this --
e.g. looking through Dudeney recently, I added about 15 entries. New sections are often so noted to indicate
that they may not be as complete as other sections.
Some of the sources cited are lengthy
and I originally added notes as to which parts might be usable in a book of
readings -- these notes have now been mostly deleted, but I may have missed a
few.
I would like to think that I am about
75% of the way through the relevant material.
However, I recently did a rough measurement of the material in my study
-- there is about 8 feet of read but unprocessed material and about 35 feet of
unread material, not counting several boxes of unread Rubik Cube material and
several feet of semi-read material on my desk and table. I recently bought two bookshelves just to
hold unread material. Perhaps half of
this material is relevant to this work.
In particular, the unread material
includes several works of Folkerts and Sesiano on medieval MSS, a substantial
amount of photocopies from Schott, Schwenter and Dudeney (400 columns), some
2000 pages of photocopies recently made at Keele, some 500 pages of photocopies
from Martin Gardner's files, as well as a number of letters. Marcel Gillen has made extracts of all US,
German and EURO patents and German registered designs on puzzles -- 26 volumes,
occupying about two feet on my shelves.
I have recently acquired an almost complete set of Scripta
Mathematica (but I have previously read about half of it),
Schwenter-Harsdörffer's Deliciæ Physico‑Mathematicae, Schott's Joco-Seriorum
and Murray's History of Board Games Other Than Chess. I have recently acquired the early issues of Eureka,
but there are later issues that I have not yet read and they persist in not
sending the current copies I have paid for!
I have not yet seen some of the
earlier 19C material which I have seen referred to and I suspect there is much
more to be found. I have examined some
18C & 19C arithmetic and algebra books looking for problem sections --
these are often given the pleasant name of Promiscuous Problems. There are so many of these that a reference
to one of them probably indicates that the problem appears in many other
similar books that I have not examined.
My examination is primarily based on those books which I happen to have
acquired. There are a few 15-17C books
which I have not yet examined, notably those included at the end of the last
paragraph.
In working on this material, it has
become clear that there were two particularly interesting and productive eras
in the 19C. In the fifteen years from
1857, there appeared about a dozen books in the US and the UK: The
Magician's Own Book (1857); Parlour
Pastime, by "Uncle George" (1857); The Sociable (1858);
The Boy's Own Toymaker, by Landells (1858); The Book of 500 Curious Puzzles
(1859); The Secret Out
(1859); Indoor and Outdoor Games for
Boys and Girls (c1859); The
Boy's Own Conjuring Book (1860); The
Illustrated Boy's Own Treasury (1860, but see below); The Parlor Magician (1863); The Art of Amusing, by Bellew
(1866); Parlour Pastimes
(1868); Hanky Panky (1872); Within Doors, by Elliott (1872); Magic No Mystery (1876), just to name
those that I know. Most of these are of
uncertain authorship and went through several editions and versions. The Magician's Own Book, The Book
of 500 Curious Puzzles, The Secret Out, The Sociable, The
Parlor Magician, Hanky Panky, and Magic No Mystery seem to be
by the same author(s). I have recently
had a chance to look at a number of previously unseen versions at Sotheby's and
at Edward Hordern's and I find that sometimes two editions of the same title are
essentially completely different! This
is particularly true for US and UK editions.
Many of the later UK editions say 'By the author of Magician's Own Book
etc., translated and edited by W. H. Cremer Jr.' From the TPs, it appears that they were written by Wiljalba
Frikell (1818‑1903) and then translated into English. However, BMC and NUC generally attribute the
earlier US editions to George Arnold (1834-1865), and some catalogue entries
explicitly say the Frikell versions are later editions, so it may be that
Frikell produced later editions in some other language (French or German ??)
and these were translated by Cremer. On
the other hand, the UK ed of The Secret Out says it is based on Le
Magicien des Salons. This is
probably Le Magicien des Salons ou le Diable Couleur de Rose, for which
I have several references, with different authors! -- J. M. Gassier, 1814; M.
[Louis Apollinarie Christien Emmanuel] Comte, 1829; Richard (pseud. of A. O. Delarue), 1857 and earlier. There was a German translation of this. Some of these are at HPL but ??NYS. Items with similar names are: Le Magicien de Société, Delarue,
Paris, c1860? and Le Manuel des Sorciers (various Paris
editions from 178?-1825, cf in Common References). It seems that this era was inspired by these earlier French
books. To add to the confusion, an
advertisement for the UK ed. of Magician's Own Book (1871?) says it is
translated from Le Magicien des Salons which has long been a standard in
France and Germany. Toole Stott opines
that Frikell had nothing to do with these books -- as a celebrated conjuror of
the times, his name was simply attached to the books. Toole Stott also doubts whether Le Magicien des Salons
exists -- but it now seems pretty clear that it does, though it may not have
been the direct source for any of these works, but see below.
Christopher 242 cites the following
article on this series.
Charles L. Rulfs. Origins of some conjuring works. Magicol 24 (May 1971) 3-5. He discusses the various books, saying that
Cremer essentially pirated the Dick & Fitzgerald productions. He says The Magician's Own Book draws
on Wyman's Handbook (1850, ??NYS), Endless Amusement, Parlour
Magic (by W. Clarke?, 1830s, ??NYS), Brewster's Natural Magic
(??NYS). He says The Secret Out
is largely taken, illustrations and all, from Blismon de Douai's Manuel du
Magicien (1849, ??NYS) and Richard & Delion's Magicien des salons ou
le diable couleur de rose (1857 and earlier, ??NYS).
Christopher 622 says Harold Adrian
Smith [Dick and Fitzgerald Publishers; Books at Brown 34 (1987) 108-114]
has studied this series and concludes that Williams was the author of Magician's
Own Book, assisted by Wyman.
Actually Smith simply asserts: "The book was undoubtedly [sic]
written by H. L. Williams, a "hack writer" of the period, assisted by
John Wyman in the technical details."
He gives no explanation for his assertion. He later says he doubts whether Cremer ever wrote anything. He suggests The Secret Out book is
taken from DeLion. He states that The
Boy's Own Conjuring Book is a London pirate edition.
Several of the other items are
anonymous and there was a tremendous amount of copying going on -- problems are
often reproduced verbatim with the same diagram or sometimes with minor
changes. In some cases, the same error
is repeated in five different books! I
have just discovered some earlier appearances of the same material in The
Family Friend, a periodical which ran in six series from 1849 to 1921 and
which I have not yet tracked down further.
However, vol. 1-3 of 1849‑1850 and the volume for Jul‑Dec
1859 contain a number of the problems which appear repeatedly and identically
in the above cited books. Toole Stott
407 is an edition of The Illustrated Boy's Own Treasury of c1847 but the
BM copy was destroyed in the war and the other two copies cited are in the
US. If this date is correct, then this
book is a forerunner of all the others and a major connection between Boy's
Own Book and Magician's Own Book.
I would be most grateful to anyone who can help sort out this material
-- e.g. with photocopies of these or similar books or magazines.
The other interesting era was about
1900. In English, this was largely
created or inspired by Sam Loyd and Henry Dudeney. Much of this material first appeared in magazines and
newspapers. I have seen much less than
half of Loyd's and Dudeney's work and very little of similar earlier material
(but see below). Consequently problems
due to Loyd or Dudeney may seem to first appear in the works of Ball (1892, et
seq.), Hoffmann (1893) and Pearson (1907).
Further examination of Loyd's and Dudeney's material will be needed to
clarify the origin and development of many problems. Though both started puzzle columns about 1896, they must have
been producing material for a decade or more previously which does not seem to
be known. I have just obtained
photocopies of 401 columns by Dudeney in the Weekly Dispatch of
1897-1903, but have not had time to study them. Will Shortz and Angela Newing have been studying Loyd and Dudeney
respectively and turning up their material.
The works of Lucas (1882‑1895),
Schubert (1890s) and Ahrens (1900‑1918) were the main items on the
Continent and they interacted with the English language writers. Ahrens was the most historical of these and
his book is one of the foundations of the present work. All of these also wrote in newspapers and
magazines and I have not seen all their material.
I would be happy to hear from anyone
with ideas or suggestions for this bibliography. I would be delighted to hear from anyone who can locate missing
information or who can provide copies of awkward material. I am particularly short of information about
recreations in the Arabic period. I
prepared a separate file, 'Queries and Problems in the History of Recreational
Mathematics', which is about 23 pages, and has recently been updated. I have also prepared three smaller letters
of queries about Middle Eastern, Oriental and Russian sources and these are
generally more up-to-date.
I have prepared a CD containing this
and much else of my material. I divided
Sources into four files when I used floppy discs as it was too big to
fit on one disc, and I have not yet changed this. The files are: 1:
Introductory material and list of abbreviations/references; 2: Sections 1 - 6; 3: Section 7; 4: Sections 8 - 11. It is
convenient to have the first file separate from the main material, but I might
combine the other three files. (I have
tried to send it by email in the past, but this document is very large
(currently c4.1MB and the Word version will be longer) and most people who
requested it by email found that it overflowed their mailbox and created chaos
in their system -- this situation has changed a bit with larger memories and
improved transmission speeds.)
This file started on a DEC-10, then
was transferred to a VAX. It is now on
my PC using Script Professional, the development of LocoScript on the
Amstrad. Even in its earliest forms, this
provided an easy and comprehensive set of diacritical marks, which are still
not all available nor easy to use in WordPerfect or Word (except perhaps by
using macros and/or overstriking??). It
also provides multiple cut and paste buffers and easy formatting, though I have
learned how to overcome these deficiencies in Word.
Script provides an ASCII output, but
this uses IBM extended ASCII which has 8-bit codes. Not all computers will accept or print such characters and
sometimes they are converted into printer control codes causing considerable
confusion. I have a program that
converts these codes to 7-bits -- e.g. accents and umlauts are removed. However, ASCII loses a great deal of the
information, such as sub- and superscripts, so this is not a terribly useful
format.
Script also provides WordStar and
"Revisable-Form-Text DCA" output, but neither of these seems to be
very successful (DCA is better than WordStar).
Script later added a WordPerfect exporting facility. This works well, though some (fairly rare)
characters and diacritical marks are lost and the output requires some
reformatting. (Nob Yoshigahara reports
that Japanese WordPerfect turns all the extended ASCII characters into Kanji
characters!)
Reading the WordPerfect output in Word
(you may need to install this facility) gives a good approximation to my text,
but in Courier 10pt. Selecting All and changing to Times New
Roman 12pt gives an better
approximation. (Some files use a
smaller font of 10pt and I may have done some into 9pt.) You have to change this in the Header
separately, using View Header and Footer. The page layout is awkward as my page numbering header gets put
into the text, leaving a large gap at the top.
I go into Page Setup and set the
Paper Size to A4 and the Top, Bottom and Header Margins to 15mm and the Left and Right Margins to
25mm. (It has taken me some time to
work this out and some earlier files may have other settings.) However, I find that lines are a bit too
close together and underlines and some diacritical marks are lost, so one needs
to also go into Format Paragraph
Spacing -- Line Spacing and
choose At least and
12pt (or 10pt). I use hanging indentation in most of the
main material and this feature is not preserved in this conversion. By selecting a relevant section and going
into Format Paragraph Indentation --
Special and selecting Hanging,
it should automatically select
10.6mm which corresponds to my
automatic spacing of five characters in 12pt.
Further, I use second level hanging indentation in quite a number of
places. You need to create a style
which is the basic style with the left hand margin at 10.6mm (or 10 or 11 mm).
When second level indenting is needed, select the desired section and
apply this style to it.
However, this still leaves some
problems. I use em dashes a bit,
i.e. –, which gets converted into an underline, _. In Word, this is
obtained by use of CTRL and the
- sign on the numeric
keypad. One can use the find and
replace feature, EXCEPT that a number of other characters are also converted
into underlines. In particular,
Cyrillic characters are all converted into underlines. This is not insuperable as I always(?) give
a transliteration of Cyrillic (using the current Mathematical Reviews
system) and one can reconstruct the original Cyrillic from it. I notice that the Cyrillic characters are
larger than roman characters and hence may overlap. One can amend this by selecting the Cyrillic text and going
into Format Font Character
Spacing Spacing and choosing Expanded By 2 pt (or thereabout). But a number of characters with unusual
diacritical marks are also converted to underlines or converted to the unmarked
character and not all of these are available in Word. E.g. ĭ, which is the transliteration of й
becomes just i. I am slowly forming a Word file containing
the Word versions of entries having the Cyrillic or other odd characters, and I
will include this file on my CD, named
CYRILLIC.DOC. For diacritical
marks not supported by Word, I use an approximation and/or an explanation.
It is very tedious to convert the
underlines back to em dashes, so I will convert every em dash to a double
hyphen --.
Finally, I have made a number of
diagrams by simple typing without proportional spacing and Word does not permit
changing font spacing in mid-line and ignores spaces before a right-alignment
instruction. The latter problem can be
overcome by using hard spaces and the former problem is less of a problem, and
I think it can be overcome.
Later versions of Script support
Hewlett-Packard DeskJets and I am now on my second generation of these, so the
7th and future editions will be better printed (if they ever are!). However, this required considerable
reformatting as the text looks best in proportional spacing (PS) and I found I
had to check every table and every mathematical formula and diagram. Also, to set off letters used as
mathematical symbols within text, I find PS requires two spaces on each side of
the letter -- i.e. I refer to x
rather than to x. (I find this
easier to do than to convert to italics.)
I also sometimes set off numbers with two spaces, though I wasn't
consistent in doing this at the beginning of my reformatting. The conversion to proportional spacing
reduced the total length from 587 to
488 pages, a reduction of 16.87%
which is conveniently estimated as
1/6. The percentage of reduction
was fairly consistent throughout the conversion process.
The printing of Greek characters went
amiss in the second part of the 6th Preliminary Edition, apparently due to the
printer setting having been changed without my noticing -- this happens if an
odd character gets sent to the printer, usually in DOS when trying to use or
print a corrupted file, and there is no indication of it. I was never able to reproduce the effect!
The conversion to (Loco)Script
provided many improved features compared to my earlier DEC versions. I am using an A4 page (8¼
by 11⅔ inches) rather than an 8½ by 11 inch page, which gives
60 lines of text per page, four
more or 7% more than when using the DEC or VAX.
[SIXTH
EDITION: 1: Fibonacci, 1: Montucla;
3.B; 4.A.1.a, 4.B.9, 4.B.10,
4.B.11, 4.B.12; 5.R.1.a, 5.W.1, 5.AA,
5.AB; 6.AS.1.b, 6.AS.2.a, 6.AS.5,
6.AW.4, 6.BP, 6.BQ, 6.BR; 7.I.1, 7.Y.2,
7.AY, 7.AZ; 7.BA; 8.I, 8.J; 9.E.2, 9.K;
10.A.4, 10.A.5, 10.U, 10.V, 10.W;
11.K.6, 11.K.7, 11.K.8.]
In
the last edition, I had 8.K instead of 8.J in the list of New Sections and in
the Contents.
1:
Pacioli, Carroll, Perelman; 4.B.13,
4.B.14, 4.B.15; 5.B.2, 5.H.3 (the
previous 5.H.3 has been renumbered 5.H.4), 5.K.3, 5.R.1.b, 5.X.4, 5.AC, 5.AD,
5.AE, 5.AF, 5.AG.1, 5.AG.2; 6.AJ.4,
6.AJ.5, 6.AS.3.a, 6.AT.8, 6.AT.9, 6.AY.2, 6.BF.4, 6.BF.5, 6.BS, 6.BT, 6.BU,
6.BV, 6.BW; 7.H.6, 7.H.7 (formerly part
of 7.H.5), 7.M.4.a, 7.M.4.b, 7.M.6, 7.R.4, 7.AC.3.a, 7.AC.7, 7.AH.1, 7.AJ.1,
7.BB, 7.BC; 8.K, 8.L; 10.A.6, 10.A.7, 10.A.8, 10.D has become
10.D.1, 10.D.2, 10.D.3, 10.E.4, 10.X, 10.Y, 10.Z, 10.AA, 10.AB, 10.AC, 10.AD,
10.AE; 11.N, 11.O, 11.P, 11.Q, 11.R,
11.S. (65 new sections)
I am immensely indebted to many
mathematicians, historians, puzzlers, bookdealers and others who have studied particular
topics, as will be apparent.
I have had assistance from so many
sources that I have probably forgotten some, but I would like to give thanks
here to the following, and beg forgiveness from anyone inadvertently omitted --
if you remind me, I will make amendment.
In some cases, I simply haven't got to your letter yet! Also I have had letters from people whose
only identification is an undecipherable signature and phone messages from
people whose name and phone number are unintelligible.
Sadly, a few of these have died since
I corresponded with them and I have indicated those known to me with †.
André
Allard, Eric J. Aiton†, Sue Andrew,
Hugh Ap Simon, Gino Arrighi, Marcia Ascher, Mohammad Bagheri,
Banca Commerciale Italiana,
Gerd Baron, Chris Base, Rainier [Ray] Bathke, John Beasley, Michael Behrend, Jörg
Bewersdorff, Norman L. Biggs, C. [Chris] J. Bouwkamp, Jean Brette, John Brillhart, Paul
J. Campbell,
Cassa di Risparmio di Firenze, Henry Cattan, Marianna Clark,
Stewart Coffin,
Alan & Philippa Collins,
John H. Conway,
H. S. M. Coxeter,
James Dalgety,
Ann E. L. Davis,
Yvonne Dold, Underwood Dudley, Anthony W. F. Edwards, John Ergatoudis, John Fauvel†, Sandro Ferace,
Judith V. Field,
Irving Finkel,
Graham Flegg,
Menso Folkerts,
David Fowler,
Aviezri S. Fraenkel,
Raffaella Franci,
Gregory N. Frederickson,
Michael Freude,
Walter W. Funkenbusch,
Nora Gädeke,
Martin Gardner,
Marcel Gillen, Leonard
J. Gordon, Ron Gow, Ivor Grattan‑Guinness, Christine Insley Green, Jennifer Greenleaves (Manco), Tom Greeves, H. [Rik] J. M. van Grol, Branko Grünbaum, Richard K. Guy, John Hadley, Peter Hajek, Diana
Hall, Joan Hammontree, Anton Hanegraaf†, Martin Hansen, Jacques Haubrich, Cynthia Hay,
Takao Hayashi,
Robert L. Helmbold,
Hanno Hentrich,
Richard I. Hess,
Christopher Holtom,
Edward Hordern†,
Peter Hosek,
Konrad Jacobs,
Anatoli Kalinin,
Bill Kalush, Michael
Keller,
Edward S. Kennedy,
Sarah Key (The Haunted Bookshop), Eberhard Knobloch, Don Knuth,
Bob Koeppel,
Joseph D. E. Konhauser,
David E. Kullman,
Mogens Esrom Larsen,
Jim Lavis (Doxa (Oxford)), John Leech†, Elisabeth Lefevre, C. Legel, Derrick [Dick] H. Lehmer†, Emma Lehmer, Leisure Dynamics,
Hendrik W. Lenstra,
Alan L. Mackay,
Andrzej Makowski,
John Malkevitch,
Giovanni Manco,
Tatiana Matveeva,
Ann Maury, Max Maven, Jim McArdle, Patricia McCulloch,
Peter McMullen,
Leroy F. Meyers†, D.
P. Miles, Marvin Miller, Nobuo Miura, William O. J. Moser, Barbara Moss,
Angela Newing,
Jennie Newman,
Tom and Greta O'Beirne††, Owen O'Shea,
Parker Brothers, Alan
Parr, Jean J. Pedersen, Luigi Pepe,
William Poundstone, Helen
Powlesland, Oliver Pretzel, Walter Purkert, Robert A. Rankin†, Eleanor Robson, David J. A. Ross, Lee Sallows, Christopher Sansbury,
Sol Saul, William L. Schaaf, Doris Schattschneider, Jaap Scherphuis, Heribert Schmitz,
Š. Schwabik,
Eileen Scott†, Al
Seckel, Jacques Sesiano, Claude E. Shannon†, John Sheehan, A. Sherratt,
Will Shortz,
Kripa Shankar Shukla,
George L. Sicherman, Deborah
Singmaster, Man‑Kit Siu,
Gerald [Jerry] K. Slocum, Cedric A. B. Smith† (and Sue Povey & Jim
Mallet at the Galton Laboratory for letting me have some of Cedric's
books), Jurgen Stigter, Arthur H. Stone, Mel Stover†, Michael Stueben,
Shigeo Takagi†, Michael Tanoff, Gary J. Tee, Andrew Topsfield, George Tyson†, Dario Uri, Warren Van Egmond,
Carlo Viola,
Kurt Vogel†,
Anthony Watkinson,
Chris Weeks,
Maurice Wilkes,
John Winterbottom,
John Withers,
Nob. Yoshigahara,
Claudia Zaslavsky.
I would also like to thank the
following libraries and museums which I have used:
University
of Aberdeen;
University of Bristol;
Buckleys Shop Museum, Battle, East Sussex; University of Calgary; University of Cambridge; Marsh's Library, Dublin;
FLORENCE:
Biblioteca Nazionale; Biblioteca Riccardiana;
University of Keele -- The Turner Collection(†)
and its librarian Martin Phillips;
Karl‑Marx‑Universität, Leipzig:
Universität Bibliothek and Sektion Mathematik Bibliothek,
especially Frau Letzel at the latter;
LONDON:
Birkbeck College; British Library (at Bloomsbury and
then at St. Pancras; also at Colindale);
The London Library; School of Oriental and African
Studies, especially Miss Y. Yasumara, the Art Librarian; Senate House, particularly the Harry Price
Library; South Bank
University;
Southwark Public Library;
University College London, especially the Graves
Collection and the Rare Book Librarians Jill Furlong, Susan Stead and their
staff; Warburg Institute;
MUNICH:
Deutsches Museum; Institut für
Geschichte der Naturwissenschaften;
NEW
YORK:
Brooklyn Public Library;
City College of New York; Columbia University;
Newark Public Library, Newark, New
Jersey;
University
of Newcastle upon Tyne -- The Wallis Collection and its librarian Lesley
Gordon;
OXFORD:
Ashmolean Museum; The Bodleian Library;
Museum of the History of Science, and its librarian
Tony Simcock;
University
of Reading; University of St. Andrews;
SIENA:
Biblioteca Comunale degli
Intronati;
Dipartimento di Matematica, Università di Siena;
University of Southampton; Mathematical Institute, Warsaw.
I would like especially to thank the
following.
Interlibrary Loans (especially Brenda
Spooner) at South Bank University and the British Library Lending Division for
obtaining many strange items for me.
Richard Guy, Bill Sands and the Strens
bequest for a most useful week at the Strens/Guy Collection at Calgary in early
1986 and for organizing the Strens Memorial Meeting in summer 1986 and for
printing the first preliminary edition of these Sources.
Gerd Lassner, Uwe Quasthoff and the
Naturwissenschaftlich‑Theoretisches Zentrum of the Karl‑Marx‑Universität,
for a very useful visit to Leipzig in 1988.
South Bank University Computer Centre
for the computer resources for the early stages of this project, and especially
Ann Keen for finding this file when it was lost.
My School for printing these preliminary
editions.
Martin Gardner for kindly allowing me
to excavate through his library and files.
James Dalgety, Edward Hordern, Bill
Kalush, Chris Lewin, Tom Rodgers and Will Shortz for allowing me to rummage
through their libraries.
John Beasley, Edward Hordern, Bill
Kalush, Will Shortz and Jerry Slocum for numerous photocopies and copies from
their collections.
Menso Folkerts, Richard Lorch, Michael
Segre and the Institut für Geschichte der Naturwissenschaft, Munich, for a most
useful visit in Sep 1994 and for producing a copy of Catel.
Raffaella Franci and the Dipartimento
di Matematica and the Centro Studi della Matematica Medioevale at Università di
Siena for a most useful visit in Sep 1994.
Takao Hayashi for much material from
Japan and India.
My wife for organizing a joint trip to
Newcastle in Sep 1997 where I made use of the Wallis Collection.
Finally, I would like to thank a large
number of publishers, distributors, bookdealers and even authors who have
provided copies of the books and documents upon which much of this work is
based. Bookdealers have often let me
examine books in their shops. Their
help is greatly appreciated. There are
too many of these to record here, but I would like to mention Fred Whitehart
(†1999), England's leading dealer in secondhand scientific books for many years
who had a real interest in mathematics.
DIACRITICAL MARKS AND NOTATION
Before
converting to LocoScript, I used various conventions, given below, to represent
diacritical marks. Each symbol
(except ') occurred after the letter it
referred to. I have now converted these
and all mathematical conventions into correct symbols, so far as possible, but
I may have missed some, so I am keeping this information for the present.
Common
entries using such marks are given later in this section and only the
abbreviated or simplified form is used later -- e.g. I use Problemes for
Bachet's work rather than Problèmes.
(Though this may change??)
Initially,
I did not record all diacritical marks, so some may be missing though I have
checked almost all items. I may omit
diacritical marks which are very peculiar.
Transliterations
of Arabic, Sanskrit, Chinese, etc. are often given in very different
forms. See Smith, History, vol. 1, pp. xvii-xxii
for a discussion of the problems. The
use of ^ and ˉ seems interchangeable and I have used ^
when different versions use both
^ and ˉ , except when quoting a title or passage when I use the
author's form. [Smith, following Suter,
uses ^ for Arabic, but
uses ˉ for Indian. Murray uses ˉ
for both. Wieber uses ˉ
for Arabic. Van der Linde
uses ´
for Arabic. Datta & Singh
use ^
for Indian.]
There
are two breathing marks in Arabic -- ayn
‘ and alif/hamzah ’ --
but originally I didn't have two forms easily available, so both were
represented by '. I have now converted almost all of these
to ‘
and ’. These don't seem to be as distinct in the printing as on my
screen.
French
practice in accenting capitals is variable and titles are often in capitals, so
expected marks may be missing. Also,
older printing may differ from modern usage -- e.g. I have seen: Liège and
Liége; Problèmes, Problêmes and
Problémes. When available, I have
transcribed the material as printed without trying to insert marks, but many
places insert the marks according to modern French spelling.
Greek
and Cyrillic titles are now given in the original with an English
transliteration (using the Amer. Math. Soc. transliteration for Cyrillic).
I
usually ignore the older usage of
v for u and i
for j, so that I give
mathematiqve as mathematique and xiij as
xiii.
I
used a1, a2, ..., ai, etc. for subscripted variables, though I
also sometimes used a(1), a(2),
..., a(i), etc. Superscripts or exponents were indicated by
use of ^, e.g. 2^3 is 8. These have been converted to ordinary sub-
and superscript usage, but ^ may be used when the superscript is
complicated -- e.g. for 2^ai or
9^(99).
Greek
letters were generally spelled out in capitals or marked with square brackets,
e.g. PI, [pi], PHI, but these have probably all been converted.
My
word processor does not produce binomial coefficients easily, so I use BC(n, k)
for n!/k!(n‑k)!
Many
problems have solutions which are sets of fractions with the same denominator
and I abbreviate a/z, b/z,
c/z as (a, b, c)/z. Notations
for particular problems are explained at the beginning of the topic.
Rather
than attempting to italicise letters used as symbols, I generally set them off
by double-spaces on each side -- see examples above. Other mathematical notations may be improvised as necessary and
should be obvious.
Recall
that the symbols below occurred after the letter they referred to, except
for ' .
" denoted umlaut or diaeresis in general, e.g.: ä, ë, ï, ö, ü.
/ was used after a letter for accent acute, ́, after l for ł
in Polish, and after o for
ø in Scandinavian.
\ denoted accent grave,
̀.
^ denoted the circumflex,
^, in Czech, etc.; the overbar
(macron) ˉ or
^ for a long vowel in Sanskrit,
Hindu, etc.; and the overbar used to indicate omission in medieval MSS.
@ denoted the cedilla (French
ç and Arabic ş)
and the ogonek or Polish hook (Polish
ą).
. denoted the underdot in
ḥ, ḳ, ṇ, ṛ,
ṣ, ṭ, in Sanskrit, Hindu, Arabic.
These are sometimes written with a following h -- e.g. k
may also be written kh and I may sometimes have used this. (It is difficult to search for ḥ. , etc., so not all of these
may be converted.) This mark vanishes
when converted to WordPerfect.
* denoted the overdot for
ġ, ṁ, ṅ, in Sanskrit, Hindu,
Arabic. This vanishes over m
and n in WordPerfect.
~ denoted the Spanish tilde
~ and the caron or hachek ˇ,
in ğ, š. The breve is a curved version, ˘,
of the same symbol and is essentially indistinguishable from the
caron. It occurs in Russian й,
which is translitereated as
ĭ.
_ denoted the underbar in
ḏ , j, ṯ (I cannot find a j
with an underbar in Arial). This mark vanishes
in WordPerfect.
' denotes breathing marks in Arabic, etc. There are actually two forms of this --
ayn ’
and alif/hamzah ‘ -- but I didn't have two forms easily
available and originally entered both as apostrophe ' . These normally occur between letters and I
placed the ' in the same space. I have
converted most of these.
Commonly
occurring words with diacritical marks are: Académie, arithmétique,
bibliothèque, Birkhäuser, café, carré, école, Erdös, für, géomètre, géométrie,
Göttingen, Hanoï -- in French only, ‑ième, littéraire, mathématique,
mémoire, ménage, misère, Möbius, moiré, numérique, Pétersbourg, probabilités,
problème (I have seen problêmes??), Rätsel, récréation, Sändig, siècle,
société, Thébault, théorie, über, umfüllung.
I
have used ?? to indicate uncertainty and points where further work needs to be
done. The following symbols after ??
indicate the action to be done.
* check for diacritical marks,
etc.
NX no Xerox or other copy
NYS not yet seen
NYR not yet read
o/o on order
SP check spelling
Other
comments may be given.
ABBREVIATIONS
OF JOURNALS AND SERIES.
See: AMM, CFF, CM,
CMJ, Family Friend, G&P,
G&PJ, HM, JRM,
MG, MiS, MM,
MS, MTg, MTr, M500,
OPM, RMM, SA,
SM, SSM in Common References below.
See: AMS, C&W, CUP,
Loeb Classical Library, MA, MAA,
NCTM, OUP in Common References below.
ABBREVIATIONS
OF MONTHS. All months are given by their first three letters in
English: Jan, Feb, ....
PUBLISHERS'
LOCATIONS. The following publisher's locations will not be cited each
time. Other examples may occur and can
be found in the file PUBLOC.
AMS (American Mathematical
Society), Providence, Rhode Island,
USA.
Chelsea Publishing, NY, USA.
CUP (Cambridge University
Press), Cambridge, UK.
Dover, NY, USA.
Freeman, San Francisco, then NY, USA.
Harvard University Press, Cambridge, Massachusetts, USA.
MA (Mathematical
Association), Leicester, UK.
MAA (Mathematical Association of
America), Washington, DC, USA.
NCTM (National Council of
Teachers of Mathematics), Reston,
Virginia, USA.
Nelson, London, UK.
OUP (Oxford University
Press), Oxford, UK (and also NY, USA).
Penguin, Harmondsworth, UK.
Simon & Schuster, NY, USA.
NOTES. When referring to items below, I will
usually include the earliest reasonable date, even though the citation may be
to a much later edition. For example, I
would say "Canterbury Puzzles, 1907", even though I am citing problem
numbers or pages from the 1958 Dover reprint of the 1919 edition. Sometimes the earlier editions are hard to
come by and I have sometimes found that the earlier edition has different
pagination -- in that case I will (eventually) make the necessary changes.
Edition
information in parentheses indicates items or editions that I have not seen,
though I don't always do this when the later version is a reprint or facsimile.
Abbaco. See:
Pseudo-dell'Abbaco.
Abbot Albert. Abbot Albert von Stade. Annales Stadenses. c1240.
Ed. by J. M. Lappenberg.
In: Monumenta Germaniae Historica, ed. G. H. Pertz, Scriptorum
t. XVI, Imp. Bibliopolii Aulici Hahniani, Hannover, 1859 (= Hiersemann,
Leipzig, 1925), pp. 271‑359.
(There are 13 recreational problems on pp. 332‑335.) [Vogel, on p. 22 of his edition of the
Columbia Algorism, dates this as 1179, but Tropfke gives 1240, which is more in
line with Lappenberg's notes on variants of the text. The material of interest, and several other miscellaneous
sections, is inserted at the year 1152 of the Annales, so perhaps Vogel was
misled by this.] I have prepared an
annotated translation of this: The problems of Abbot Albert (c1240). I have numbered the problems and will cite
this problem number.
Abraham. R. M. Abraham.
Diversions and Pastimes.
Constable, London, 1933
= Dover, 1964 (slightly amended and with different pagination,
later retitled: Tricks and Amusements with Coins, Cards, String, Paper and
Matches). I will cite the Constable
pages (and the Dover pages in parentheses).
Ackermann. Alfred S. E. Ackermann. Scientific Paradoxes and Problems and Their
Solutions. The Old Westminster Press,
London, 1925.
D. Adams. New Arithmetic. 1835.
Daniel
Adams (1773-1864). ADAMS NEW
ARITHMETIC. Arithmetic, in which the
principles of operating by numbers are analytically explained, and
synthetically applied; thus combining the advantages to be derived both from
the inductive and synthetic mode of instructing: The whole made familiar by a great variety of useful and
interesting examples, calculated at once to engage the pupil in the study, and
to give him a full knowledge of figures in their application to all the
practical purposes of life. Designed
for the use of schools and academies in the United States. J. Prentiss, Keene, New Hampshire, 1836,
boarded. 1-262 pp + 2pp publisher's
ads, apparently inserted backward.
[Halwas 1-6 lists 1st ed as 1835, then has 1837, 1838, 1839, 1842,
c1850.] This is a reworking of The
Scholar's Arithmetic of 1801.
D. Adams. Scholar's Arithmetic. 1801.
Daniel
Adams (1773-1864). The Scholar's
Arithmetic; or, Federal Accountant: Containing.
I. Common arithmetic, .... II.
Examples and Answers with Blank Spaces, ....
III. To each Rule, a Supplement, comprehending, 1. Questions .... 2. Exercises. IV. Federal Money, ....
V. Interest cast in Federal Money, ....
VI. Demonstration by engravings ....
VII. Forms of Notes, .... The
Whole in a Form and Method altogether New, for the Ease of the Master and the
greater Progress of the Scholar. Adams
& Wilder, Leominster, Massachusetts, 1801; 2nd ed, 1802. 3rd ed ??.
4th ed, by Prentiss, 1807; 6th ed, 1810; 10th ed, 1816; Stereotype
Edition, Revised and Corrected, with Additions, 1819, 1820, 1824; John Prentiss,
Keene, New Hampshire, 1825. [Halwas
8-14.] I have the 1825, whose Preface
is for the 10th ed of 1816, so is probably identical to that ed. The Preface says he has now made some
revisions. The only change of interest
to us is that he has added answers to some problems. So I will cite this as 1801 though I will be giving pages from
the 1825 ed. The book was thoroughly
reworked as Adams New Arithmetic, 1835.
M. Adams. Indoor Games. 1912.
Morley
Adams, ed. The Boy's Own Book of Indoor
Games and Recreations. "The Boy's
Own Paper" Office, London, 1912; 2nd ptg, The Religious Tract Society,
London (same address), 1913. [This is a
major revision of: G. A. Hutchison, ed.; Indoor Games and
Recreations; The Boy's Own Bookshelf; New ed., Religious Tract Society,
London, 1891 (possibly earlier) -- see 5.A.]
M. Adams. Puzzle Book. 1939.
Morley
Adams. The Morley Adams Puzzle
Book. Faber & Faber, London, 1939.
M. Adams. Puzzles That Everyone Can Do. 1931.
Morley
Adams. Puzzles that Everyone Can Do. Grant Richards, London, 1931, boarded.
AGM. Abhandlungen zur Geschichte der Mathematischen
Wissenschaften mit Einschluss ihrer Anwendungen. Begründet von Moritz Cantor.
Teubner, Leipzig. The first ten
volumes were Supplements to Zeitschrift für Math. u. Physik, had a slightly
different title and are often bound in with the journal volume.
Ahrens, Wilhelm Ernst Martin
Georg (1872-1927). See: A&N,
MUS, 3.B, 7.N.
al‑Karkhi. Aboû Beqr Mohammed Ben
Alhaçen Alkarkhî [= al‑Karagi
= al‑Karajī].
Untitled MS called Kitāb al-Fakhrī (or just Alfakhrî) (The
Book Dedicated to Fakhr al-Din).
c1010. MS 952, Supp. Arabe de la
Bibliothèque Impériale, Paris. Edited
into French by Franz Woepcke as: Extrait du Fakhrî. L'Imprimerie Impériale, Paris, 1853; reprinted by Georg Olms
Verlag, Hildesheim, 1982. My page
citations will be to Woepcke. Woepcke
often refers to Diophantos, but his numbering gets ahead of Heath's.
Alberti. 1747. Giuseppe
Antonio (or Giusepp-Antonio) Alberti (1715-1768). I Giochi Numerici Fatti Arcani Palesati da Giuseppe Antonio
Alberti. Bartolomeo Borghi, Bologna,
1747, 1749. Venice, 1780, 1788(?). 4th ed., adornata di figure, Giuseppe
Orlandelli for Francesco di Niccolo' Pezzana, Venice, 1795 (reprinted: Arnaud, Florence, 1979), 1813. Adornata di 16 figure, Michele Morelli,
Naples, 1814. As: Li Giuochi Numerici
Manifestati, Edizione adorna di Figure in rame, Giuseppe Molinari, Venice,
1815.
The
editions have almost identical content, but different paginations. I have compared several editions and seen
little difference. The 1747 ed. has a
dedication which is dropped in the 2nd ed. which also omits the last paragraph
of the Prefazione. I only saw one other
point where a few words were changed. I
will give pages of 1747 (followed by 1795 in parenthesis). Much of Alberti, including almost all the
material of interest to us and many of the plates, is translated from vol. 4 of
the 1723 ed. of Ozanam.
(Serge
Plantureux's 1993 catalogue describes a 1747-1749 ed. with Appendice al
Trattato de' Giochi Numerici (1749, 72 pp) & Osservazioni all'Appendice de'
Giochi Numerici (38 pp), ??NYS. The
copy in the Honeyman Collection had the Appendice. Christopher 3 has the Osservazioni. The Appendice is described by Riccardi as a severe criticism of
Alberti, attributed to Giovanni Antonio Andrea Castelvetri and published by
Lelio dall Volpe, Bologna, 1749. The
Osservazioni are Alberti's response.)
Alcuin (c735-804).
Propositiones
Alcuini doctoris Caroli Magni Imperatoris ad acuendos juvenes. 9C.
IN:
B. Flacci Albini seu Alcuini, Abbatis et Caroli Magni Imperatoris
Magistri. Opera Omnia: Operum pars
octava: Opera dubia. Ed. D. Frobenius,
Ratisbon, 1777, Tomus secundus, volumen secundum, pp. 440‑448. ??NYS.
Revised and republished by J.‑P. Migne as: Patrologiae Cursus
Completus: Patrologiae Latinae, Tomus 101, Paris, 1863, columns 1143‑1160.
A
different version appears in: Venerabilis Bedae, Anglo‑Saxonis
Presbyteri. Opera Omnia: Pars Prima,
Sectio II -- Dubia et Spuria: De Arithmeticus propositionibus. Tomus 1, Basel, 1563. (Rara, 131, says there were earlier
editions: Paris, 1521 (part), 1544‑1545 (all), 1554, all ??NYS.) Revised and republished by J.‑P. Migne
as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 90, Paris, 1904,
columns 665‑672. Incipiunt aliae
propositiones ad acuendos juvenes is col. 667‑672. A version of this occurs in Ens'
Thaumaturgus Mathematicus of 1636 -- cf under Etten.
The
Alcuin has 53 numbered problems with answers.
The Bede has 3 extra problems, but the problems are not numbered, there
are only 31 1/2 answers and there are several transcription errors. The editor has used the Bede to rectify the
Alcuin.
There
is a recent critical edition of the text by Folkerts -- Die älteste mathematische
Aufgabensammlung in lateinischer Spräche: Die Alkuin zugeschriebenen
Propositiones ad Acuendos Iuvenes; Denkschriften der Österreichischen Akademie
der Wissenschaften, Mathematische‑naturwissenschaftliche Klasse 116:6
(1978) 13‑80. (Also separately
published by Springer, Vienna, 1978.
The critical part is somewhat revised as: Die Alkuin zugeschriebenen
"Propositiones ad Acuendos Iuvenes"; IN: Science in Western and
Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D.
Lohrmann; Birkhäuser, Basel, 1993, pp. 273-281.) He finds that the earliest text is late 9C and is quite close to
the first edition cited above. He uses
the same numbers for the problems as above and numbers the extra Bede problems
as 11a, 11b, 33a. I use Folkerts for
the numbering and the titles of problems.
John
Hadley kindly translated Alcuin for me some years ago and made some amendments
when Folkerts' edition appeared. I
annotated it and it appeared as: Problems to Sharpen the Young, MG 76 (No. 475)
(Mar 1992) 102-126. A slightly
corrected and updated edition, containing some material omitted from the MG
version, is available as Technical Report SBU-CISM-95-18, School of Computing,
Information Systems, and Mathematics, South Bank University, Oct 1995, 28pp.
Menso
Folkerts and Helmuth Gericke have produced a German edition: Die Alkuin
zugeschriebenen Propositiones ad Acuendos Juvenes (Aufgabe zur Schärfung des
Geistes der Jugend); IN: Science in Western and Eastern Civilization in
Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; Birkhäuser, Basel,
1993, pp. 283-362.
See
also: David Singmaster. The history of some of Alcuin's Propositiones. IN: Charlemagne and his Heritage 1200 Years of Civilization and Science in
Europe: Vol. 2 Mathematical Arts; ed.
by P. L. Butzer, H. Th. Jongen & W. Oberschelp; Brepols, Turnhout, 1998,
pp. 11‑29.
AM. 1917. H.
E. Dudeney. Amusements in
Mathematics. Nelson, 1917. (There were reprintings in 1919, 1920, 1924,
1925, 1927, 1928, 1930, 1932, 1935, 1938, 1939, 1941, 1943, 1946, 1947, 1949,
1951, but it seems that the date wasn't given before 1941?) = Dover, 1958.
AMM. American Mathematical Monthly.
AMS. American Mathematical Society.
Les Amusemens. 1749.
Les
Amusemens Mathématiques Precedés Des Elémens d'Arithmétique, d'Algébre & de
Géométrie nécessaires pour l'intelligence des Problêmes. André‑Joseph Panckoucke, Lille,
1749. Often listed with Panckoucke as
author (e.g. by the NUC, the BNC and Poggendorff), but the book gives no such
indication. Sometimes spelled
Amusements. There were 1769 and 1799
editions.
Apianus. Kauffmanss Rechnung. 1527.
Petrus
Apianus (= Peter Apian or Bienewitz or Bennewitz) (1495‑1552). Eyn Newe Unnd wolgegründte underweysung
aller Kauffmanss Rechnung in dreyen Büchern / mit schönen Regeln uň
[NOTE: ň denotes an n with an overbar.] fragstucken
begriffen. Sunderlich was fortl unnd
behendigkait in der Welschē Practica uň Tolletn gebraucht wirdt / des
gleychen fürmalss wider in Teützscher noch in Welscher sprach nie
gedrückt. durch Petrum Apianū von
Leyssnick / d Astronomei zů Ingolstat Ordinariū / verfertiget. Georgius Apianus, Ingolstadt, (1527),
facsimile, with the TP of the 1544 ed. and 2pp of publication details added at
the end, Polygon-Verlag, Buxheim-Eichstätt, 1995, with 8pp commentary leaflet
by Wolfgang Kaunzner. (The TP of this
has the first known printed version of Pascal's Triangle.) Smith, Rara, pp. 155-157. (The
d is an odd symbol, a bit like
a δ or an 8, which is used regularly for der
both as a single word and as the ending of a word, e.g. and
for ander.) Smith notes that Apianus follows Rudolff
(1526) very closely.
AR. c1450. Frater
Friedrich Gerhart (attrib.). Latin
& German MSS, c1450, known as Algorismus Ratisbonensis. Transcribed and edited from 6 MSS by Kurt
Vogel as: Die Practica des Algorismus Ratisbonensis; C. H. Beck'sche
Verlagsbuchhandlung, Munich, 1954.
(Kindly sent by Prof. Vogel.)
Vogel classifies the problems and gives general comments on the
mathematics on pp. 155‑189. He
gives detailed historical notes on pp. 203‑232. When appropriate, I will cite these pages before the specific
problems. He says (on p. 206) that
almost all of Munich 14684 (see below) is included in AR.
Arnold, George. See:
Book of 500 Puzzles, Boy's Own
Conjuring Book, Hanky Panky.
Arrighi, Gino. See:
Benedetto da Firenze,
Calandri, Pseudo-Dell'Abbaco, della Francesca, Gherardi, Lucca
1754, P. M. Calandri.
Aryabhata. Āryabhata (I))
[NOTE: ţ denotes a t with a dot under it and ş
denotes an s with a dot under it.] (476-
). Āryabhatīya.
499. Critically edited and
translated into English by Kripa Shankar Shukla, with K. V. Sarma. Indian National Science Academy, New Delhi,
1976. (Volume 1 of a three volume series
-- the other two volumes are commentaries, of which Vol. 2 includes the
commentary Āryabhatīya-Bhāşya, written by Bhaskara I in
629. Aryabhata rarely gives numerical
examples, so Bhaskara I provided them and these were later used by other Indian
writers such as Chaturveda, 860. The
other commentaries are later and of less interest to us. Prof. Shukla has sent a photocopy of an
introductory booklet, which is an abbreviated version of the introductory
material of Vol. 1, with some extensions relating Aryabhata to other
writers.) The material is organized
into verses. There is an older
translation by Walter Eugene Clark as:
The Âryabhaţîya of
Âryabhaţa; Univ. of Chicago Press, Chicago, 1930. (There was an Aryabhata II, c950, but he
only occurs in 7.K.1.)
A&N. Wilhelm Ahrens. Altes und Neues aus der Unterhaltungsmathematik. Springer, Berlin, 1918.
Bachet, Claude‑Gaspar
(1581-1638). See: Problemes.
Bachet-Labosne. See:
Problemes.
Badcock. Philosophical Recreations, or, Winter
Amusements. [1820].
Philosophical
Recreations, or, Winter Amusements.
Thomas Hughes, London, nd [1820].
[BCB 18-19; OCB, pp. 180 & 197.
Heyl 22-23. Toole Stott
75-77. Christopher 54-56. Wallis 34 BAD, 35 BAD. These give dates of 1820, 1822, 1828.] HPL [Badcock] RBC has three versions with
slightly different imprints on the title pages, possibly the three dates
mentioned.
Wallis
34 BAD has this bound after the copy of:
John Badcock; Domestic Amusements, or Philosophical Recreations ...; T.
Hughes, London, nd [1823], and it is lacking its Frontispiece and TP -- cf in
6.BH. HPL [Badcock] has both books,
including the folding Frontispieces.
The earlier does not give an author, but its Preface is signed
J. B. and the later book does give his name and calls itself a sequel to
the earlier. Toole Stott 75-80 clearly
describes both works. Some of the
material is used in Endless Amusement II.
Baker. Well Spring of Sciences.
1562?
Humfrey
Baker (fl. 1557-1587). The Well Sprynge
of Sciences Which teacheth the perfect worke and practise of Arithmeticke both
in whole numbers and fractions, with such easye and compendious instruction
into the sayde arte, .... Rouland Hall
for James Rowbotham, London, 1562.
[Smith, Rara, p. 327, says it was written in 1562 but wasn't actually
printed until 1568, but a dealer says the 1st ed. was 1564 and there was a 4th
ed. in 1574, which I have examined.]
Apparently much revised and extended, (1580). Reprinted, with title: The Wel [sic] Spring of Sciences: Which
teacheth the perfect worke and practise of Arithmetike; Thomas Purfoote,
London, 1591. I have seen Thomas
Purfoot, London, 1612, which is essentially identical to 1591. I have also seen: Christopher Meredith, London, 1646; Christopher Meredith, London, 1650; R. & W. L. for Andrew Kemb, London, 1655; which are all the same, but differently
paged than the 1591. I have also seen
Baker's Arithmetick, ed. by Henry Phillippes, Edward Thomas, London, 1670,
which has different pagination and some additional problems compared to the
1646/1655 ed. [Smith, Rara, 327-330
& 537, says it was rewritten in 1580, but there is little difference
between the 1580 and the many later editions, so the 1591 ed. is probably close
to the 1580 ed. The copy of the 1562 in
the Graves collection ends on f. 160r, but an owner has written a query as
to whether the book is complete.
Neither Smith nor De Morgan seems to have seen a 1562 so they don't give
a number of pages for it. (STC records
no copies of the 1562, 1564, 1576, 1584, 1607 editions, but there was a 1576 by
[T. Purfoote], apparently the 5th ed., of c500pp, in the Honeyman
Collection.) Almost all the problems of
interest occur on ff. 189r-198r of the 1591 ed. and hence are not in the Graves
copy of the 1562 ed., but H&S 61 refers to one of these problems as being
in Baker, 1568. The 1574 ends at fol.
200 (misprinted as 19?, where the ? is an undecipherable blob) and Chapter 16,
which is headed: The 16 Chapter
treateth of sportes and pastime, done by number, is on ff. 189r-200v, and contains just a few recreations, as in
Recorde. So I will date the book as
1562?, but most of the later material as 1580?. The problems of 7.AF.1 and 10.A may be in Graves copy of the 1562
ed. -- ??check. I will cite the 1580?,
1646 and 1670 editions, e.g. 1580?:
ff. 192r 193r; 1646: pp. 302-304; 1670: pp. 344-345.] Bill Kalush has recently sent a CD with
1574, 1580, 1591, 1598, 1602, 1607, 1612, 1617, 1650, 1655 on it -- ??NYR.
Bakhshali MS. The
Bakhshālī Manuscript, c7C.
This MS was found in May 1881 near the village of Bakhshālī,
in the Yusufzāī district of the Peshawer division, then at the
northwestern frontier of India, but apparently now in Pakistan. This is discussed in several places, such as
the following, but a complete translation has only recently appeared. David Pingree says it is 10C, but his
student Hayashi opts for 7C which seems pretty reasonable and I will adopt c7C.
1. A. F. Rudolf Hoernle. Extract of his report in some journal of the
previous year. The Indian Antiquary 12 (Mar
1883) 89-90. A preliminary report,
saying it was found near Bakhshâlî in the Yusufzai District in the Panjâb.
2. A. F. Rudolf Hoernle. On the Bakhshālī Manuscript. Berichte des VII. Internationalen
Orientalisten‑Congresses, Wien, 1886.
Alfred Hölder, Vienna, 1889.
Arische Section, p. 127-147 plus three folding plates. Cf next item. I will cite this as Hoernle, 1886.
3. A. F. Rudolf Hoernle. The Bakhshali manuscript. The Indian Antiquary 17 (Feb 1888) 33‑48
& Plate I opp. p. 46; 275‑279
& Plates II & III opp. pp. 276 & 277. This is essentially a reprint of the
previous item, with a few changes or corrections, but considerable additional
material. He dates it c4C. I will cite this as Hoernle, 1888.
4. G. R. Kaye. The Bakhshālī Manuscript – A Study
in Medieval Mathematics. Archæeological
Survey of India – New Imperial Series XLIII: I-III, with parts I & II as
one volume, (1927‑1933).
(Facsimile reprint in two volumes, Cosmo Publications, New Delhi, 1981 –
this is a rather poor facsimile, but all the text is preserved. I have a letter detailing the changes
between the original and this 'facsimile'.)
I will only cite Part I – Introduction, which includes a discussion of
the text. Part II is a discussion of
the script, transliteration of the text and pictures of the entire MS. Part III apparently was intended to deal
with the language used, but Kaye died before completing this and the published
Part III consists of only a rearranged version of the MS with footnotes
explaining the mathematics. Gupta,
below, cites part III, as Kaye III and I will reproduce these citations. He dates it c12C.
5. B. Datta. The Bakhshâlî mathematics.
Bull. Calcutta Math. Soc. 21 (1929) 1‑60. This is largely devoted to dating of the MS and
of its contents. He asserts that the MS
is a copy of a commentary on some lost work of 4C or 5C (?).
6. R. C. Gupta. Some equalization problems from the
Bakhshālī manuscript. Indian
Journal of the History of Science 21 (1986) 51-61. Notes that Hoernle gave the MS to the Bodleian Library in 1902,
where it remains, with shelf mark MS. Sansk. d.14. He follows Datta in believing that this is a commentary on a
early work, though the MS is 9C, as stated by Hoernle. He gives many problems from Kaye III,
sometimes restoring them, and he discusses them in more detail than the
previous works.
7. Takao Hayashi. The Bakhshālī Manuscript An ancient Indian mathematical
treatise. Egbert Forsten, Groningen,
Netherlands, 1995. (Based on his PhD
Dissertation in History of Mathematics, Brown University, May 1985,
774pp.) A complete edition and
translation with extensive discussion of the context of the problems. He dates it as 7C.
Ball, Walter William Rouse
(1850-1925). See: Ball‑FitzPatrick; MRE.
Ball‑FitzPatrick.
French
translation of MRE by J. Fitz‑Patrick, published by Hermann, Paris.
1st
ed., Récréations et Problèmes Mathématiques des Temps Anciens &
Modernes. From the 3rd ed, 1896, of
MRE, 'Revue et augmentée par l'auteur'.
1898. The Note says 'M. Ball ...
a bien voulu apporter à la troisième édition anglaise des additions et des
modifications importantes.' 352pp.
2nd
ed., Récréations et Problèmes Mathématiques des Temps Anciens et Modernes. From the 4th ed, 1905, of MRE, 'et enrichie
de nombreuses additions'.
As three volumes, 1907‑09. [I have vol. 1, 1907, which is 356pp. Pp. 327‑355 is a note by A. Hermann,
Comptabilité d'une persone qui dépense plus que son revenu. I have not yet seen the other volumes to
compare with the 1926 reprint, but Strens's notes in his copy indicate that
they are identical.]
Reprinted in one vol., Gabay, Paris, 1992,
544pp.
Reprinted, 1926-1927. The only copies that I have seen are bound
as one volume, but with separate pagination.
My copy has left out the title pages of vols. 2 & 3. The copy in the Strens Collection has these
title pages, but its vol. II is 1908.
The 1926 vol. 1 says Nouvelle édition française, but the 1927 vol. 3
says Deuxième édition française.
[Vol.
1 is 326pp, omitting the note by Hermann.
Vol. 2 is 363pp (pp. 322‑355 is a historical note on the cubic,
based on Cossali (1797)). Vol. 3 is
363pp including: Notes diverses de M. Aubry, pp. 137‑206 (or 340? -- the
Table des Matières and the page set up do not make it clear if Aubry's Notes
end on p. 206); Note de M. Fitz‑Patrick,
La géométrie par le pliage et découpage du papier, pp. 341‑360; A. Margossian, De l'ordonnance des nombres
dans les carrés magiques impairs, pp. 1‑60 (pp. 61-64 is a Note on the
same subject, presumably part of Margossian's material); Capt. Reinhart, some geometric notes, pp.
130-136.]
Barnard. 50 Observer Brain-Twisters. 1962.
Douglas
St. Paul Barnard. Fifty Observer Brain‑Twisters A Book of Mathematical and Reasoning
Problems. Faber, 1962. US ed.:
A Book of Mathematical and Reasoning Problems: Fifty Brain
Twisters; Van Nostrand, 1962. The editions have identical pagination.
Bartl. c1920. János
Bartl. Preis-Verzeichnis von Bartl's
Akademie für moderne magische Kunst.
Hamburg, c1920. Reprinted by
Olms Verlag, Zürich, 1983, as: Zauberkatalog Bartl. References are to the section: Vexier- und Geduldspiele,
pp. 305‑312.
Bartoli. Memoriale.
c1420.
Francesco
Bartoli ( -1425). Memoriale (= Notebook) containing some 30
mathematical problems copied during 1400?-1425. Ms 1 F 54 of the Archives départementales du Vaucluse,
France. ??NYS -- described and quoted
in: Jacques Sesiano; Les problèmes
mathématiques du Memoriale de F. Bartoli; Physis 26:1 (1984) 129-150.
BC. Binomial Coefficient, i.e. BC(n, k) = n!/k!(n-k)!.
BCB. See: Hall, BCB.
BDM. See under DSB.
Bede, The Venerable
(c672-735). (Now St. Bede.) See:
Alcuin.
Benedetto da Firenze. c1465.
Benedetto
da Firenze. Trattato d'Abacho. c1465.
This was a popular treatise and Van Egmond's Catalog 356 lists 18 copies
under Benedetto. Six show B as
author, one has Benedetto, one has Benedetto da Firenze, one has Po Ma and one has Filipo Chalandri, so it seems
Benedetto is the most likely author.
The MSS date from c1465 to c1525 and contain 9 to 25 chapters.
The
version in Cod. Acq. e doni 154, Biblioteca Medicea Laurenziana, Florence,
c1480. has been transcribed and edited
by Gino Arrighi as: Pier Maria
Calandri; Tractato d'Abbacho; Domus Galilaeana, Pisa, 1974. The incipit names Po Ma
as author. Cf Van Egmond's
Catalog 96. This version has 23
chapters.
Benson. 1904. J.
K. Benson. The Book of Indoor Games for
Young People of All Ages. C. Arthur
Pearson, London, 1904. [This copies a lot
from Hoffmann (or a common ancestor?).]
Much
of the material of Indoor Games is repeated in: J. K. Benson, ed.; The Pearson
Puzzle Book; C. Arthur Pearson, London, nd [1921 -- BMC]. This is not in BMC or NUC under Benson --
but I have seen an ad listing this as by Mr. X and it is listed under Mr. X in
BMC. Puzzle Book pp. 1-96 = Indoor Games pp. 189-257; Puzzle Book pp. 109-114 =
Indoor Games pp. 258-262. The
only different material in Puzzle Book is pp. 97-108. Neither book refers to the other. Cf Mr. X in Section 4.A.1
Berkeley & Rowland. Card Tricks & Puzzles. 1892.
"Berkeley"
[Peel, Walter H.] & Rowland, T. B. Card Tricks and Puzzles.
The Club Series, George Bell
& Sons, London, 1892 -- according to BMC, but my copy is 1897. Card Puzzles, etc., pp. 1-74 is by Berkeley;
Arithmetical Puzzles, pp. 75-120 is by Rowland.
Berlekamp, Elwyn R. (1940- )
See: Winning Ways.
Bestelmeier. 1801-1803.
G.
H. [Georg Hieronimus] Bestelmeier.
Magazin von verschiedenen Kunst‑ und andern nützlichen Sachen
.... [Toy catalogues.] Nuremberg, 1801‑1803.
Eight
issues and cumulative classified index reprinted by Olms, Zurich, 1979. Issue VII is 1801; the others are 'neue
verbesserte Auflage', 1803. This
includes items numbered 1 through 1111.
Selections,
with English translations. Daniel S.
Jacoby, ed. The Amazing Catalogue of
the Esteemed Firm of George Hieronimus Bestelmeier. Selected Excerpts from Editions of 1793 and 1807. [A comment inside makes me wonder if
1793-1807 is meant??] Merrimack
Publishing Corp., NY, 1971, 82pp. The
numeration is the same as in the Olms edition, but the Jacoby continues to item
1321. Obviously these later items come
from the 1807 edition, but we cannot tell if they might date from 1805, say,
nor whether all the earlier items go back to 1793. Jerry Slocum uses Jacoby in his Compendium and has kindly
provided photocopies of Jacoby's pp. 70-82 containing all the items after 1111
and some examples of the earlier items.
Jacoby does not translate the texts, but just provides English labels
for each picture and these labels are sometimes unconnected with the text.
Many
of Bestelmeier's items are taken from Catel; Kunst-Cabinet; 1790. Sometimes the figure is identical (often
reversed) or is a poor copy. Texts are
often copied verbatim, or slightly modified, but often abbreviated. E.g. Catel often explains the puzzle and
this part is frequently omitted in Bestelmeier. Bestelmeier was the successor to Catel, qv. The booklet by Slocum & Gebhardt (qv
under Catel) gives precise datings for the various parts of these catalogues,
but I have not yet entered these details.
Bhaskara I. 629.
Bhāskara
I.
Āryabhaţīya-Bhāşya. [NOTE: ţ denotes a
t with a dot under it and ş
denotes an s with a dot under it.] 629.
Critically edited, including an English Appendix of the numerical
examples used, by Kripa Shankar Shukla.
Indian National Science Academy, New Delhi, 1976. (Vol. 2 of a three volume series devoted to
the Āryabhaţīya (499) of Aryabhata (476- ), qv.)
Bhaskara I repeats and exposits Aryabhata verse by verse, but
Aryabhata rarely gives numerical examples, so Bhaskara I provided them and
these were later used by other Indian writers such as Chaturveda, 860. His earlier Maha-Bhaskariya (Mahā‑Bhāskarīya)
of c629 is cited in 7.P.2. Shukla's
Appendix is sometimes brief, but sometimes very detailed, e.g. on the 26
examples of Chinese remainder problems.
Bhaskara II (1114-c1185).
Bhâskara
II (1114-c1185, see Colebrooke).
Biggs, Norman L. See:
BLW.
Bijaganita. Bîjaganita of Bhaskara II,
1150 (see Colebrooke).
The Bile Beans Puzzle Book. 1933.
Bile
Beans (C. E. Fulford, Ltd., Leeds, England).
The Bile Beans Puzzle Book.
1933.
Birtwistle. Math. Puzzles & Perplexities. 1971.
Claude
Birtwistle. Mathematical Puzzles and
Perplexities How to Make the Most of
Them. George Allen & Unwin, London,
1971.
Birtwistle. Calculator Puzzle Book. 1978.
Claude
Birtwistle. The Calculator Puzzle
Book. Paperfronts (Elliot Right Way
Books), Kingswood, Surrey, 1978. (There
is a US ed. by Bell, NY, 1978.)
BL(LD). British Library (Lending Division).
Blasius. 1513. Johannis
(or Joannes) Martinus Blasius (later denoted Sileceus or Sciliceus). Liber Arithmetice Practice Astrologis Phisicis
et Calculatioribus admodum utilis.
Thomas Kees for Joannis Parui & Joannis Lambert (in colophon; TP has
Jehanlambert), Paris, 1513. Facsimile
by Heffer Scientific Reprint, Cambridge, 1960.
See Smith, Rara, pp. 95-97.
The Glaisher article in 7.P.5 [Messenger of Mathematics 53 (1923-24) 1‑131]
discusses this book and says he only knows one example of it, which he has in
front of him, so I suspect this facsimile is from that copy. See Rara 95-97. The Honeyman Collection had a copy, saying it was printed for J.
Petit and J. Lambert and that copy had Petit's device on the TP while the TP
shown in Rara has Lambert's device, which is as in this facsimile. There was a reprinting in 1514 and extended
editions in 1519 (ed. by Oronce Finé) and 1526 (ed. by T. Rhaetus) [Honeyman
Collection, nos. 350-352].
BLC. British Library Catalogue, replacing BMC,
in progress since 1970s.
BLC-Ø Indicates that I could not find the item in
the BLC.
BLW. 1976. Norman L. Biggs, E. Keith Lloyd & Robin J.
Wilson. Graph Theory 1736‑1936. OUP, 1976.
Blyth. Match-Stick Magic. 1921.
Will
Blyth. Match-Stick Magic. C. Arthur Pearson, London, 1921, reprinted
1923, 1939.
BM(C). British Museum (Catalogue (of books) to
1955. c1963).
BMC65. Supplement to the above Catalogue for 1956‑1965. c1968.
BN(C). Bibliothèque National, Paris. (Catalogue, 1897-1981.)
Bodleian. The Bodleian Library, University of Oxford, or
its catalogue.
Bonnycastle. Algebra.
1782
John
Bonnycastle (??-1821). An Introduction
to Algebra, with Notes and Observations; designed for the Use of Schools, and
Other Places of Public Education.
1782. The first nine editions
appeared "without any material alterations". In 1815, he produced a 10th ed., "an
entire revision of the work" which "may be considered as a concise
abridgment" of his two volume Treatise on Algebra, 1813, (2nd ed. in
1820). The 1815 ed. had an Appendix: On
the application of Algebra to Geometry.
I have a copy of the 7th ed., 1805, printed for J. Johnson, London, and
it is identical to the 2nd ed. of 1788 except for a problem in the final
section of Miscellaneous Questions.
However, the 9th ed. of 1812 has page numbers advanced by 10 except
toward the end of the book. I also have
the 13th ed. of 1824, printed for J. Nunn and 11 other publishers, London,
1824. This version has an Addenda: A
New Method of resolving Numerical Equations, by his son Charles Bonnycastle
(1797-1840), but is otherwise identical to the 10th ed. of 1815. The earlier text was expanded by about 10%
in 1815, so a number of problems only occur in later editions. I will cite these later problems as 1815 and
will cite the earlier problems as 1782.
[Halwas 36-38 gives some US editions.]
Book of 500 Puzzles. 1859.
The
Book of 500 Curious Puzzles: Containing a Large Collection of Entertaining
Paradoxes, Perplexing Deceptions in Numbers, and Amusing Tricks in
Geometry. By the author of "The
Sociable," "The Secret Out," "The Magician's Own
Book," "Parlor Games," and " Parlor Theatricals,"
etc. Illustrated with a great Variety
of Engravings. Dick & Fitzgerald,
NY, 1859. Compiled from The Sociable
(qv) and Magician's Own Book. Pp. 1-2
are the TP and its reverse. Pp. 3‑36,
are identical to pp. 285-318 of The Sociable; pp. 37-54 are identical to
pp. 199-216 of Magician's Own Book and pp. 55-116 are identical to pp. 241-302
of Magician's Own Book. [Toole Stott
103 lists it as anonymous. NUC, under
Frikell, says to see title. NUC, under
Book, also has an 1882 ed, compiled by William B. Dick. Christopher 129. C&B lists it under Cremer.]
The
authorship of this and the other books cited -- The Sociable, The Secret Out,
The Magician's Own Book, Parlor Games, and Parlor Theatricals, etc. -- is
confused. BMC & NUC generally
assign them to George Arnold (1834-1865) or Wiljalba (or Gustave) Frikell (1818
(or 1816) - 1903), sometimes with Frikell as UK editor of Arnold's US version
-- but several UK versions say they are translated and edited by W. H. Cremer
Jr, and one even cites an earlier French book (though the given title may not
exist!, but cf Manuel des Sorciers, 1825) -- see the discussion under Status of
The Project, in the Introduction, above.
The names of Frank Cahill, Henry Llewellyn Williams and Gustave Frikell
(Jr.) are sometimes associated with versions of these as authors or
coauthors. The Preface of The Sociable
says that most of the Parlor Theatricals are by Frank Cahill and George Arnold
-- this may indicate they had little to do with the parts that interest
us. Toole Stott 640 opines that this
reference led Harry Price to ascribe these books to these authors.
A
publisher's ad in the back says: "The above five books are compiled from
the "Sociable" and "Magician's Own."", referring to:
The Parlor Magician [Toole Stott 543, 544]; Book of Riddles and Five Hundred
Home Amusements [Toole Stott 107, 951]; Book of Fireside Games [possibly Toole
Stott 300??]; Parlor Theatricals; The Book of 500 Curious Puzzles. However, [Toole Stott 951] is another version
of The Book of Riddles and Five Hundred Home Amusements "by the author of
"Fireside Games" [Toole Stott 300], "The Parlor Magic"
[perhaps Toole Stott 543, 544], "Parlor Tricks with Cards" [Toole
Stott 1056 lists this as by Frikell, "abridged from The Secret Out"
(see also 547, 1142)], ..."; Dick & Fitzgerald, 1986 [sic, but must
mean 1886??].
See
Magician's Own Book for more about the authorship.
See
also: Boy's Own Book, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Indoor and Outdoor, Landells: Boy's Own Toy-Maker, The Secret Out, Hanky Panky, The
Sociable.
Book of Merry Riddles. 1629?
The
Book of Merry Riddles. London,
1629. [Santi 235.]
Several
reprints. Also known as Prettie
Riddles.
A
Booke of Merry Riddles; Robert Bird, London, 1631. [Mark Bryant; Dictionary of Riddles; Routledge, 1990, p. 100.]
Booke
of Merry Riddles; John Stafford & W. G., London, 1660.
Reprint
of the 1629 in: J. O. Halliwell; The literature of the sixteenth and
seventeenth centuries; London, 1851, pp. 67‑102. [Santi 235.]
Reprint
of the 1660 in: J. O. Halliwell; The Booke of Merry Riddles, together with
proper questions, and witty proverbs, to make pleasant pastime. Now first reprinted from the unique edition
printed at London in the year 1660. For
the author, London, 1866. This was a
printing of 25 copies. There is a copy
at UCL and a MS note at the end says 15 copies were destroyed on 9 Apr 1866,
signed: J. O. H., with Number 9 written below.
[Santi 307.] I have seen this,
but some of the riddles are quoted by other authors and I will date all items
as 1629? until I examine other material.
Reprint
of the 1629 in: Alois Brandl; Shakespeares Book of Merry Riddles und die
anderen Räthselbücher seiner Zeit; Jahrbuch der deutschen
Shakespeare-Gesellschaft 42 (1906) 1-64 (with the 1631 ed on pp. 53-63). ??NYR.
[Santi 235 & 237.]
Borghi. Arithmetica. 1484.
Pietro
Borghi = Piero Borgo or Borgi (?? - ³1494). Qui comenza la nobel opera de arithmethica
ne la qual se tracta tute cosse amercantia pertinente facta & compilata p
Piero borgi da veniesia. Erhard
Ratdolt, Venice, 1484. 2 + 116 numbered
ff. This is the second commercial
arithmetic printed in Italy and was reprinted many times. See Rara 16-22. This edition was reproduced in facsimile, with notes by Kurt
Elfering, as: Piero Borghi; Arithmetica
Venedig 1484; Graphos, Munich, 1964;
in: Veröffentlichungen des Forschungsinstituts des Deutschen Museums für
die Geschichte der Naturwissenschaften und der Technik, Reihe C -- Quellentexte
und Übersetzunge, Nr. 2, 1965.
The 3rd ed of 1491 had a title: Libro
dabacho. From the 4th ed of 1501, the
title was Libro de Abacho, so this is sometimes used as the title for the first
editions also. Rara indicates that the
printing was revised to 100 numbered ff by the 4th ed. of 1491. I have examined a 1509 ed. by Jacomo Pentio,
Venice, ??NX. This has 100 numbered ff,
but the last three ff contain additional material, though Rara doesn't mention
this until the 11th ed of 1540. H&S
discusses a problem and the folio in the 1540 ed is the same as in the 1509
ed. The locations of interest in the
1509 ed. are c18ff before the corresponding locations of the 1484. Van Egmond's Catalog 293-297 lists 13
Venetian editions from 1484 to 1567.
It
has been conjectured that this was a pseudonym of Luca Pacioli, but there is no
evidence for this [R. Emmett Taylor; No Royal Road Luca Pacioli and His Times; Univ. of North Carolina Press, Chapel
Hill, 1942, pp. 60 & 349].
See
also: D. E. Smith; The first great
commercial arithmetic; Isis 8 (1926) 41-49.
Bourdon. Algèbre.
7th ed., 1834.
Louis
Pierre Marie Bourdon (1779-1854).
Élémens d'Algèbre. 7th ed.,
Bachelier, Paris, 1834. (1st ed, 1817;
5th, 1828; 6th, 1831; 8th, 1837; 1840.
Undated preface in the 7th ed. describes many changes, so I will cite
this as 1834, though much of the material would have occurred earlier.)
Boy's Own Book. 1828.
William
Clarke, ed. The Boy's Own Book. The bibliography of this book is extremely
complex -- by 1880, it was described as having gone through scores of
editions. My The Bibliography of Some Recreational Mathematics Books has 11 pages listing 76 English (40 UK, 37
US, 1 Paris) versions and a Danish version, implying 88 English (50 UK, 37
US, 1 Paris) versions, and 10 (or 11) related versions, and giving a detailed
comparison of the versions that I have seen.
Because of the multiplicity of versions, I have cited it by title rather
than by the original editor's name, which is not in any of the books (except
the modern facsimile) though this attribution seems to be generally
accepted. I have examined the following
versions, sometimes in partial photocopies or imperfect copies.
Vizetelly,
Branston and Co., London, 1828, 448pp.;
2nd ed., 1828, 462pp.; 3rd ed.,
1829, 464pp (has an inserted advertisement sheet); 6th ed??, c1830, 462pp?? (my copy lacks TP, pp. 417-418, 431-436,
461-462); 9th ed., 1834, 462pp. Longman, Brown & Co., London, 24th ed.,
1846, 462pp. [The latter five are
identical, except for a bit in the Prelude (and the extra sheet in 3rd ed), so
I will just cite the first of these as 1828‑2. It seems that all editions from the 2nd of 1828 through the 29th
of 1848, 462pp. are actually identical except for a bit of the Prelude (and the
advertisement sheet in the 3rd ed.)]
First
American Edition. Munroe & Francis,
Boston & Charles S. Francis, NY, 1829, 316pp. Facsimile by Applewood Books, Bedford, Massachusetts, nd
[1998?]. This is essentially an
abridgement of the 2nd ed of 1828, copying the Prelude and adding "So far the London Preface. The American publishers have omitted a few
articles, entirely useless on this side of the Atlantic, ...." The type is reset, giving some reduction in
pages. A number of the woodcuts have
been omitted. The section title pages
are omitted. Singing Birds, Silkworms,
White Mice, Bantams, Magnetism, Aerostatics, Chess and Artificial Fireworks are
omitted. Angling, Rabbits, Pigeons,
Optics are reduced. Rosamond's Bower is
omitted from Paradoxes and Puzzles.
Surprisingly, The Riddler is increased in size. The 2pp Contents is omitted and an 8pp Index
is added.
Baudry's
European Library & Stassin & Xavier, Paris, 1843, 448pp. [The existence of a Paris edition was
previously unknown to the vendor and myself, but it is Heyl 354 and he cites
Library of Congress. It is very
different than the English and US editions, listing J. L. Williams as
author. Even when the topic is the
same, the text, and often the topic's name, are completely rewritten. See my
The Bibliography of Some Recreational Mathematics Books for details -- in it I have found it
generally necessary to treat this book separately from all other editions. I will cite it as 1843 (Paris). Much of this, including almost all of the
material of interest is copied exactly in
Anon: Boy's Treasury, 1844, qv,
and in translated form in
de Savigny, Livre des Écoliers, 1846, qv. The problem of
finding the number of permutations of the letters of the alphabet assumes 24
letters, which makes me wonder if these books are based on some earlier French
work. Heyl 355 is probably the same
book, with slight variations in the title, by Dean and Munday, London, c1845.]
David
Bogue, London, 1855, 611pp. [It seems
that this version first appears in 1849 and continues through about 1859, when
two sections were appended.]
[W.
Kent (late D. Bogue), London, 1859, 624pp??.
For almost all material of interest, this is identical to the 1855 ed,
so I will rarely (if ever?) cite it.]
[Lockwood
& Co., London, 1861, 624pp.
Identical to the 1859 ed., so I will not cite it.]
Lockwood
& Co., London, 1868, 696pp.
[Lockwood
& Co., London, 1870, 716pp.
Identical to 1868 with 20pp of Appendices, so page numbers for material
of interest are the same as in 1868, so I will not cite it.]
[Crosby
Lockwood & Co., London, 1880, 726pp.
Identical to 1870, but having the Appendices and 20 more pages
incorporated into a new section. For
almost all material of interest, the page numbers are 30 ahead of the 1868
& 1870 page numbers, so I will not cite it except when the page numbers are
not as expected.]
[5th
(US?) ed., Worthington, NY, 1881, 362pp.
For almost all material of interest, this is identical to the 1829 (US)
ed., so I will rarely (if ever?) cite it.]
I
will cite pages with edition dates and edition numbers or locations if needed
(e.g. 1828-2: 410 or
1829 (US): 216). See also: Book of 500 Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury.
Anonymous. The Riddler; A Collection of Puzzles,
Charades, Rebusses, Conundrums, Enigmas, Anagrams, &c. for the Amusement of
Little Folks. S. Babcock, New Haven,
Connecticut, 1835. 22pp. My copy has leaf 11/12 half missing and leaf
17/18 missing; NUC & Toole Stott 1392 say it should be 24pp, so presumably
leaf 23/24 is also missing here. [Toole
Stott 1392 has The Riddler: or, Fire-Side Recreations; a collection ..., 1838,
also listed in NUC.] Paradoxes and
Puzzles section consists of the introduction and 11 puzzles copied almost
exactly from the Paradoxes and Puzzles section of Boy's Own Book, 2nd ed. of
1828 and this material is all in the first American edition of 1829. Other material is charades, etc. and is all
in both these versions of Boy's Own Book.
Shortz states that this is the first American book with puzzles -- but
there were at least five American versions of Boy's Own Book before this and
all the material in The Riddler, except some woodcuts, is taken from Boy's Own
Book, so this pamphlet seems to be a pirate version. NUC also lists a 1838 version.
Boy's Own Conjuring Book. 1860.
The
Boy's Own Conjuring Book: Being a Complete Hand-book of Parlour Magic; and
Containing over One Thousand Optical, Chemical, Mechanical, Magnetical, and
Magical Experiments, Amusing Transmutations, Astonishing Sleights and Subtleties,
Celebrated Card Deceptions, Ingenious Tricks with Numbers, Curious and
Entertaining Puzzles, Charades, Enigmas, Rebuses, etc., etc., etc. Illustrated with nearly two hundred
engravings. Intended as a source of
amusement for one thousand and one evenings.
Dick and Fitzgerald, NY, 1860.
384pp. [Toole Stott 115,
corrected, lists this as (1859), and under 114, describes it as an extended
edition of The Magician's Own Book -- indeed the running head of the book is
The Magician's Own Book! -- but see below.
Toole Stott 481 cites a 1910 letter from Harris B. Dick, of the
publishers Dick & Fitzgerald. He
describes The Boy's Own Conjuring Book as a reprint of Magician's Own Book
"evidently gotten up and printed in London, but singularly enough it had
printed in the book on the title-page -- New York, Dick &
Fitzgerald." Indeed, all the
monetary terms are converted into British.
Harold Adrian Smith [Dick and Fitzgerald Publishers; Books at Brown 34
(1987) 108-114] states that this is a London pirate edition. BMC has 384pp, c1860. NUC has a 384pp version, nd. Christopher 145-149 are five versions from
1859 and 1860, though none has the blue cover of my copy. Christopher 145 says the 1859 versions were
printed by Milner & Sowerby, Halifax, and describes it as an extraction
from Magician's Own Book, but see below.
Christopher 148 cites Smith's article.]
I also have a slightly different version with identical contents except
omitting the date and frontispiece, but with a quite different binding, probably
Christopher 149. [NUC lists 334pp, nd;
416pp, nd and 416pp, 1860. Toole Stott
114 is a 416pp version, 1861. Toole
Stott 959 is a 534pp version, 1861.
C&B cite a New York, 1859 with 416pp, a New York, nd, 334pp and
London, c1850 (surely too early?).]
I
have now compared this with The Magician's Own Book of 1857 and it is
essentially a minor reworking of that book.
The Magician's Own Book has 17 chapters and an answers chapter and a
miscellaneous chapter of items which are almost all also listed in the Contents
under earlier sections. All together,
there are some 635 items. The Boy's Own
Conjuring Book copies about 455 of these items essentially directly, completely
omitting the chapters on Electricity, Galvanism, Magnetism, Geometry, Art, Secret
Writing and Strength, and almost completely omitting the chapter on
Acoustics. Of the 488 items in the
other chapters, 453 are copied into the Boy's Own Conjuring Book, and this has
in addition two of the acoustic problems, 125 new miscellaneous problems and
38pp of charades, riddles, etc. (The
later UK edition of Magician's Own Book is very different from the US
edition.) Many of the problems are
identical to the Boy's Own Book or the Illustrated Boy's Own Treasury. See also:
Book of 500 Puzzles, Boy's Own
Book, Illustrated Boy's Own
Treasury, Landells: Boy's Own Toy‑Maker.
Boy's Treasury. 1844.
Anonymous. The Boy's Treasury of Sports, Pastimes, and
Recreations. With four hundred
engravings. By Samuel Williams. [The phrasing on the TP could be read as
saying Williams is the author, but the NUC entry shows he was clearly listed as
the designer in later editions and his name appears on the Frontispiece.] D. Bogue, London, 1844. Despite the similarity of title, this is
quite different from Illustrated Boy's Own Treasury and the similar books of
c1860. [Toole Stott 116. Toole Stott 117 is another ed., 1847,
'considerably extended'. Toole Stott
gives US editions: 959; 960; 118; 199 & 961-965 are 1st, 1847; 2nd, 1847;
3rd, 1848; 6 versions of the 4th, 1850, 1848, 1849, 1852, 1854, 1848. Hall, BCB 37 is a US ed. of 1850 = Toole
Stott 119. Christopher 151 is a US
version of 1850? NUC lists 9 versions,
all included in Toole Stott. Toole
Stott cites some BM copies, but I haven't found this in the BMC. A section of this, with some additional
material, was reissued as Games of Skill and Conjuring: ..., in 1860, 1861,
1862, 1865, 1870 -- see Toole Stott 312-317.]
I
have now found that much of this, including all the material of interest, is
taken directly from the 1843 Paris edition of
Boy's Own Book, qv, by J. L.
Williams, including many of the illustrations - indeed they have the same Frontispiece,
with S. Williams' name on it.
BR. c1305. Greek
MS, c1305, Codex Par. Suppl. Gr. 387, fol. 118v‑140v. Transcribed, translated and annotated by
Kurt Vogel as: Ein Byzantinisches Rechenbuch des frühen 14.Jahrhunderts; Wiener
Byzantinistische Studien, Band VI; Hermann Böhlaus Nachf., Wien, 1968. I will cite problem numbers and pages from
this -- Vogel gives analysis of the methods on pp. 149‑153 and historical
comments on pp. 154‑160, but I will not cite these.
Brahmagupta, c628. See:
Brahma‑sphuta‑siddhanta;
Colebrooke.
Brahma‑sphuta‑siddhanta.
Bráhma‑sphuta‑siddhânta
of Brahmagupta, 628 (see Colebrooke).
He only states rules, which are sometimes obscure. It appears from Colebrooke, p. v, and Datta
(op. cit. under Bakhshali, p. 10), that almost all the illustrative examples
and all the solutions are due to Chaturveda Prthudakasvâmî in 860. Brahmagupta's rules are sometimes so general
that one would not recognise their relevance to these examples and I have often
not cited Brahmagupta. E.g. cistern
problems are given as examples to Brahmagupta's verse on how to add and
subtract fractions. (See also Datta
& Singh, I, p. 248.) Some of these
comments are taken from Bhaskara I in 629.
Brush. Hubert Phillips.
Brush Up Your Wits. Dent,
London, 1936.
BSHM. British Society for the History of
Mathematics. The produce a useful
Newsletter.
Buteo. Logistica. 1559.
Johannes
Buteo (= Jean Borrel, c1485-c1560 or c1492-1572). Ioan. Buteonis Logistica, quæ & Arithmetica vulgò dicitur in
libros quinque digesta: quorum index summatim habetur in tergo. Gulielmus Rovillius, Lyons, 1559. Most of the material is in books IV and
V. H&S cites some problems in the
1560 ed with the same pages as in the 1559 ed, so these editions are presumably
identical. See Rara 292-294.
c. circa, e.g. c1300. Also
c= means "approximately
equal", though @ will be used in mathematical contexts.
C. Century, e.g. 13C, -5C.
Calandri. Arimethrica. 1491.
Philippo
Calandri. Untitled. Frontispiece is labelled "Pictagoras
arithmetrice introductor". Text
begins: "Philippi Calandri ad nobilem et studiosus Julianum Laurentii
Medicē de arimethrica opusculū." Lorenzo de Morgiani & Giovanni Thedesco da Maganza, Florence,
1491. Van Egmond's Catalog
298-299. The Graves collection has two
copies dated 1491, one with the folio number
c iiii misprinted as b iiii - cf Van Egmond for other
differences in this unique variant.
There was a reprint by Bernardo Zucchetta, Florence, 1518 -- ??NYS but
mentioned: in a handwritten note in one
of the Graves copies of the 1491 (giving Bernardo Zucchecta, 1517); in Smith, Rara, p. 48 (giving Bernardo
Zuchetta, 1518); in Riccardi [I, col.
208-209] (giving Bernardo Zuchecta, 1515)
and in Van Egmond's Catalog 299.
"It is the first printed Italian arithmetic with illustrations
accompanying problems, ...." (Smith, Rara, pp. 46‑49). There are about 50 of these illustrations,
which appear to be woodcuts, but they are quite small, about 25mm (1")
square, and the same picture is sometimes repeated for a related but
inappropriate problem. Rara reproduces
some of these, slightly reduced.
Riccardi [I, col. 208-209] says there may have been a 1490 ed. by
Bernardo Zuchecta, but Van Egmond did not find any example.
Calandri. Aritmetica.
c1485.
Filippo
Calandri. Aritmetica. c1485 [according to Van Egmond's Catalog
158-159]. Italian MS in Codex 2669,
Biblioteca Riccardiana di Firenze.
Edited by Gino Arrighi, Edizioni Cassa di Risparmio di Firenze,
Florence, 1969. 2 vol.: colour
facsimile; transcription of the text.
Copies of the facsimile were exhausted about 1980 and repeated requests
to the Cassa di Risparmio have not produced a reprint, though they usually send
a copy of the text volume every time I write!
I have now (1996) acquired a example of the 2 vol. set and I find that
copies of the text volume which are not part of a set have 8 colour plates
inserted, but these are not in the copy in the set.
I
cite folios from the facsimile volume and pages from the text volume. These are in direct correspondence with the
original except for those pages with full page illustrations. The original begins with a blank side with a
Frontispiece verso, then 9 sheets (18 pp.) of full page tables, then two blank
sheets. The numbered folios then begin
and go through 110. Ff. 1r - 32r are
pp. 3 - 65 of the text. F. 32v is a full page calculation which is
not in the text. Then ff. 33r - 110r are pp. 66 - 220 of the text. F. 110v is a full page illustration omitted in the text. The first 80 folio numbers are in elaborate
Roman numerals centred at the head of the page. (These are sometimes unusually written -- e.g. XXIIIIII.) The later folios were not originally
numbered and were later numbered in the top right corner using Hindu-Arabic
numerals.
In
Sep 1994, I examined the original MS, though it is on restricted access. The original colours are rather more
luminous than in the facsimile, but the facsimile is a first class job. The history of this codex is obscure. It is said to have belonged to Piero di
Lorenzo dei Medici and it may be the book catalogued in the library of
Francesco Pandolfini, c1515, as 'uno libretto ... di Filippo Calandri in
arithmetica'. The Riccardi family
collected continuously from their rise in the mid 15C until the library was
acquired by the city in 1813. A number
of items from the Pandolfini catalogue can be identified as being in the
Riccardiana. Van Egmond's dating may be
early as some claim this was produced for Giuliano de' Medici, who was born in
1479.
Calandri. Raccolta.
c1495.
Filippo
Calandri. Una Raccolta di Ragioni. In: Cod. L.VI.45, Biblioteca Comunale di
Siena. Ed. by D. Santini. Quaderni del Centro Studi della Matematica
Medioevale, No. 4, Univ. di Siena, 1982.
Van Egmond's Catalog 193 identifies this as ff. 75r-111v of the codex,
titles it Ragone Varie and gives a date of c1495.
Calandri. See also:
Benedetto da Firenze, P. M.
Calandri.
Cardan. Ars Magna.
1545.
Jerome
Cardan = Girolamo Cardano = Hieronymus Cardanus (1501‑1576). Artis Magnae sive de Regulis
Algebraicis Liber Unus. Joh. Petreium, Nuremberg, 1545, ??NYS Included in Vol. IV of the Opera Omnia,
Joannis Antonius Huguetan & Marcus Antonius Ravaud, Lyon, 1663, and often
reprinted, e.g. in 1967. NEVER CITED??
Cardan. Practica Arithmetice. 1539.
Jerome
Cardan = Girolamo Cardano = Hieronymus Cardanus (1501‑1576). Practica Arithmetice, & Mensurandi
Singularis. (Or: Practica Arithmeticae
Generalis Omnium Copiosissima & Utilissima, in the 1663 ed.) Bernardini Calusci, Milan, 1539. Included in Vol. IV of the Opera Omnia,
1663, see above. Some of the section
numbers are omitted in the Opera Omnia and have to be intuited. I will give the folios from the 1539 ed.
followed by the pages of the 1663 ed., e.g. ff. T.iiii.v-T.v.r (p. 113).
Cardan. De Rerum Varietate. 1557.
Jerome
Cardan = Girolamo Cardano = Hieronymus Cardanus (1501‑1576). De Rerum Varietate. Henricus Petrus, Basel, 1557; 2nd ed., 1557;
5th ed., 1581, ??NYS. Included in Vol.
III of the Opera Omnia, 1663, see above.
Cardan. De Subtilitate. 1550.
Jerome
Cardan = Girolamo Cardano = Hieronymus Cardanus (1501‑1576). De Subtilitate Libri XXI. J. Petreium, Nuremberg, 1550; Basel, 1553;
6th ed., 1560; and five other 16C editions, part ??NYS. Included in Vol. III of the Opera Omnia,
1663, see above. French ed. by Richard
Leblanc, Paris, 1556, 1584, titled: Les Livres d'Hieronymus Cardanus: De la
Subtilité et subtiles Inventions, ensemble les causes occultes et les raisons
d'icelles; 9th ed., 1611. I have seen a
note that the 1582 ed. by Henricus Petrus, Basel, was augmented by a riposte to
attacks by Scaliger with further illustrations.
Carlile. Collection.
1793.
Richard
Carlile. A Collection of One Hundred
and Twenty Useful and Entertaining Arithmetical, Mathematical, Algebraical, and
Paradoxical Questions: With the Method of Working Each. Printed by T. Brice for the author, Exeter,
1793. Wallis 227 CAR, ??NX. Includes a number of straightforward
problems covered here, but I have only entered the more unusual examples.
Carroll-Collingwood. 1899.
The
Lewis Carroll Picture Book. Stuart
Dodgson Collingwood, ed. T. Fisher
Unwin, London, 1899. = Diversions
and Digressions of Lewis Carroll, Dover, 1961.
= The Unknown Lewis Carroll, Dover, 1961(?). Reprint, in reduced format, Collins, c1910. The pagination of the main text is the same
in the original and in both Dover reprints, but is quite different than the
Collins. I will indicate the Collins
pages separately. The later Dover has
42 additional photographs.
Carroll-Gardner. c1890?
or 1996
Martin
Gardner. The Universe in a
Handkerchief. Lewis Carroll's
Mathematical Recreations, Games, Puzzles and Word Plays. Copernicus (Springer, NY), 1996. As with Carroll-Wakeling, Carroll material
will be dated as 1890?, but there is much material by Gardner which is dated
1996.
Carroll-Wakeling. c1890?
Lewis
Carroll's Games and Puzzles. Newly
Compiled and Edited by Edward Wakeling.
Dover and the Lewis Carroll Birthplace Trust, 1992. This is mostly assembled from various
manuscript sheets of Carroll's containing problems which he intended to
assemble into a puzzle book. Wakeling
has examined a great deal of such material, including a mass of Carroll's notes
to Bartholomew Price (1818‑1898) who was Sedleian Professor of Natural
Philosophy at Oxford in 1853-1898.
Price was at Pembroke College, becoming the Master, adjacent to
Carroll's Christ Church. He had tutored
Carroll (1833‑1898) and they were close friends and in continual contact
until their deaths, both in 1898.
However, few of the papers are dated and they are simply loose sheets
with no indication of being in order, so there is no way to date the undated
sheets and I have given a fairly arbitrary date of c1890? for these, though
Carroll was more active before then rather than after. Some items are taken from Carroll's youthful
magazines or his correspondence and hence are more precisely dated. The correspondence is more fully given in
Carroll-Collingwood.
In
response to an inquiry, Wakeling wrote on 28 May 2003 and said that some of the
Carroll-Price notes were typewritten 'probably using Dodgson's Hammond
typewriter, purchased in 1888.' This
gives a somewhat more precise dating than my c1890? and I will give: 1888 to 1898 for such items, unless there is other evidence.
Carroll-Wakeling II. c1890?
Rediscovered
Lewis Carroll Puzzles. Newly Compiled
and Edited by Edward Wakeling. Dover,
1995. See the notes to
Carroll-Wakeling, above.
Cassell's. 1881.
Cassell's
Book of In‑Door Amusements, Card Games, and Fireside Fun. Cassell, Peter, Gilpin & Co., London,
1881; Cassell, London, 1973. 217pp
[probably + 1p + 6pp Index] (pp. 1-8 are preliminary matter). [There was a companion volume: Cassell's
Book of Sports and Pastimes. In 1887,
the two were combined, with the spine titled
Cassell's Book of Outdoor Sports and Indoor Amusements. The front cover says Out Door Sports, the
back cover says Indoor Amusements, while the title page says Cassell's Book of
Sports and Pastimes. It contains all
the main text of Book of In‑Door Amusements, ..., advanced by 744
pages. From at least 1896, Card Games
and Parlour Magic were completely revised and later there were a few other
small changes. The title varies
slightly. Manson (qv) is a 1911
revision and extension to 340pp of main text.]
Catel. Kunst-Cabinet. 1790.
Peter
Friedrich Catel. Mathematisches und
physikalisches Kunst-Cabinet, dem Unterrichte und der Belustigung der Jugend
gewidmet. Nebst einer zweckmässigen
Beschreibung der Stücke, und Anzeige der Preise, für welche sie beim Verfassser
dieses Werks P. F. Catel in Berlin zu bekommen sind. [I.e. this is a catalogue of items for sale by post!] Lagarde und Friedrich, Berlin & Libau,
1790. [MUS #113.] P. iv says he started his business in 1780.
There
is a smaller Vol. 2, with the same title, except 'beim Verfasser dieses Werkes
P. F. Catel' is replaced by 'in der P. F. Catelschen Handlung', and the
publisher is F. L. Lagarde, Berlin, 1793.
My
thanks to M. Folkerts for getting a copy of the example in the Deutsches Museum
made for me.
All
citations are to vol. 1 unless specified.
Many
of Bestelmeier's items are taken from Catel.
Sometimes the figure is identical (often reversed) or is a poor
copy. Texts are often copied verbatim,
or slightly modified, but usually abbreviated.
E.g. Catel often explains the puzzle and this part is frequently omitted
in Bestelmeier. Bestelmeier was the
successor to Catel. Dieter Gebhardt has
searched for the various editions and associated price lists of the Catel and
Bestelmeier catalogues in German libraries and he and Jerry Slocum have
published the details in: Jerry
Slocum & Dieter Gebhardt. Puzzles
from Catel's Cabinet and Bestelmeier's Magazine 1785 to 1823. English
translations of excerpts from the German Catel-Katalog and
Bestelmeier-Katalog. Intro. by David
Singmaster. History of Puzzles
Series. The Slocum Puzzle Foundation,
PO Box 1635, Beverly Hills, California, 90213, USA, 1997. I have not yet made detailed entries from
this which gives precise dates for the various parts of these catalogues.
CFF. Cubism for Fun. This is the Newsletter of the Nederlandse
Kubus Club (NKC) (Dutch Cubists Club) which has been in English since the mid
1980s.
Chambers -- see: Fireside
Amusements.
Charades, Enigmas, and Riddles. 1859.
Charades,
Enigmas, and Riddles. Collected by A
Cantab. [BLC gives no author. "A
Cantab." was a common pseudonym.
One such author of about the right time and nature was George
Haslehurst.] (Cambridge, 1859).
3rd ed.,
J. Hall and Son, Cambridge, 1860, HB.
Half-title, 6 + 96pp.
4th
ed., Bell & Daldy, London, 1862. 8
preliminaries (i = half-title; FP facing iii = TP; v-viii = Introduction;
errata slip; two facing plates illustrating a charade for Harrowgate [sic]
Waters), 1-160, 32pp publisher's ads, dated Jan 1863; (my copy is lacking pp.
63-64). The three plates are signed
J.R.J. This is a substantial expansion
of the 3rd ed.
I
also have photocopy of part of the 5th ed., Bell and Daldy, London, 1865, and
this shows it was even larger than the 4th ed, but most of the problems of
interest have the same or similar problem numbers in the three editions that I
have seen. I will cite them as in the
following example. 1860: prob. 28, pp.
59 & 63; 1862: prob. 29, pp. 135 & 141; 1865: prob. 573, pp. 107
& 154.
Chaturveda. Chaturveda Pŗthudakasvâmî [NOTE: ŗ
denotes an r with an underdot.]. Commentator on the Brahma‑sphuta‑siddhanta
(qv), 860. Some of these comments are
taken from Bhaskara I in 629.
Shukla calls him Pŗthūdaka, but Colebrooke cites him as Ch.
Chessics. Chessics.
The Journal of Generalised Chess.
Produced by G. P. Jelliss, 5 Biddulph Street, Leicester, LE2 1BH. No. 1 (Mar 1976) -- No. 29 & 30 (1987). Succeeded by G&PJ.
Child. Girl's Own Book.
Mrs.
L. Maria Child [= Mrs. Child = Lydia
Maria Francis, later Child]. The Girl's
Own Book. The bibliography of this book
is confused. According to the Opies
[The Singing Game, p. 481], the first edition was Boston, 1831 and there was a
London 4th ed. of 1832, based on the 2nd US ed. However the earliest edition in the BMC is a 6th ed. of
1833. I have examined and taken some notes
from the 3rd ed., Thomas Tegg, London, 1832 -- unfortunately I didn't have time
to go through the entire book so I may have missed some items of interest. I have also examined the following.
Clark
Austin & Co., NY, nd [back of original TP says it was copyrighted by
Carter, Hendee, & Babcock in Massachusetts in 1833]; facsimile by Applewood Books, Bedford,
Massachusetts, nd [new copy bought in 1998 indicates it is 4th ptg, so
c1990]. The facsimile is from a copy at
Old Sturbridge Village. The back of the
modern TP says the book was first published in 1834 and the Cataloguing-in-Publication
data says it was originally published by Carter, Hendee and Babcock in
1834. However, the earliest version in
the NUC is Clark, Austin, 1833. I am
confused but it seems likely that Carter, Hendee and Babcock was the original
publisher in Boston in 1831 and that that this facsimile is likely to be from
1833 or an 1834 reprint of the same.
The pagination is different than in the 1832 London edition I have seen.
The
Tenth Edition, with Great Additions. By
Mrs. Child. Embellished with 144 Wood
Cuts. Thomas Tegg, London (& three
other copublishers), 1839. 12 + 307 pp
+ 1p publisher's ad. Has Preface to the
Second Edition but no other prefaces.
This Preface is identical to that in the 1833 NY ed, except that it
omits the final P.S. of season's greetings.
The 1833 NY essentially has the same text, but they have different
settings and different illustrations with some consequent rearrangement of
sections. However the main difference
is that the NY ed omits 41pp of stories.
There are a number of minor differences which lead to the NY ed having 9
extra pages of material.
The
Eleventh Edition, with Great Additions.
By Mrs. Child. Embellished with
124 Wood Cuts. Thomas Tegg, London
(& three other copublishers), 1842.
12 + 363 pp + 1p publisher's ad.
The Preface is identical to that in the 10th ed, but omits 'to the
Second Edition' after Preface. 90 pp of
games and 40 pp of enigmas, charades, rebuses, etc. have been added; 56 pp of
stories have been dropped.
The
Girl's Own Book of Amusements, Studies and Employments. New Edition. Considerably enlarged and modernized by Mrs. L. Valentine, and
others. William Tegg, London,
1876. This differs considerably from
the previous editions.
I
will cite the above by the dates 1832,
1833, 1839, 1842, 1876.
Various
sources list: 13th ed., 1844 [BMC,
Toole Stott 831]; Clark Austin, NY,
1845 [NUC]; 16th ed., 1853 [BMC]; 17th ed. by Madame de Chatelain, 1856 [BMC,
NUC, Toole Stott 832]; 18th ed. by
Madame de Chatelain, 1858 [BMC, Toole Stott 833]; 1858 [Osborne Collection (at Univ. of Toronto)]; rev. by Mrs. R. Valentine, 1861 [BMC,
Osborne Collection]; rev. by Mrs. R. Valentine,
1862 [BMC, NUC]; rev. by Mrs. R.
Valentine, 1864 [BMC]; rev. by Mrs. R.
Valentine, 1867 [BMC]; enlarged by Mrs.
L. Valentine, 1868 [NUC]; enlarged by
Mrs. L. Valentine, 1869 [BMC]; enlarged
by Mrs. L. Valentine, 1873 [NUC];
enlarged by Mrs. L. Valentine, 1875 [NUC]; enlarged by Mrs. L. Valentine, 1876 [BMC];
Heyl
gives the following under the title The
Little Girl's Own Book: Carter, Hendee
and Co., Boston, 1834; American
Stationers Co, John B. Russell, Boston, 1837;
Edward Kearney, NY, 1847; NY,
1849.
I think there were at least 33 editions. See my
The Bibliography of Some Recreational Mathematics Books for more details. Cf Fireside Amusements, below, which is largely copied from
Child.
Chiu Chang Suan Ching. c-150?
Chiu
Chang Suan Ching (Nine Chapters on the Mathematical Art). (Also called Chiu Chang Suan Shu and
variously transliterated. The pinyin is
Jiŭ Zhāng Suàn Shù.) c‑150? German translation by K. Vogel; Neun
Bücher arithmetischer Technik; Vieweg, Braunschweig, 1968. My citations will be to chapter and problem,
and to the pages in Vogel. (Needham
said, in 1958, that Wang Ling was translating this, but it doesn't seem to have
happened.) Some of the material dates
from the early Han Dynasty or earlier, say c-200, but Chap. 4 & 9, the most
original of all, have no indication of so early a date. A text of c50 describes the contents of all
the chapters and Høyrup suggests that Chap. 4 & 9 and the final assembly of
the book should be dated to the [early] 1C.
Christopher. 1994.
Maurine
Brooks Christopher & George P. Hansen.
The Milbourne Christopher Library.
Magic, Mind Reading, Psychic Research, Spiritualism and the Occult 1589-1900.
Mike Coveney's Magic Words, Pasadena, 1994. 1118 entries. References
are to item numbers.
Christopher II. 1998.
Maurine
Brooks Christopher & George P. Hansen.
The Milbourne Christopher Library -- II. Magic, Mind Reading, Psychic Research, Spiritualism and the
Occult 1589-1900. Mike Coveney's Magic Words, Pasadena, 1998. 3067 entries. References are to item numbers.
Recently received, ??NYR.
Chuquet. 1484. Nicolas
Chuquet. Problèmes numériques faisant
suite et servant d'application au Triparty en la science des nombres de Nicolas
Chuquet Parisien. MS No. 1346 du Fonds
Français de la Bibliothèque Nationale, 1484, ff. 148r-210r. Published in an abbreviated version as:
Aristide Marre; Appendice au Triparty en la science des nombres de Nicolas
Chuquet Parisien; Bulletino di bibliografia e di storia delle scienze
matematiche e fisiche 14 (1881) 413‑460.
(The first part of the MS was published by Marre; ibid. 13 (1880)
593-814; ??NYS) Marre generally
transcribes the text of the problem, but just gives the answer without any of
the text of the solution. I will cite
problems by number. There are 166
problems. (Much of this was used in his
student's book: Estienne de la Roche; Larismethique novellement composee par
maistre Estienne de la roche dict Villefrāche; Lyons, 1520, ??NYS. (Rara 128‑130).)
FHM Graham Flegg, Cynthia Hay & Barbara Moss. Nicolas Chuquet, Renaissance Mathematician. A study with extensive translation of Chuquet's
mathematical manuscript completed in 1484.
Reidel, Dordrecht, 1985. This
studies the entire MS, of which the above Appendice is only the second quarter. It often gives a full English translation of
the text of the problem and the solution, but it may summarize or skip when
there are many similar problems. The
problems in the first part of the MS are not numbered in FHM. I will cite this as FHM xxx, where xxx is
the page number, with 'English in FHM xxx' when the problem is explicitly
translated.
Clark. Mental Nuts. 1897, 1904,
1916.
A book
of Old Time Catch or Trick Problems
Regular old Puzzlers that kept your Grandad up at night. Copyright, 1897, by S. E. Clark,
Philadelphia. Flood & Conklin
Co. Makers of Fine Varnishes, Newark,
N.J. 100 problems and answers. 32pp + covers.
A book
of 100 Catch or Trick Problems Their
simplicity invites attack, while their cunningly contrived relations call forth
our best thought and reasoning.
Copyright, 1897, by S. E. Clark, Philadelphia. Revised 1904 Edition.
Waltham Watches, Waltham, Massachusetts. This was an promotional item and jewellers would have their
address printed on the cover. My
example has: With the compliments
of J. H. Allen Jeweler [sic] Shelbina, Mo. Thanks to
Jerry Slocum for this. In fact there
are only 95 problems; numbers 68, 75, 76, 78, 84 are skipped. 32pp + covers.
Revised
Edition 1916, with no specific company mentioned. Enlarged PHOTOCOPY from Robert L. Helmbold. 100 numbered problems, but some figures
inserted after no. 75 are the solutions to a problem in the other editions and
I have counted this as a problem (no. 75A), making 101 problems. 28pp + covers.
The editions are
considerably different. Only 40
problems occur in all three editions. There
are 50 problems common to 1897 and 1904, 42 common to 1897 and 1916 and 71
common to 1904 and 1916, though this counting is a bit confused by the fact
that problems are sometimes combined or expanded or partly omitted, etc. Solutions are brief. It includes a number of early examples or
distinct variants, which is remarkable for a promotional item. I have entered 36 of the 1897 problems plus
13 of the 1904 problems not in 1897 and 7 of the 1916 problems not in 1897 or
1904. Many others are standard examples
of topics covered in this work, but are not sufficiently early to be worth
entering.
I originally had the 1904
ed and cited the 1904 problems as 1897 on the grounds that editions of this
period do not change much, but having now seen the 1897 and 1916 eds, I realise
that the editions are very different, so I will cite the actual dates. Since only the 1897 version is paginated, I
will just cite problem numbers; the solutions are at the back.
Clarke, William. See:
Boy's Own Book.
CM. Crux Mathematicorum (originally titled
Eureka until 4:3)
CMJ. The College Mathematics Journal. Before the early 1980s, this was the Two
Year College Mathematics Journal.
Colebrooke. 1817.
Henry
Thomas Colebrooke (1765-1837), trans.
Algebra, with Arithmetic and Mensuration from the Sanscrit of
Brahmegupta and Bháscara. John Murray,
London, 1817. Contains Lîlâvatî and
Bîjaganita of Bhâskara II (1150) and Chapters XII (Arithmetic) and XIII
(Algebra) of the Bráhma‑sphuta‑siddhânta of Brahmagupta (628). There have been several reprints, including
Sändig, Wiesbaden, 1973. (Edward
Strachey produced a version: Bija Ganita: or the Algebra of the Hindus; W.
Glendinning, London, 1813; by translating a Persian translation of 1634/5.)
Collins. Book of Puzzles. 1927.
A.
Frederick Collins. The Book of
Puzzles. D. Appleton and Co., NY,
1927.
Collins. Fun with Figures. 1928.
A.
Frederick Collins. Fun with
Figures. D. Appleton and Co., NY,
1928.
Columbia Algorism. c1350.
Anonymous
Italian MS, c1350 [according to Van Egmond's Catalog 253‑254], Columbia
X511 .A1 3. Transcribed and edited by
K. Vogel; Ein italienisches Rechenbuch aus dem 14.Jahrhundert;
Veröffentlichungen des Forschungsinstituts des Deutschen Museums für die
Geschichte der Naturwissenschaften und der Technik, Reihe C, Quellentexte und
Übersetzungen, Nr. 33, Munich, 1977. My
page references will be to this edition.
Van Egmond says it has a title in a later hand: Rascioni de Algorismo.
The Algorism is discussed at length in
Elizabeth B. Cowley; An Italian mathematical manuscript; Vassar Medieval
Studies, New Haven, 1923, pp. 379‑405.
Conway, John Horton. (1937- ).
See: Winning Ways.
Cowley, Elizabeth B. See:
Columbia Algorism.
CP. 1907. H.
E. Dudeney. Canterbury Puzzles. (1907);
2nd ed. "with some fuller solutions and additional notes",
Nelson, 1919; 4th ed. = Dover, 1958. (I have found no difference between the 2nd and 4th editions,
except Dover has added an extra note on British coins and stamps. I now have a 1st ed, which has different
page numbers, but I have not yet added them.)
CR Comptes Rendus des Séances de
l'Académie des Sciences, Paris.
Crambrook. 1843. W.
H. M. Crambrook. Crambrook's Catalogue
of Mathematical & Mechanical Puzzles Deceptions and Magical Curiosities,
contained in the Necromantic Tent, Royal Adelaide Gallery, West Strand,
London. ... To which is added, a Complete Exposé [of] the Baneful Arts by
which unwary Youth too often become the prey of professed gamesters. And ... an extract from The Anatomy of
Gambling. Second Edition, Corrected
& Enlarged. T. C. Savill, 107 St.
Martin's Lane, 1843. 23pp. Photocopy provided by Slocum. [According to: Edwin A. Dawes; The Great
Illusionists; Chartwell Books, Secaucus, New Jersey, 1979, p. 138, this is the
first known magical catalogue. It has a
list of about 100 puzzles on pp. 3-5, with the rest devoted to magic
tricks. Unfortunately there are no
pictures. Comparison with Hoffmann
helped identify some of the puzzles, but I can not identify many of them. I have marked almost all these entries with
?? or check??, but the only way one can check is if actual examples or an
illustrated catalogue turn up. Some of
the names are so distinctive that it seems certain that the item does fit where
I have cited it; others are rather speculative. There are several names which may turn up with more
investigation. Toole Stott 190 says
there should be 48pp, though the later pages may be the added material on
gambling.]
Cremer, William Henry, Jr. See under: Book of 500 Puzzles, Hanky
Panky, Magician's Own Book.
CUP. Cambridge University Press.
Cyclopedia. 1914.
Sam
Loyd's Cyclopedia of 5,000 Puzzles, Tricks and Conundrums (ed. by Sam Loyd
Jr). Lamb Publishing, 1914 = Pinnacle or Corwin, 1976. This is a reprint of Loyd's "Our Puzzle
Magazine", a quarterly which started in June 1907 and ran till 1908. See OPM for details.
C&B. 1920. Sidney
W. Clarke & Adolphe Blind. The Bibliography of Conjuring
And Kindred Deceptions. George
Johnson, London, 1920. Facsimile by
Martino Fine Books, Mansfield Centre, Connecticut, nd [obtained new in 1998].
C&W. Chatto & Windus, London.
Datta & Singh. Bibhutibhusan Datta & Avadhesh
Narayan Singh. History of Hindu
Mathematics. Combined edition of Parts I
(1935) and II (1938), Asia Publishing House, Bombay, 1962. NOTE: This book makes some contentious
assertions. Readers are referred to the
following reviews.
O.
Neugebauer. Quellen und Studien zur
Geschichte der Mathematik 3B (1936) 263-271.
S.
Gandz. Isis 25 (1936) 478-488.
Datta, B. See:
Bakhshali MS; Datta & Singh.
De Morgan (1806-1871). See:
Rara.
De Viribus. See:
Pacioli.
dell'Abbaco. See:
Pseudo-dell'Abbaco.
Depew. Cokesbury Game Book.
Arthur
M. Depew. The Cokesbury Game Book. Abingdon-Cokesbury Press, NY &
Nashville, 1939. [The back of the TP
says it is copyright by Whitmore & Smith -- ?? The Acknowledgements say material has been assembled from various
sources and colleagues who have been collecting and writing over the previous
thirty years.]
Dickson. Leonard Eugene Dickson (1874-1954). History of the Theory of Numbers, 3
vols. Carnegie Institution of
Washington, Publication 256, 1919-1923;
facsimile reprint by Chelsea, 1952.
Dilworth. Schoolmaster's Assistant.
Thomas
Dilworth. The Schoolmaster's
Assistant: Being a Compendium of
Arithmetic, both Practical and Theoretical.
(1743); 11th ed., Henry Kent,
London, 1762 (partly reproduced by Scott, Foresman, 1938.) 20th ed., Richard & Henry Causton,
London, 1780. De Morgan suggests the
1st ed. was 1744 or 1745, but the testimonials are dated as early as Jan 1743,
so I will assume 1743. Comparison of a
1762 ed. (Wallis 321 DIL) with my 1780 ed. shows the 1780 ed. is identical to
the 1762 ed., except the section on exchange is much expanded, so the page
numbers of all material of interest are increased by 12pp. I will cite the pages of the 1762 ed., but
give the date as 1743. [Wallis also has: 14th ed., 1767; 15th ed., 1768;
1783; 22nd ed., 1785; 1791;
24th ed., 1792; 1793; 33rd ed., 179-; 1799; 1800; 1804.]
[Halwas 149‑162 are some US editions.]
Diophantos. c250.
Diophantos. Arithmetica. c250. In: T. L. Heath;
Diophantos of Alexandria; 2nd ed., (OUP, 1910); Dover, 1964. Note: Bachet edited a Greek and Latin
version of Diophantos in 1620, which inserted 45 problems from the Greek
Anthology at the end of Book V. (It was
in Fermat's copy of this work that Fermat wrote the famous marginal note now
called his Last Theorem; Fermat's son published an edition with his father's
annotations in 1670, but the original copy was lost in a fire.)
DNB. Leslie Stephen, ed. The Dictionary of National Biography. Smith, Elder and Co., London, 1885‑1901
in 22 volumes. OUP took it over in
1917. Decennial Supplements were
added.
Compact
Edition, with Supplement amalgamating the six decennial supplements to 1960,
OUP, 1975. The Compact Ed. shows the
original volumes and pages so I will cite them in ( ), followed by the pages in
the Compact Ed.
Dodson. Math. Repository. (1747?); 1775.
James
Dodson. The Mathematical
Repository. Containing Analytical
Solutions of near Five Hundred Questions, mostly selected from Scarce and
Valuable Authors. Designed As Examples
to Mac-Laurin's and other Elementary Books of Algebra; And To conduct Beginners
to the more difficult Properties of Numbers.
2nd ed., J. Nourse, London, 1775, HB.
(I have now acquired vols. II & III (1753 & 1755), but these are
largely concerned with annuities, etc., except the beginning of vol. II has a
section on indeterminate equations, entered in 7.P.1. From references in these volumes, it seems that the 1775 ed. of
volume I is pretty close to the first ed. of c1747, but has been a little
rearranged, so I have redated the entries as above.)
Doubleday -- n. 1969, etc.
Eric
Doubleday. Test Your Wits, Vols. 1 -
5. Ace Publishing, NY, 1969; 1971;
1972; 1969[sic]; (1969), revised 1973.
[Vols. 1 - 3 are good collections, with a number of novel variations of
standard problems. Vols. 4 & 5 are
vol. 1 split into two parts and much padded by putting each answer on a
separate page! The books refer to
Doubleday as puzzle setter for a London newspaper and one of the best known
setters in the English speaking world.
However, none of the older puzzle setters/editors in England have ever
heard of him and there is no book by him in the British Library Catalogue. Surprisingly, there is also no book by him
in the Library of Congress Catalogue! I
am beginning to think the author is a deception, but the first three books are
better than scissors and paste hack work.]
DSB. Dictionary of Scientific Biography. Ed. by Charles C. Gillespie for the American
Council of Learned Societies.
Scribner's, NY, 1970-1977, in 18 volumes. I will give the volume and the pages.
The
mathematical material has been reprinted in four volumes as: Biographical Dictionary of
Mathematicians Reference Biographies
from the Dictionary of Scientific Biography. Scribner's, NY, 1990?
This has new pagination, continuous through the four volumes. If I don't have the DSB details, I will cite
this as BDM.
Dudeney, Henry Ernest
(1857-1930). See: AM,
CP, MP, PCP,
536. I also cite his columns or
contributions in The Captain, Cassell's Magazine, Daily Mail, London Magazine,
The Nineteenth Century, The Royal Magazine, Strand Magazine, Tit-Bits, The
Tribune, The Weekly Dispatch.
Eadon. Repository. 1794.
John
Eadon. The Arithmetical and
Mathematical Repository, Being a New Improved System of Practical Arithmetic,
in all its Branches; Designed for the Use of Schools, Academies,
Counting-Houses, and Also for the Benefit of private Persons who have not the
Assistance of a Teacher. In Four
Volumes. Volume 1. In Three Books. Printed for the author, and sold by G. G. and J. Robinson,
Paternoster Row, London, 1794.
EB Encyclopædia Britannica. I tend to use my 1971 ed.
Endless Amusement I. c1818.
Anonymous. Endless Amusement; A Collection of Nearly
400 Entertaining Experiments In various
Branches of Science; ..., All the Popular Tricks and Changes of the Cards,
.... 3rd ed.(?), Thomas Boys, London,
nd [1825]. Frontispiece & TP are
missing, but James Dalgety has inserted a photocopy of the TP of the 3rd
ed. [BMC lists 2nd ed., 1819?; 3rd ed., 1825? BMC65
lists 1st ed. by Thorp & Burch, c1818;
2nd ed., c1820. Toole Scott
255-267 lists 3rd ed., c1820; 4th ed., 1822; 5th ed., 1830; 6th ed.,
1834; 7th ed., 1839. Hall, BCB 116-123 are: 1st ed., c1815; 2nd ed., c1820; 3rd ed.,
c1820; 3rd ed., Philadelphia,
1822; 4th ed., c1825; 5th ed., c1830; 6th ed., 1834;
Philadelphia, 1847. Heyl
110-115, 121 are 1819; 2nd & 3rd ed., M. Carey & Sons,
Philadelphia, 1821 & 1822; 3rd ed,
C. Tilt, London, 1825; Borradaile, NY,
1831; Henry Washburne & Thomas
Tegg, London, 1839; Lea &
Blanchard, Philadelphia, 1847. Almost
all of these are listed as 2 + 216 pp, so the editions are probably all the
same as the 1st ed., except that Hall notes that the 1st ed. title is slightly
different: Endless Amusement; A Collection of Upwards of 400 Entertaining and
Astonishing Experiments. Among a
Variety of other Subjects, are Amusements in Arithmetic, Mechanics, Hydraulics
.... All the Popular Tricks and Changes
of the Cards, ..., and Heyl gives a
similar title for the 1825 ed. and the 1831 NY ed. has some variations. Christopher 330-338 are 2nd ed., Philadelphia, 1821; 3rd ed., c1820; 3rd ed., Philadelphia, 1822;
4th ed., c1822; 4th ed., c1822
(slightly different to preceding; 5th
ed., c1830; 6th ed., 1834; 7th ed., 1839; Philadelphia, 1847.
C&B list it under Thorp and Burch, the publishers, with no dates.] [There is a Recreations in Science, ..., by
the author of Endless Amusement, 1830.]
Endless Amusement II. 1826?
Anonymous. A Sequel to the Endless Amusement,
Containing Nearly Four Hundred Interesting Experiments, In various Branches of
Science, ..., to Which are Added, Recreations with cards, and a Collection of
Ingenious Problems. Thomas Boys,
London, nd [1826?]. Pp. 203-216 are
missing, but James Dalgety has inserted photocopies. [BMC lists one ed., 1826?
Hall, BCB 252 gives c1825. Heyl
says this refers to Thomas Boys ... and Thorp and Burch, London (1825). Toole Stott 623 gives 1825.]
=
Anonymous. The Endless Amusement. New Series Containing Nearly Four Hundred
Interesting Experiments, ... (as above).
Thomas Tegg & Sons, London, 1837.
Angela Newing has provided a photocopy of the interesting parts of this
and it is virtually identical to the 1826? ed., though it has been reset, resulting
in an extra word fitting on some lines, and it has rather poorer pictures. One problem has been replaced by
another. [Heyl 122.] 21 problems, including the replacement
problem, are copied in The New Sphinx.
=
Anonymous. A Companion to the Endless
Amusement. James Gilbert, London,
1831. [Toole Stott 172 says this is a
reprint of A Sequel ..., from the same type, with new TP, and this is clear
from examination of the example Wallis 187.5 COM. Heyl 66 dates it as c1820?]
Some
of the material is taken from Badcock.
van Etten/Leurechon. 1624.
Recreation
Mathematicque.
The
bibliography of this book is very complicated.
I have now made a separate bibliography of this, augmented by many
contributions from Voignier, which is now (Aug 2001) 19pp, listing 50 French
editions, 5 English editions, 4 Latin editions and 8 (or 9) Dutch editions -- a
total of 67 (or 68) editions from 1624 to 1706, though at least 10 of the
French editions may be 'ghosts'. This
is part of my The Bibliography of Some
Recreational Mathematics Books.
This
book has traditionally been attributed (since 1643) to Père Jean Leurechon, SJ
(c1591-1670), who was probably van Etten's university teacher, but the book
specifically names van Etten and there seems to be very little real evidence
for Leurechon's authorship. Trevor
Hall's booklet and chapter are a substantial study of this question and he
concludes that there is no real reason to doubt van Etten's authorship, though
he may well have had help or inspiration from his teacher. Hall has also shown that van Etten and his
uncle, the book's dedicatee, were real people.
(Toole Stott 429‑431 dismisses Hall's work as a result of
completely misunderstanding it!)
However, the book was amended, revised and translated many times, so
that versions may occur under the following names: Hendrik van Etten; Jean
Leurechon; D.H.P.E.M. = Denis (or Didier) Henrion, professeur en
mathématique (or Professeur ès Mathematiques), a pseudonym of Clément
Cyriaque de Mangin, who also called himself Pierre Hérigone; Claude Mydorge; Caspar (or Gaspar) Ens;
Wynant van Westen; William
Oughtred; William Leake; not to mention
Anonymous and versions of the title -- I have found it under Recreations or
Récréations and under Vermakelijkheden. E.g. Lucas, RM1, 239-240, has 7 entries for this book under five
different authors and twice under Récréations.
Jacques
Voignier; Who was the author of "Recreation Mathematique" (1624)?; The
Perennial Mystics #9 (1991) 5-48 (& 1-2 which are the cover and its
reverse). [This journal is edited and
published by James Hagy, 2373 Arbeleda Lane, Northbrook, Illinois, 60062,
USA.] This is the second serious study
of this book. He points out evidence
for Leurechon's connection with the book, which makes it seem more likely, but
definite evidence is still lacking, so I am suggesting that it may have been
some kind of joint production and I will change my references it to van Etten/Leurechon. The work of Hall and Voignier form the basis
of the following discussion, supplemented by the standard catalogues and
personal inspection of about a half of the French and English editions --
generally after 1630.
Henrik
van Etten. Recreation
Mathematicque. Composee de Plusieurs
Problemes Plaisants et Facetieux. En faict
d'Arithmeticque Geometrie, Mechanicque, Opticque, & autres parties de ces
belles sciences. Jean Appier Hanzelet,
Pont‑a‑Mousson, 1624 [taken from facsimile of the 1626 ed.]. 155pp., ??NYS.
2nd
ed., 1626, ibid. = recent facsimile
with no details, but with 'Pont à Mousson
13 ‑ 10 ‑ 54' written inside the back cover. [An apparently identical copy at the Museum
of the History of Science, Oxford, has a small insert saying it was reissued by
La Compagnie de Pont-à-Mousson, printed by l'Imprimerie Berger-Levrault,
nd.]. 91 problems on xiv + 144 =
158pp. [The extra pages include
questions V and VI of problem 91 -- these questions occur in no other edition,
except probably in the 1629 reissue in Pont-à-Mousson.]
After
the two Hanzelet editions, there were three editions in Paris in 1626, by Rolet
Boutonné (2nd ed.), by Antoine Robinot (2nd ed.) and by Jean Moreau &
Guillaume Loyson (3rd ed.). Boutonné
and Robinot were closely associated and their output was interchangeable. Their 2nd eds. appear to be essentially the
1624 ed. The Moreau & Loyson has
Notes added to the problems and 8pp. of Additions. This was the first to put the illustrations as woodcuts in the text
rather than using copperplates for five separate sheets of 8 figures. The Notes are signed D.A.L.G., but are due
to Claude Mydorge. (NUC indicates the
Robinot had further comments signed D.H.P.E.M., later identified as Denis (or
Didier) Henrion Professeur En Mathématique (though Henrion is a pseudonym of
Clément Cyriaque de Mangin!) -- but this seems to be a confusion.) In the next few years, editions appeared in
Paris, Rouen and Lyon. In 1627,
Boutonné issued a '4th ed.' with "Nottes sur les recreations mathematiques
... Par D. H. P. E. M." and the D.A.L.G. notes were omitted. In 1627, Claude Rigaud & Claude Obert,
Lyon, issued a version with 9pp of Additions as in the 1626 Moreau &
Loyson.
In
1628, Charles Osmont, Rouen, issued a version in three parts: Récréations
mathématiques ... 1re et 2de partie. La 3e partie contient un recueil
de plusieurs gentilles et récréatives inventions de feux d'artifice .... Part 1 was van Etten's 91 problems, with
questions V & VI of prob. 91 omitted, omitting the D.A.L.G. notes and the
Additions. Part 2 had 45 new problems,
often attributed to Mydorge and/or Henrion, but they had no connection with
this and the authorship of these problems is unknown, though Voignier suggests
the printer, Osmont. Part 3 is an
independent treatise on fireworks which Hall attributes to Hanzelet. This edition was reissued in Rouen by
various publishers in 1628, 1629, 1630, 1634, 1638 and in Lyon in 1642-1643,
1653, 1656, 1658, 1669, 1680.
In
1630, Boutonné and Robinot (their printing is indistinguishable and volumes
often have parts from both of them, indeed the Privilege is issued to them
jointly) issued an extended version in four parts, titled Examen du Livre des
Recreations Mathematiques, stated to be by Mydorge. Part 1 is van Etten's 91 problems, with parts V & VI of prob.
91 omitted, with many problems being followed by an Examen signed D.A.L.G. These are by Mydorge and are a revision of
his material in the 1626 Moreau & Loyson.
Part 2 has its own TP and had 45 new problems, taken from the 1628 Rouen
ed. Part 3 has its own TP, but doesn't
state the publisher, and is the independent treatise on fireworks, also taken
from the 1628 Rouen ed. Part 4 again
has its own TP and is Nottes [sic] sur les Recreation Mathematiques by
D.H.P.E.M. and are additions to 27 of van Etten's problems, taken or extended
from the Notes in the 1627 4th ed. The
book is also described as 3 parts with the Nottes, but Parts 2 and 3 are
consecutively paged, leading to some descriptions of the book as being in 3
parts. The parts were probably issued
separately as they sometimes are catalogued separately and different copies of
the whole work often have a mixture of the Boutonné and Robinot printings. This most extended form was reissued by
various publishers in Paris: 1634(??), 1638, 1639, and in Rouen: 1639 (two publishers), 1643 (two publishers ??),
1648?, 1649?
In
1659, Cardin Besonge, Paris, issued Les
Récréations mathématiques, avec l'examen de ses problèmes en arithmétique,
géométrie, .... Premièrement reveu par
D. Henrion, depuis par M. Mydorge, et tout nouvellement corrigé et
augmenté, 5e et dernière édition.
The Nottes are incorporated in the text (or perhaps omitted??). The entire text is consecutively
page-numbered. Reissued in Paris: 1660,
1661 and in Rouen as the 6th ed.: 1660?, 1663?, 1664, 1669 (seven publishers!).
The
1630 Paris ed. and the 1626 ed. have the same problem numbers for the first 91
problems, as do almost all French editions.
I will cite the problem number and the pages of the 1626 ed. I will add reference to the 1630 Paris ed.,
when the latter has additional information.
Only one of the additional problems in part 2 (prob. 7) is of any
interest to us, but several of Henrion's Nottes give corrections, extensions,
additional references and even additional problems. I didn't find any of the D.A.L.G. notes of any interest.
English editions.
Mathematicall
Recreations. Or a Collection of sundrie
[1653 has: many] Problemes, extracted out of the Ancient and Moderne
Philosophers, as secrets in nature, and experiments in Arithmeticke, Geometrie,
Cosmographie, Horologographie, Astronomie, Navigation, Musicke, Opticks,
Architecture, Staticke, Machanicks, Chimestrie, Waterworkes, Fireworks,
&c. Not vulgarly made manifest
untill this time: Fit for Schollers, Students, and Gentlemen, that desire to know
the Philosophicall cause of many admirable Conclusions. Usefull for others, to acuate and stirre
them up to the search of further knowledge; and serviceable to all for many
excellent things, both for pleasure and Recreation. Most of which were written first in Greeke and Latine, lately
compiled in French, by Henry Van Etten Gent.
And now delivered in the English tongue, with the Examinations,
Corrections, and Augmentations. Printed
by T. Cotes for Richard Hawkins, London, 1633.
328pp. (This ed. is 'excessively
rare'.)
Reissued
by William Leake, London: 2nd ed., 1653; (1667(??)); 1674. 344pp.
[Hall, OCB, says the 2nd ed. is similar to the 1633 edition, but with an
extra 16pp description (of 1636) of some dials by Oughtred (which led to the
book or the translation often being attributed to Oughtred). Hall also states that the English editions
are based on the Rouen ed. of 1628.
Sadly some interesting problems were omitted in the English, leading to
confusion in plate numbers. However, I
have just noticed that Prob. 63 is about two pages longer than the
corresponding Prob. 70 of the French editions.] The 1633, 1653 and 1674 editions are identical except for the
additional English material in the later editions. I will add citations to the English editions in parentheses. I now have an imperfect copy of the 1674 ed,
covers missing and lacking the Frontispiece and pp. 273-282 and later
material. Heyl 311 is a 1753 ed., which
must be an error for 1653.
W.
Leybourn, qv, takes several sections directly from the English editions.
Latin editions.
In
1628(??), Caspar (or Gaspar) Ens made a Latin translation but added some other
material, e.g. 49 problems from Alcuin.
I have only studied the 1636 ed. carefully.
Thaumaturgus
Mathematicus, Id est, Admirabilium Effectorum e Mathematicarum Disciplinarum
Fontibus Profluentium Sylloge. Casparo
Ens L. Collectore & Interprete.
1628. [Taken from 1636 TP. MUS #30 says this is only a translation of
van Etten. There is some doubt whether
the 1628 edition exists!]
Reissued
in 1636 and 1651. It has 89 of van
Etten's problems (omitting 38 & 46) and adding 25 new problems, with some
numbering errors so the last is numbered 113.
This is followed by 55 problems of Alcuin, using the Bede version of 56
problems, but omitting 18.
Thaumaturgus
mathematicus Gasparo Ens lectore collectore, & interprete, Nunc denuò
Correctus, & Auctus. Apollonius
Zambonus, Venice, 1706. 113 problems +
49 from Alcuin (check??). There are
some differences between this and the 1636 ed.
[MUS
#30 gives Köln, 1651, and further editions.]
Dutch editions.
In
1641, Wynant van Westen translated van Etten into Dutch. The title is: Het eerste [- derde] deel van
de Mathematische vermaecklyckheden. Te
samen ghevoeght van verscheyden ghenuchlijcke ende boertige werckstucken, soo
uyt arithmetica, geometria, astronomie, geographia, cosmographia, musica,
physica, optica, catoptrica, architectonica, sciotetica, als uyt andere
ongehoorde mysterien meer.
Ghetranslateert uyt het fransch in nederduytsche tale: ende verrijckt,
vermeerdert, ende verbetert met verscheyden observatien ende annotatien,
dienende tot onderrichtinge van eenige duystere questien, ende mis-slaghen in
den franschen druck. Door Wynant van
Westen .... op nieus oversien verbetert.
Jacob van Biesen, Arnhem, 1641.
3 parts with separate title pages and pagination, perhaps in 3 vols, but
later in 1 vol.
This
was reissued: Van Biesen, Arnhem, 1641,
??, 1644, 1662, 1671-72; Lootsman and Jacobsz, Amsterdam, 1673. I haven't examined any of these.
Euler. Algebra. 1770.
Leonard
Euler (1707-1783). Vollständig
Anleitung zur Algebra. Royal Academy of
Sciences at Petersburg, 1770. [A
Russian translation appeared in 1768.]
Translated into French by John III Bernoulli, with additions by
Bernoulli and La Grange (pp. 463-593 here), 1774. Translated from French into English as Element of Algebra, with
further notes, by Rev. John Hewlett, with a Memoir of Euler by Francis Horner
[Horner actually did the translation; Hewlett edited it.], (1797), 5th ed., Longman, Orme, and Co., London,
1840. Reprinted, with Introduction by
C. Truesdell (1972), omitting 4 pp of Horner, Springer, NY, 1984 [hidden on
back of title page]. I will cite part,
section, chapter, article and the pages from the Springer ed. (Part II has no sections.) Unfortunately these numbers seem to have
little connection with other editions.
[Though most of the recreational material in Euler is much older than
Euler, I have included it as a representative 18C text.] [Halwas 175-176 are some US editions -- the
1818 edition was the first example of a translated algebra in the US.]
Family Friend. The Family Friend. This was a magazine founded by Robert Kemp
Philp in 1849. The dating is awkward --
vol. 1 is dated 1850 on the cover, but the Preface is dated 15 Nov 1849 and
refers to the success of the past year, when it appeared monthly. It also says the magazine will henceforth
appear twice a month with two volumes per year, due on the first of June and
December. The Gardening section of vol.
1 goes from Jan to Dec. The Preface of
Vol. 2 is dated 10 Jun 1850 and its gardening section covers Jan - Jun. The Preface of Vol. 3 is dated 15 Dec 1850
and its Gardening section goes Jul - Dec.
BMC shows Philp left in 1852 and the magazine continued with two volumes
per year through a fifth series, ending in 1867, then restarted with one volume
a year from 1870 until 1921. I have
vols. 1 - 3 & the second half of 1858, which is dated 1858-9, but appears
to be Jul-Dec. None of the text is
signed. At the back of volumes are
included answers to correspondents. The
puzzles are often identical to those in The Magician's Own Book or The
Illustrated Boy's Own Treasury, etc., but are considerably earlier.
FHM. Graham Flegg, Cynthia Hay & Barbara Moss -- see under Chuquet.
Fibonacci. Leonardo Pisano, called
Fibonacci (c1170->1240). Liber
Abbaci. (1202); 2nd ed., 1228. In: Scritti di Leonardo Pisano; vol. I, ed.
and pub. by B. Boncompagni; Tipografia delle Scienze Matematiche e
Fisiche, Rome, 1857. The title pages
give 'abbaci', but Boncompagni's text begins 'Incipit liber Abaci ... Anno
MCCII.', while the c1275 MS starts 'Incipit abbacus'. Both forms are used, sometimes even in the same article -- e.g.
Loria's biographical article, see in Section 1.
Richard
E. Grimm was working on a critical edition of this and he kindly gave me some
details. There are 15 known MSS, all of
the 1228 2nd ed. Six of these consist
of 1½ to 3 chapters only; five of the others lack Chapter 10 and the second
half of Chapter 9; one lacks Chapter 10 and one lacks much of Chapter 15,
leaving two essentially complete texts.
The last four MSS mentioned are the most important: Siena L.IV.20, c1275, lacking much of Chap.
15, "the oldest and best";
Siena L.IV.21, 1463 [Grimm said c1465 -- there are dates up through 1464
in interest calculations, but the Incipit specifically says 1463], which
includes much other material from later writers, so it is at least double the
size of L.IV.20; Vatican Palatino
#1343, end of 13C, lacking Chap. 10;
Florence Bibl. Naz. Conventi Soppressi C. 1. 2616, early 14C, "handsome
but frequently badly faded" so "that a later hand found it necessary
to rewrite what he saw there."
When I examined it in Sep 1994, the black ink was indeed sometimes badly
faded but the numbers were in a clear red -- perhaps these are what was
rewritten?? L.IV.20 has the beginning
sentence ending "et correctus ab eodem a MCCXXVIII", but Grimm says
all the others are also of the 1228 ed even if they do not carry this addition
or the extra initial dedication. Sadly,
I heard in Aug 1998 that Grimm had Alzheimer's disease and was in a nursing
home. Inquiry has revealed no trace of
the photocopies of all the Liber Abbaci MSS which he said he had obtained and
in summer 2000 I heard he had died.
Boncompagni
used only one MS, then denoted Codex Magliabechiana, C. I, 2616, Badia
Fiorentina, no. 73, now Conventi Soppressi, C. I. 2616, the badly faded fourth
MS described above.
In
Sep 1994 and Mar 1998, I examined Siena L.IV.20 and 21 and Conv. Soppr.
C.1.2616. I have slides of the Incipit
& Fibonacci numbers from all of these and some other material.
The
dates of 1202 and 1228 are based on the Pisan calendar.
Fibonacci-Sigler.
Liber Abaci. Translated by
Laurence E. Sigler as: Fibonacci's
Liber Abaci A Translation into Modern
English of Leonardo Pisano's Book of Calculation. Springer, New York, 2002.
I have added page references to this, denoted S, after the Boncompagni
pages, e.g. pp. 397-398 (S:
543-544). I have given Sigler's English
wherever I previously had just quoted the Latin.
Fibonacci. Flos
and Epistola.
Leonardo
Pisano, called Fibonacci. MS of c1225
which begins "Incipit flos Leonardi bigolli pisani ...", Biblioteca
Ambrosiana, Milan, E. 75. In: Scritti
di Leonardo Pisano, vol. II, ed. and pub. by B. Boncompagni, Rome, 1862, pp.
227-252.
Part
of the MS has a separate heading: "Epistola suprascripsit Leonardi ad
Magistrum Theodorum phylosophum domini Imperatoris" and is sometimes
considered a separate work. It occupies
pp. 247-252 of the printed version. For
an English description, see: A. F.
Horadam; Fibonacci's mathematical letter to Master Theodorus; Fibonacci
Quarterly 29 (1991) 103-107.
Italian
translation (including the Epistola) and commentary: E. Picutti; Il 'Flos' di
Leonardo Pisano; Physis 25 (1983) 293-387.
Fireside Amusements. 1850.
Fireside
Amusements. Chambers's Library for
Young People. William and Robert
Chambers, Edinburgh, 1850, 188pp. The
BMC has this under Fireside Amusements and refers to Chambers for the Library,
which was 19 vols, 1848-1851. Pp. 187+
are missing in the copy I have seen, but it seems that just one page of
solutions is missing -- the NUC gives 188pp.
The NUC lists a 1870 reprint.
[The
BMC lists an 1880 ed with 159pp, part of Chambers's Juvenile Library, NYS.]
Fireside
Amusements A Book of Indoor Games. W. & R. Chambers, London and Edinburgh,
nd, 128pp. The BMC lists this as
1890[1889]. Though laid out entirely
differently, almost all the material is taken from the 1850 ed. I will cite both editions.
Much
of the material of interest is taken from Child: Girl's Own Book.
Folkerts. Aufgabensammlungen. 13-15C.
Menso
Folkerts. Mathematische
Aufgabensammlungen aus dem ausgehenden Mittelalter. Sudhoffs Archiv 55 (1971) 58-75.
He examines 33 anonymous Latin manuscript problem collections from 13-15
C in Oxford, London, Berlin, Munich, Vienna and Erfurt and catalogues the
problems therein. Of these, only Munich
14684 is published (cf below). He notes
that many more such sources exist. His
catalogue covers 14 of my topics. I
will not try to cite the individual MSS, since many of the topics occur in over
a dozen of them. I will simply say he
has n
sources, though some of the sources have several examples.
Folkerts, Menso. See:
Alcuin.
della Francesca. Trattato.
c1480.
Piero
della Francesca (1412-1492). Trattato d'Abaco. Italian MS in Codex Ashburnhamiano 359*
[291*] - 280 in the Biblioteca Mediceo-Laurenziana, Florence. c1480 [according to Van Egmond's Catalog 84,
based on watermarks in the paper which date from 1470 to 1500, but Davis,
below, p. 16, says c1450]. Transcribed
and annotated by Gino Arrighi, Testimonianze di Storia della Scienze 6, Domus
Galilæana, Pisa, 1970. Arrighi
uses c. (for carta) instead of
f. (for folio), but I will
use f.
for consistency with other usage, followed by the pages in Arrighi in (
). Arrighi reproduces many of the
diagrams, but he doesn't say anything about whether he has included all of
them. This MS appears to be that which
was in the possession of Piero's descendents until 1835 when it was reported as
having disappeared. Guglielmo Libri,
the noted historian of mathematics, who was also a shady bookdealer,
transcribed part of this MS in vol. 3 of his Histoire de la Mathématique en
Italie in 1840 as an anonymous work, then sold it to Lord Ashburnham in 1847
(recorded in his collection in 1881) whose collection was bought for the
Laurentian Library in 1884. There are
three different catalogue numbers - I use the format used in Van Egmond's
Catalog. The MS had passed out of
common knowledge until it was rediscovered in the Laurentian Library in 1917 by
Girolamo Mancini who recognised the handwriting as Piero's.
This
work and Piero's Libellus de Quinque Corporibus Regularibus are the subject of
a long standing plagiarism argument.
Giorgio Vasari [Le Vite de' più eccellenti pittori, scultori e
architetti; 1550; The Essential Vasari,
ed. by Betty Burroughs from the 1850 translation of Mrs. Jonathan Foster, Unwin
Books, London, 1962] states: "...
Piero della Francesca, who was a master of perspective and mathematics but who
first went blind and then died before his books were known to the public. Fra Luca di Borgo, who should have cherished
the memory of his master and teacher, Piero, did his best, on the contrary, to
obliterate his name, taking to himself all the honour by publishing as his own
work that of that good old man.
... Maestro Luca di Borgo caused
the works of his master, Piero della Francesca, to be printed as his own after
Piero died." The mathematical
works of Piero were unknown until they were rediscovered in 1850/1880 and
1917. Examination shows that Pacioli
certainly used 105 problems, many unusual, from Piero in the Summa. But he does praise Piero in the Epistola (f.
2r) of the Summa, as "the monarch of painting of our times". It has been suggested that Pacioli had a
large hand in the writing of Piero's works and hence was just reusing his own
material and he frequently expands on it.
However, there is no evidence that Pacioli was ever a student of Piero. Entire books have been written on the question,
so I will not try to say any more.
See: Margaret Daly Davis; Piero
della Francesca's Mathematical Treatises
The "Trattato d'abaco" and "Libellus de quinque
corporibus regularibus"; Longo Editore, Ravenna, 1977, for detailed comparisons and the work of R.
E. Taylor in Section 1: Pacioli. Davis
identifies 139 problems in the Libellus, of which 85
(= 61%) are taken from the
Trattato. Davis notes that Pacioli's
Summa, Part II, ff. 68v ‑ 73v, prob. 1-56, are essentially
identical to della Francesca's Trattato, ff. 105r ‑ 120r. See also section 6.AT.3 where the Libellus
and the Pacioli & da Vinci: De Divina Proportione are discussed.
The
work is discussed and 42 problems are given in English in: S. A. Jayawardene; The 'Trattato d'Abaco' of Piero della Francesca; IN: Cecil H. Clough, ed.; Cultural Aspects of the Italian
Renaissance Essays in Honour of Paul
Oskar Kristeller; Manchester Univ.
Press, Manchester, nd [1976?]; pp.
229-243. I will note 'English in
Jayawardene.' when relevant.
Frikell, Wiljalba (1818 (or
1816) - 1903). (The given name Gustave
sometimes occurs -- I thought Gustave might be a son of Wiljalba, but the son
was named Adalbert ( -1889) and his
name was pirated by a clumsy imposter in England.)
See
the discussion at: Book of 500
Puzzles, Boy's Own Conjuring Book, Hanky Panky, Magician's Own Book, The
Secret Out. Frikell was a noted
conjuror of the time and his name has been associated with the UK versions of
these books, but there is no evidence he had anything to do with them. The Art of Amusing, by Frank Bellew, Hotten,
London, 1866?, op. cit. in 5.E, has a note on the back of the TP saying The
Secret Out is a companion volume, just issued, by Hermann Frikell. C&B, under Williams, Henry Llewellyn ("W. Frikell") lists:
Hanky Panky; Magician's Own
Book, London & New York; (Magic No
Mystery); The Secret Out and says to also see Cremer.
Gamow & Stern. 1958.
George
Gamow & Marvin Stern. Puzzle‑Math. Macmillan, London, 1958.
Gardner. Martin Gardner (1914- ).
Many references are to both his SA column, cited by (month & year),
e.g. SA (Mar 1982), and to the appearance of the column as a chapter in one of
his books, abbreviated as shown below.
In general, I will only give the chapter reference as the various
editions and translations are differently paginated. Answers, comments and extensions appeared in succeeding issues of
SA, usually in Gardner's column, but sometimes in the Letters. All this material is collected in the book
chapter, sometimes by rewriting of the article, sometimes as notes or an
Addendum at the end of the chapter.
Since many years usually passed before the book version, the Addenda
often contain material that never appeared in SA, as well as references to work
done as a result of the SA article. I
have not tried to enter all of Gardner's references here, so anyone interested
in a topic that Gardner has considered should consult the book version of
Gardner's column. Currently some of the
earlier books are being reissued in new editions, with further extensions and
updating. See also the next entry.
For
years from at least 1950, SA appeared in two volumes per year, each of six
issues. In year 1950 + n,
vol. 182 + 2n covers Jan-Jun
and vol. 183 + 2n covers
Jul-Dec.
1st Book The Scientific American Book
of Mathematical Puzzles and Diversions.
Simon & Schuster, 1959.
UK version: Mathematical Puzzles and
Diversions from Scientific American.
Bell, London, 1961; Penguin (without the words 'from Scientific American'),
1965.
2nd Book The Second
Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster, 1961.
UK version: More Mathematical Puzzles and
Diversions from Scientific American.
Bell, London, 1963. Penguin
(without the words 'from Scientific American'), 1966. (The UK versions omit Chapter 20: "The Mysterious Dr.
Matrix". The dust wrapper of the
HB has a sentence referring to this chapter which has been blacked out. ??)
New MD
Martin Gardner's New Mathematical Diversions from Scientific American. Simon & Schuster, 1966.
Unexpected The
Unexpected Hanging and Other Mathematical Diversions. Simon & Schuster, 1969.
UK version: Further Mathematical
Diversions. Allen & Unwin, London,
1970; Penguin, 1977.
6th Book Martin
Gardner's Sixth Book of Mathematical Games from Scientific American. Freeman, 1971.
Carnival Mathematical
Carnival. Knopf, NY, 1975; Penguin,
1978.
Magic Show Mathematical
Magic Show. Random House, NY, 1978.
Circus Mathematical
Circus. Knopf, NY, 1979.
Wheels Wheels,
Life and Other Mathematical Amusements.
Freeman, 1983.
Knotted Knotted
Doughnuts and Other Mathematical Entertainments. Freeman, 1986.
Time Travel Time
Travel and Other Mathematical Bewilderments.
Freeman, 1988.
Penrose Tiles Penrose
Tiles to Trapdoor Ciphers. Freeman,
1989.
Fractal Fractal
Music, Hypercards and More ....
Freeman, 1992.
Last The
Last Recreations Hydras, Eggs, and
Other Mathematical Mystifications.
Copernicus (Springer), NY, 1997.
??NYR.
Magic Numbers The
Magic Numbers of Dr. Matrix.
Prometheus, Buffalo, 1985.
Chaps. 1-18 previously appeared as: The Incredible Dr. Matrix;
Scribner's, NY, 1976. Chaps. 1‑7
& 9 previously appeared as: The Numerology of Dr. Matrix; Simon &
Schuster, NY, 1967. In contrast to his
other books above, the answers and comments occur at the end of this book
instead of following the original articles.
Workout A
Gardner's Workout Training the Mind and
Entertaining the Spirit. A. K.
Peters, Natick, Massachusetts, 2001.
This comprises 41 chapters of articles written after his retirement from
SA.
Gardner. MM&M.
1956.
Martin
Gardner. Mathematics, Magic and
Mystery. Dover, NY, 1956.
General Trattato. 1556.
Nicolo
Tartaglia (c1499-1557). (La Prima Parte
del) General Trattato di Numeri et Misure.
Curtio Troiano, Venice, 1556.
(Modern Italian spells his given name as Niccolò, but it appears as
Nicolo on the title page.) Six parts
actually appeared in 1556-1560. All
references are to Part 1. Unless
otherwise specified, reference is to Book 16 (of Part 1), but I also have
references to Books 12 and 17. CAUTION
-- the running head in Book 17 says Libro Decimosesto for several pages before
changing to Libro Decimosettimo. Since
it is hard to find the beginnings of books, this can cause confusion. See Rara 275-279; Van Egmond's Catalog
345-346.
In
1578, Guillaume Gosselin produced an annotated translation of parts 1 & 2
into French as: L'Arithmetique de Nicolas Tartaglia -- cf Van Egmond's Catalog
347.
Ghaligai. Practica D'Arithmetica. 1521.
Francesco
Ghaligai. Practica D'Arithmetica di
Francesco Ghaligai Fiorentino.
Nuovamente Rivista, & con somma Diligenza Ristampata. I Giunti, Florence, 1552. Smith, Rara, says that this is identical to
the first (Latin?) edition by Bernardo Zucchetta, Florence, 1521, except that
edition was titled Summa De Arithmetica, so I will date the entries as
1521. See Rara 132; Van Egmond's
Catalog 316-317.
Gherardi. Libro di ragioni and Liber habaci. 1328 & c1310.
Paolo
Gherardi. Two Italian MSS in Codici
Magliabechiani Classe XI, no. 87 & 88 in Bib. Naz. di Firenze. Van Egmond's Catalog 115-116. The first is dated 1327 (but see
below). The second is undated, but
clearly of a similar date which I originally denoted 1327? - see below. Transcribed by Gino Arrighi; Collana di
Storia della Scienza e della Tecnica, No. 2; Maria Pacini Fazzi, Lucca, 1987. See also:
Warren Van Egmond; The earliest vernacular treatment of algebra: the Libro
di ragioni of Paolo Gerardi (1328); Physis 20 (1978) 155-189. Van Egmond notes that the date of 30 Jan
1327 is in our year 1328 and uses this in his Catalog. He doubts whether Liber habaci is actually
by Gherardi and his Catalog assigns no author to it, so I will put Gherardi? as
author. He dates it to c1310. His paper is concerned with the quadratic
and cubic equations and hence of little interest to us.
Good, Arthur. See:
Tom Tit.
Gori. Libro di arimetricha.
1571.
Dionigi
Gori. Libro di arimetricha. 1571.
Italian MS in Biblioteca Comunale di Siena, L. IV. 23. ??NYS.
Extensively quoted and discussed in: R. Franci & L. Toti Rigatelli;
Introduzione all'Aritmetica Mercantile del Medioevo e del Rinascimento;
Istituto di Matematica dell'Università di Siena, nd [1980?]. (Later published by Quattroventi, Urbino,
1981.) (I will quote Gori's folios and
also give the pages of this Introduzione.)
Van Egmond's Catalog 191-192.
Graves. The Graves Collection of early mathematical
books at University College London (UCL).
Guy, Richard Kenneth (1916- ).
See: Winning Ways.
G4Gn Gathering for Gardner n, held in
Atlanta. 1: Jan 1993; 2: Jan 1996; 3: Jan 1998; 4: Feb
2000; 5: Apr 2002.
G&P. Games & Puzzles. The first version ran from 1972 through
1981. The second series started in Apr
1994 and finished with No. 16 in Jul 1995.
G&PJ. Games and Puzzles Journal. Successor to Chessics. Ran through 12 issues, Sep 1987 -- Dec 1989,
then restarted intermittently in May 1996.
Haldeman-Julius. 1937.
E.
Haldeman-Julius. Problems, Puzzles and
Brain-teasers. Haldeman-Julius
Publications, Girard, Kansas, 1937.
Facsimile (I believe) presented by Bob Koeppel at IPP13, 1993.
Hall. BCB. 1957.
Trevor
H. Hall. A Bibliography of Books on
Conjuring in English from 1580 to 1850.
(Carl Waring Jones, Minneapolis, 1957); Palmyra Press, Lepton,
W. Yorks., 1957. 323 entries. I will cite item numbers. A Supplement is in Hall, OCB. See Heyl for a list of items not in BCB.
Hall. OCB. 1972.
Trevor
H. Hall. Old Conjuring Books. Duckworth, London, 1972. This covers books in English up through 1850
and it includes a Supplement to his BCB and should be checked for further
information on items in BCB. This
contains 39 new items and additional notes to 36 previous items. New items are given interpolated item
numbers, e.g. 24.5. OCB also includes a
slightly revised version of his booklet on van Etten, see Section 1 below.
Halwas. Robin Halwas, Ltd. List XV. American
Mathematical Textbooks 1760-1850. Catalogue of 511 items being sold as a
collection. London, 1997, 144pp. Quite a number of English works and a few
French works had US editions which are detailed in this.
Hanky Panky. 1872.
Hanky
Panky A Book of Easy and Difficult
Conjuring Tricks Edited by W. H.
Cremer, Jun. (John Camden Hotten,
London, 1872 [BMC & Toole Stott 193, listed under Cremer, while C&B,
under Cremer, give London, 1872]; Hotten was succeeded by Chatto & Windus
c1873 and they produced several editions [NUC has 1872, Toole Stott 1017 &
Shortz have 1875].) My copy says: A new
edition with 250 practical illustrations.
John Grant, Edinburgh, nd [Toole Stott 1016 gives 1874; NUC gives
1875? Christopher 235, under Cremer, is
c1890. NUC says it is also attributed
to Henry Llewellyn Williams, but their entry under Williams says 'supposed
author'. C&B also list it under
Williams. (This has been attributed to
Frikell, but Toole Stott doubts that Frikell had anything to do with this. I may put this under Cremer.)
HB.XI.22. 1488.
Stuttgart
Landesbibliothek German MS HB.XI.22, 1488.
Brief description by E. Rath;
Über einen deutschen Algorismus aus dem Jahr 1488; Bibl. Math. (3) 14 (1913‑14)
244‑248.
Heath, Sir Thomas L. See:
Diophantos; HGM.
Heyl. 1963. Edgar
Heyl (1911-1993). A Contribution to
Conjuring Bibliography. English
Language 1580 to 1850. Edgar Heyl conjuring books, Baltimore,
1963. Facsimile edition of 100 copies
by Maurizio Martino Fine Books, PO Box 373, Mansfield Center, Connecticut,
06250, nd [1998?]. 360 entries +
appendix of 14 more, almost all not in Hall, BCB.
HGM. 1921. Sir Thomas L.
Heath. A History of Greek Mathematics,
2 vols. (OUP, 1921); corrected reprint,
Dover, 1981.
HM. Historia Mathematica.
Hoernle, A. F. Rudolf. See:
Bakhshali MS.
Hoffmann. 1893. Professor
Louis Hoffmann [pseudonym of Angelo John Lewis (1839‑1919)]. Puzzles Old and New. Warne, London, 1893. Reprinted with Foreword by
L. E. Hordern; Martin Breese, London, 1988.
In
1984, Hordern published a limited edition (15 copies) of "The Hordern
Collection of Hoffmann Puzzles 1850‑1920",
which gives colour photos of examples from his collection and the appropriate
text. I often cite these pictures as
they often differ from those in the following item, with the heading Hordern
Collection. Generally, the next item
gives more specific dating and/or older examples.
In
1993, Hordern produced a corrected edition of all of Hoffmann as: Hoffmann's
Puzzles Old & New; published by himself.
This has colour photos of all puzzles for which known examples
exist. I will cite this as
Hoffmann-Hordern. The 1893 edition
gives solutions for each chapter in a following chapter, but both of Hordern's
illustrated versions give each solution immediately after the problem, with
colour picture nearby. A small section
on Elementary Properties of Numbers is omitted from the 1993 edition.
See
also: Tom Tit.
Honeyman Collection.
The
Honeyman Collection of Scientific Books and Manuscripts. Sold by Sotheby's [Sotheby Parke Bernet],
1978-1981. Seven volumes -- details
given in Section 3.B.
Hordern, L. Edward
(1941-2000). See under Hoffmann and in
5.A.
HPL. The Harry Price Library, Senate House,
University of London OR its catalogues.
Harry
Price (1881-1948). Short-Title
Catalogue and Supplementary Catalogue of Works on Psychical Research,
Spiritualism, Magic, Psychology, Legerdemain and Other Methods of Deception,
Charlatanism, Witchcraft, and technical Works for the Scientific Investigation
of Alleged Abnormal Phenomena from Circa 1450 A.D. to 1935 A.D. Compiled by Harry Price. (The first part was originally ... to 1929 A. D.; Proc. National Lab. of
Psychical Research 1:2 (1929); National Laboratory of Psychical Research,
London, 1929. The second part was
originally: Short-Title Catalogue of the Research Library, for 1472 A.D. to the
Present Day; Bull. Univ. of London Council for Psychical Investigation 1
(1935); Univ. of London Council for Psychical Investigation, 1935.) New Introduction by R. W. Rieber and Andy
Whitehead With an appendix entitled
"The St. Louis Magnet" (which originally appeared in 1845) by T. J. McNair
and J. F. Slafter. Da Capo Press
(Plenum), NY, 1982. NOTE: The works
listed here are now in the Harry Price Library, though the editors have added
36 items in their Introduction.
Hummerston. Fun, Mirth & Mystery. 1924.
R.
A. Hummerston. The Book of Fun, Mirth
& Mystery A feast of delightful
entertainment, including games, tricks, puzzles and solutions, "how to
makes," and various other means of amusement. Pearson, London, 1924.
Hunt. 1631 & 1651.
Nich.
Hunt. Newe Recreations or The Mindes
release and solacing. Aug. Math. for
Luke Fawne, 1631.
Nich.
Hunt. New Recreations or A Rare and
Exquisite Invention. J. M. for Luke
Fawn, London, 1651. This edition
contains a few more pages and several problems of interest and is differently
paginated. I will give both page
numbers for problems in both editions.
Bill Kalush has sent both texts on a CD.
Hutton. A Course of Mathematics. 1798?
Charles
Hutton (1737-1823). A Course of
Mathematics. Composed for the Use of
the Royal Military Academy. (In 2 vols,
plus a third, 1798-1811.) A New
Edition, entirely Remodelled. By William
Ramsay, B. A., Trinity College, Cambridge.
T. T. & J. Tegg, London, and Richard Griffin & Co.,
Glasgow, 1833 (in one volume). 8 +
822 pp.
Hutton-Rutherford. A Course of Mathematics. 1841?
Charles
Hutton. A Course of Mathematics, Composed for the Use of The Royal Military
Academy. By Charles Hutton, LL.D.,
F.R.S., Late Professor of Mathematics in that Institution. A new and carefully corrected Edition,
Entirely Re-modelled, and Adapted to the Course of Instruction Now Pursued in
the Royal Military Academy. By William
Rutherford, F.R.A.S. Royal Military
Academy. William Tegg, London, 1857
[Preface dated Nov 1840, so probably identical or nearly identical to the 1841
ed]. 8 + 895 pp.
[Two
volume versions: 1798/1801; 3rd, 1800/1801; 4th, 1803/1804; 5th, [1810, NUC gives 1806- and 1807]; 6th, [1810-1811 -- NUC].
Three volume versions -- apparently the early forms were just the
earlier 2 volumes with an additional third vol: 6th, 1811; 1813; 7th, 1819-1820; 8th, 1824;
9th, 1827/1828. 10th ed by
Olinthus Gregory, in 3 vols., 1827-1831.
New ed by William Ramsay in one vol., 1833; 1838. 11th ed by Gregory in 2 vols,
1836-1837. 12th ed, revised by Thomas
Stephens Davies, 2 vols., 1841-1843.
Ed by William Rutherford in one vol, 1841; 1843; 1846; 1849; 1851; 1853;
1857; 1860.
This
is a pretty straightforward text, but it well illustrates the situation in
early 19C England, outside Oxford and Cambridge. Almost all the material of interest is in the first two sections:
Arithmetic and Algebra and is identical in these two editions. I imagine most of these problems appeared in
the first edition, so I will date this as 1798?, citing pages as 1833 and
1857. However the 1857 has an
additional three pages on Practical Questions in Arithmetic which has 44
problems, some of which are recreational, and another new problem. Assuming the 1857 is essentially the same as
the 1841, I will cite this as 1841?.]
See
also: Ozanam‑Hutton.
H&S. 1927. Vera
Sanford. The History and Significance of
Certain Standard Problems in Algebra.
(Teachers College, Columbia University, NY, Contributions to Education,
No. 251, 1927) = AMS Press, NY, 1972.
Illustrated Boy's Own
Treasury. c1847.
The
Illustrated Boy's Own Treasury of I. - Science, II. - Drawing, III. - Painting,
IV. - Constructive Wonders, V. - Rural Affairs, VI. - Wild and Domesticated
Animals, Outdoor Sports & Indoor Pastimes forming a Complete Repository of
Home Amusements & Healthful Recreations embellished with five hundred descriptive
engravings. (John & Robert Maxwell,
London, c1847 [Toole Stott 407]). Ward
and Lock, 1860 [Toole Stott 1091]. (3rd
ed, for the Proprietors, 1865? [Toole Stott 408].) [Toole Stott's descriptions make it seem that these editions are
identical.] See also: Boy's Own Book, Boy's Own Conjuring Book.
[Many of the problems are the same as in the other two books, but the
illustrations here occasionally omit some labels, so these must be errors in
copying from some earlier source. If
the c1847 date is correct, then this considerably changes the chronology of
these problems, with this book being the major known intermediate between Boy's
Own Book and Magician's Own Book. I
will hold off making these changes until I see the c1847 ed -- this may take
some time as the BM copy was lost in the war and the other two copies cited are
in the US. Hall, BCB 187 is: The
Illustrated Boys' Own Treasury of Indoor Pastimes; Robson, London, c1845. This may be related to this book.]
Indoor & Outdoor. c1859.
Indoor
and Outdoor Games for Boys and Girls: Comprising Parlour Pastimes, Charades,
Riddles, Fireside Games, Chess, Draughts, &c, &c. With a Great Variety of Athletic Sports,
Parlour Magic, Exercises for Ingenuity, and Much That is Curious, Entertaining,
and Instructive. James Blackwood,
London, nd [c1859]. This is a
combination of two earlier books, comprising two separately paginated
parts. The earlier books are Parlour
Pastime (1857 -- qv) and Games for All Seasons [Toole Stott 311 & BMC give
1858]. There is a later version of the
first part -- Parlour Pastimes, qv, which the BMC dates as 1868. Also there is another version of the
combined ed "with additions by Oliver Optic", as Sports and Pastimes
for Indoors and Out, G. W. Cottrell, Boston, 1863 [Toole Stott 1186, which
identifies Optic as William Taylor Adams].
Both BMC and NUC say Indoor & Outdoor is by George Frederick
Pardon. Hence the problems in the first
part will be cited as: Parlour Pastime, 1857
= Indoor & Outdoor, c1859, Part 1
= Parlour Pastimes, 1868. Many
of the problems are identical to Book of 500 Puzzles.
IPPn n-th International Puzzle Party. 10 = London, 1989; 13 = Amsterdam, 1993; 16 = Luxembourg, 1996;
19 = London, 1999;
20 = Los Angeles, 2000;
22 = Antwerp, 2002.
This are the ones I have attended, but some material has appeared at
other IPPs.
Jackson. Rational Amusement. 1821.
John
Jackson. Rational Amusement for Winter
Evenings; or, A Collection of above 200 Curious and Interesting Puzzles and
Paradoxes relating to Arithmetic, Geometry, Geography, &c. With Their Solutions, and Four Plates. Designed Chiefly for Young Persons. By John Jackson, Private Teacher of the
Mathematics. London: Sold by J. and A.
Arch, Cornhill; and by Barry & Son, High‑Street; and P. Rose; Bristol. 1821.
[Other copies, apparently otherwise identical, say: London: Sold by
Longman, Hurst, Rees, Orme, and Brown; G. and W. B. Whittaker; and Harvey and
Darton. And Barry and Son, High-Street,
Bristol. 1821. [Heyl 185.
Toole Stott 413.] Will Shortz
says this is the first English-language book devoted to non-word puzzles.]
JRM. Journal of Recreational Mathematics.
Kanchusen. Wakoku Chiekurabe. 1727.
Tagaya
Kanchusen [pseud. of Fuwa Senkuro].
Wakoku Chiekurabe [Japanese Wisdom Competition -- in Japanese]. 2 vols, 1727, 12 & 29 pp. PHOTOCOPY from Shigeo Takagi's copy sent by
Naoaki Takashima. Edited into modern
Japanese, with commentary on Kanchusen, by Shigeo Takagi, 1991, 42pp, present
from Takagi. Translated into English by
Hiroko Dean, 1999, 15pp plus annotations on the 42pp. Takagi and Takashima are working on a translation and annotation
into modern Japanese. We intend to
produce an English version from Dean's translation with commentary on the
puzzles. I will cite pages from
Takagi's edition. (Partly reproduced in
Akira Hirayama; Tôzai Sûgaku Monogatari [Mathematical Stories from East and
West]; (1973), 3rd ed., 1981, p. 208, ??NYS, from which it has been reproduced
in the exhibition Horizons Mathématiques at La Villette, Paris, and elsewhere.)
Kaye, George R. See:
Bakhshali MS.
King. Best 100. 1927
Tom
King. The Best 100 Puzzles. W. Foulsham, London, nd [1927, according to
BMC -- my copy says 'Wartime reprint'.]
A selection of these are reproduced in a booklet: Foulsham's Games and
Puzzles Book; W. Foulsham, London, nd [c1930].
I will indicate this by =
Foulsham's, no. & pp.
Knott, Cargill G. See under:
Tom Tit.
Labosne. See under:
Problemes.
Ladies' Diary. See under: T. Leybourn.
Landells. Boy's Own Toy-Maker. 1859.
E[benezer]
Landells. The Boy's Own Toy-Maker: A
Practical Illustrated Guide to the Useful Employment of Leisure Hours. Griffith & Farran, London, 1859(1858);
Shepard, Clark & Brown, Boston, 1859; Griffith & Farran, 3rd ed., 1860;
D. Appleton, NY, 1860; Griffith & Farran, 6th ed., 1863 [Toole Stott 1286‑1290]. [BMC has 1859(1858) and a longer 10th ed,
1881. NUC has the latter four of the
versions given by Toole Stott.] [The
Preface to the Second Edition, reproduced in the 3rd ed., says it appeared just
two months after the first edition.
Toole Stott indicates that all the versions he cites are identical. I have 3rd ed., 1860. Shortz has Appleton, 1860.] The date 1859(1858) indicates that the book
appeared in late 1858 to catch the Christmas trade, but was postdated 1859 to
seem current for the whole of 1859, so I will date this as 1858. The 2nd ed. must be 1859. Comparison shows that the section on
Practical Puzzles is essentially an exact subset of the material in Boy's Own
Conjuring Book.
Leeming. 1946. Joseph
Leeming. Fun with Puzzles. (Lippincott, Philadelphia, 1946); Comet
Books (Pocket Books), NY, 1949.
Lemon. 1890. "Don
Lemon" [= "The Sphinx" =
Eli Lemon Sheldon], selector.
Everybody's Illustrated Book of Puzzles. Saxon & Co., London, 1890 (with 1891 on the back cover) and
1892. The 1892 ed. omits the text on
the back cover and adds some pages of publisher's advertisements, but is
otherwise identical. 794 problems,
about 100 being mathematical, on 125pp.
This looks like a UK reprint of a US book, but the NUC only lists London
editions, so perhaps it is just selected from US publications. Some of the problems are attributed to
Golden Days, Good Housekeeping, St. Nicholas, etc.
I
also have an undated edition which says 'Selected by the Sphinx'. This has 744 problems on 122pp, about 45% of
which come from the other edition. The
NUC dates this as 1895. I will refer to
this edition as: Sphinx. 1895.
Leopold. At Ease!
1943.
Jules
Leopold. At Ease! Ill. by Warren King. Whittlesey House (McGraw‑Hill),
1943. [This appears to be largely drawn
from Yank, The Army Weekly, over the previous few years.]
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Marie
Leske. Illustriertes Spielbuch für Mädchen Unterhaltende und anregende Belustigungen,
Spiele and Beschäftigungen für Körper und Geist, im Zimmer sowie im
Freien. (1864; 19th ed., 1904); 20th ed., Otto Spamer, Leipzig, 1907. There is no indication of any updating in
the Foreword to the 19th ed. which is included here, and it was common to
describe new printings as new editions, so I will date this as 1864? This book is jammed with material of all
sorts, including lots of rebuses, riddles and puzzles. It is a bit like Boy's Own Book. My copy is lacking pp. 159-160 and 211‑212.
Leurechon, Jean
(c1591-1670). See: van Etten.
T. Leybourn.
Thomas
Leybourn, ed. The Mathematical
Questions, proposed in the Ladies' Diary, and Their Original Answers, Together
with some New Solutions, from its commencement in the year 1704 to 1816. 4 vols., J. Mawman, London, and two
co-publishers, 1817. I have only
examined vols. I & II so far. The
problems are proposed each year with solutions in the following year. Leybourn puts the solutions just after the
problem and numbers almost all the problems consecutively, though I don't know
if these numbers are in the Ladies' Diary.
Problems do not have any names and sometimes have pseudonyms or vague
names, e.g. Mr. Deare. I will give the
names of the proposer and solver(s), followed by Ladies' Diary and the two
years involved, then = T. Leybourn, his
volume and pages and his question number.
E.g. Chr. Mason, proposer; Rob.
Fearnside, solver. Ladies' Diary,
1732-33 = T. Leybourn, I: 223, quest. 168.
W. Leybourn. Pleasure with Profit. 1694.
William
Leybourn. Pleasure with Profit:
Consisting of Recreations of Divers Kinds, viz. Numerical, Geometrical,
Mechanical, Statical, Astronomical, Horometrical, Cryptographical, Magnetical,
Automatical, Chymical, and Historical.
Published to Recreate Ingenious Spirits; and to induce them to make
farther scrutiny into these (and the like) Sublime Sciences. And
To divert them from following such Vices, to which Youth (in this age)
are so much Inclin'd. To this work is
also Annext, A Treatise of Algebra, ..., by R. Sault. Richard Baldwin and John Dunton, London, 1694. The text consists of several parts, labelled
Tract. I, Tract. II, ..., which are separately paginated. All material is from Tract. I unless otherwise
specified. Several sections are taken
from the English editions of van Etten.
[Santi 371.]
Li & Du. 1987. Li
Yan & Du Shiran. Chinese
Mathematics: A Concise History. (In
Chinese: Commercial Press, Hong Kong, c1965.)
English translation by John Crossley & Anthony W.‑C. Lun. OUP, 1987.
Libbrecht. 1973. Ulrich
Libbrecht. Chinese Mathematics in the
Thirteenth Century. MIT Press,
Cambridge, Mass., 1973.
Lilavati. 1150. Lîlâvatî
of Bhaskara II, 1150 (see Colebrooke).
Lloyd, E. Keith. See:
BLW.
Loeb Classical Library.
Published
by Harvard Univ. Press, or Putnam's, NY, & Heinemann, London.
Loyd, Sam (1841-1911) (&
Sam Loyd Jr. (1873-1934).
See: Cyclopedia, MPSL,
OPM, SLAHP.
Lucas, Édouard (1842-1891). See:
RM and the following.
Lucas. L'Arithmétique Amusante. 1895.
Édouard
Lucas. L'Arithmétique Amusante. Ed. by H. Delannoy, C.‑A. Laisant
& E. Lemoine. (Gauthier-Villars,
Paris, 1895.) = Blanchard, Paris, 1974.
Lucca 1754. c1330.
Scuola
Lucchese. Libro d'abaco. c1390.
Dal Codice 1754 (sec. XIV) della Biblioteca Statale di Lucca. Edited by Gino Arrighi. Cassa di Risparmio di Lucca, 1973. Arrighi gives folio numbers and I will cite
these and the pages of his edition.
Arrighi has c1390, but Van Egmond's Catalog 163-164 gives c1330.
MA. Mathematical Association (UK).
MAA. Mathematical Association of America.
Magician's Own Book. 1857.
The
Magician's Own Book, or The Whole Art of Conjuring. Being a Complete Hand-Book of Parlor Magic, and Containing over
One Thousand Optical, Chemical, Mechanical, Magnetical, and Magical
Experiments, Amusing Transmutations, Astonishing Sleights and Subtleties,
Celebrated Card Deceptions, Ingenious Tricks with Numbers, Curious and
Entertaining Puzzles, Together with All the Most Noted Tricks of Modern
Performers. The Whole Illustrated with
over 500 Wood Cuts, and Intended as a Source of Amusement for One Thousand and
One Evenings. Dick & Fitzgerald,
NY, ©1857. 12 + 362 pp. + 10 pp.
publisher's ads. My thanks to Jerry
Slocum for providing a copy of this.
[Toole Stott 481 lists this as anonymous and entirely different from the
UK ed. He cites a 1910 letter from
Harris B. Dick who says H. L. Williams may have edited it, but both Dick's
father and John Wyman may also have had a hand in it. Toole Stott 929, 930, 1378, 931 lists Dick & Fitzgerald,
1862, 1866, 1868, 1870, all apparently identical to the 1857. 929-930 are listed under Arnold and he there
cites Cushing's Anonyms as saying the book is by Arnold and Cahill. Christopher 622-625 are all Dick &
Fitzgerald; 622-623 are 1st ed., 624-625 are reprints of about the same time
and my copy seems most likely to be 625.
C&B, under Cremer, say "It is believed that they were all
written by H. L. Williams, a prolific hack writer of the period." Christopher 622 says Harold Adrian Smith
[Dick and Fitzgerald Publishers; Books at Brown 34 (1987) 108-114] has studied
this book and concludes that Williams was the author, assisted by Wyman. Actually Smith simply asserts: "The
book was undoubedly [sic] written by H. L. Williams, a "hack writer"
of the period, assisted by John Wyman in the technical details." He gives no explanation for his assertion,
but it may be based on C&B. NUC
lists this as by George Arnold (1834-1865) and Frank Cahill, under both Arnold
and Cahill. C&B list it under
Cremer, attributed to Arnold & Cahill, but they give a date of 1851, which
must be a transcription error. C&B
also list it under Magician's, from New York, 1857, and under Williams, but as
London, 1857.] See the discussion under
Status of the Project in the Introduction for the sources of the material.
Boy's
Own Conjuring Book, qv, appears to be a UK pirate edition largely drawn from
this. Book of 500 Puzzles copies about
80 pages of this. See the comments under
Book of 500 Puzzles. A fair number of
the problems are identical to or similar to the Boy's Own Book and a woodcut, a
poem and the introduction to a section are taken directly from Boy's Own Book. Otherwise I had thought that this book was
the source for the spate of puzzle books in the following 15 years, but I have
found that some of the identical puzzles appeared in The Family Friend
c1850.
Magician's Own Book (UK
version). 1871.
This
is quite different than the previous book.
The
Magician's Own Book. By the Author of
"The Secret Out," "The Modern Conjuror," &c. Edited by W. H. Cremer, Jun. Containing Ample Instructions for
Recreations in Chemistry, Acoustics, Pneumatics, Legerdemain, Prestidigitation,
Electricity (with and without apparatus).
(In the middle of the page is an illustration of a wizard in white on
red.) Performances with Cups and Balls,
Eggs, Hats, Flowers, Coin, Books, Cards, Keys, Rings, Birds, Boxes, Bottles,
Handkerchiefs, Glasses, Dice, Knives, &c., &c. With 200 Practical Illustrations. John Camden Hotten, nd [1871]. This has a two page list of Very Important
New Books at the beginning on pp. i-ii.
This lists Magician's Own Book as by the Author of "The Secret
Out" and The Secret Out as by the Author of the "Magician's Own
Book". But a further note says
"Under the title of "Le Magicien des Salons" the first has long
been a standard Magic Book with all French and German Professors of the
Art." -- see the discussion under Status of the Project in the
Introduction, above. This list is
followed by a half-title, p. iii, whose reverse (p. iv) has the printer's
colophon, then a blank page v, backed by a Frontispiece, p. vi, comprising
Figures 105 & 110 from the text.
The TP is p. vii,, backed by a blank p. viii. The Preliminary on pp. ix-x has the address Piccadilly
at the end and states this is "an Entirely New Edition" and is
by the same author as The Secret Out.
It refers to Cremer's display of Toys of the World at the recent
International Exhibition (possibly the 1862??) and to [Frank Bellew's]
"The Art of Amusing" (of 1866 and published by Hotten in 1870) and
Clara Bellew's "The Merry Circle".
Contents are given on pp. xi‑xii and then the text on pp.
13-326. [Toole Stott 194 lists this
under Cremer and says there are 30pp of publisher's catalogue at the end, dated
1872 -- I didn't record these details.
Christopher 239 lists this as "An entirely new edition" with
200 illustrations, 325 pp. + 30 pp. publisher's ads.]
I
have also seen a John Grant, Edinburgh, ed. which omits pp. i-vi, and has a
simplified title page, saying it is A New Edition, and drops the address Piccadilly.
Otherwise the text appears to be identical. It has no date but Toole Stott 1015 gives the date 1871. C&B, under Cremer, have London, 1871
with no indication of the New York ed.
C&B also list it under Williams.
I
have a Chatto & Windus ed., which omits pp. i-vi, and whose TP has only
slight changes from the Hotten TP, saying it is A New Edition, but the text
appears to be identical, except for dropping the address Piccadilly
from the end of the Preliminary.
It is dated 1890, with separately paginated publisher's catalogue of
32pp. dated Apr 1893.
Mahavira. 850. Mahāvīrā(cārya). Gaņita‑sāra‑sangraha
[NOTE: ņ denotes an n with a dot under it and ń
denotes an n with a dot over it.] (= Gaņita‑sāra‑samgraha
[The m
should have a dot over it.] = Ganitasar Samgrha). 850.
Translated by M. Rańgācārya. Government Press, Madras, 1912.
The sections in this are verses.
I will refer to the integral part of the first verse of the
problem. E.g. where he uses 121½ ‑ 123, I will use v. 121. [This work is described by David Eugene
Smith; The Ganita-Sara-Sangraha of Mahāvīrācārya;
Bibliotheca Mathematica (3) (1908/09) 106-110.
In: [G. R. Kaye; A brief
bibliography of Hindu mathematics; J.
Asiatic Society of Bengal (NS) 7:10 (Nov 1911) 679-686], this work is cited as
1908 with the note: "This is really an advance copy of a work not yet
actually published, kindly supplied to me by the author." See the entry under Pearson, 1907, in 7.E.]
Mair. 1765?.
John
Mair. Arithmetic, Rational and
Practical. Wherein The Properties of Numbers are clearly
pointed out, the Theory of the science deduced from first principles, the
methods of Operation demonstratively explained, and the whole reduced to
Practice in a great variety of useful Rules.
Consisting of Three Parts, viz.
I. Vulgar Arithmetic. II.
Decimal Arithmetic. III. Practical
Arithmetic. A. Kincaid & J. Bell,
Edinburgh, in three vols, 1765-1766 (Turner G1.14/1-3); 2nd ed., A. Kincaid & W. Creech, and J.
Bell, Edinburgh, 1772; 3rd ed, John
Bell and William Creech, Edinburgh, 1777.
I have the 3rd ed and have seen the 2nd ed. All the material of interest is in part 3, which first appeared
as vol. 3 in 1765 and books of this era often had little or no change between
editions, so I will date entries as 1765?
Manson. Indoor Amusements. 1911.
J.
A. Manson, compiler. Indoor
Amusements. Cassell & Co., London,
1911. FP + 8pp + 340pp + 8pp Index
(341-348). This is an extension of
Cassell's Book of In-Door Amusements ..., expanding the earlier 209pp of main
text to 340pp. This is partly due to
using larger type, getting 47 lines per page instead of 54. The material which was in Cassell's is
generally unchanged.
Manuel des Sorciers. 1825.
Manuel
des Sorciers ou Cours de Récréations Physiques,
Mathématiques, Tours de Cartes et de Gibecière, suive Des Petits Jeux de Société, et de Leurs Pénitences. (Conort, Paris, 178?; 2nd ed, Metier & Levacher, Paris, 1802
[Christopher 642, C&B]; 4th ed,
Ferra Jeune, Paris, 1815 [C&B]; 5th
ed, Ferra Jeune, 1820 [C&B]); 6th
ed, Augmentée d'une Notice sur la Magie noire, Ferra Jeune, Paris, 1825
[Christopher 643, C&B, HPL]. There
are a great many French books with similar titles from this era. They seem to be the predecessors of
Magician's Own Book, etc. -- cf the discussion under Status of the Project in
the Introduction and under Book of 500 Puzzles and Magician's Own Book. This is the first that I have found and
examined carefully, at HPL. It is
likely that most of the material dates back to the first ed of 178?, but until
I see some earlier editions, I'll date it as 1825, Gaidoz, in Section 7.B, cites the 2nd ed, but the material is on
a different page than in the 1825.
Marinoni, Augusto. See:
Pacioli. De Viribus. c1500.
McKay. At Home Tonight. 1940.
Herbert
McKay. At Home Tonight. OUP, 1940.
Section V: Puzzles and problems, pp. 63-88.
McKay. Party Night. 1940.
Herbert
McKay. Party Night. OUP, 1940.
Sections on Dinner-Table Tricks, pp. 134-171; Some Tricks in English, pp. 174-175; Arithmetical Catches and Puzzles, pp. 176-184.
Metrodorus. c510. In: The Greek Anthology, W. R. Paton,
trans. Loeb Classical Library, 1916‑1918. Vol. 5, Book 14. This contains 44 mathematical problems, most of which are
attributed to Metrodorus, though he is clearly simply a compiler and some may
be much older. I have cited the pages
of the English translation -- the Greek is on the previous page. Paton gives English answers, but they are
not in the Greek.
The
Greek Anthology is the modern name for a combination of two anthologies which
were based on earlier compilations dating back to the time of Alexander (-4C),
e.g. the Garlands of Meleager (c-95) and Philippus of Thessalonika (c40) and
the Cycle of Agathias (c570). In the
late 9C or early 10C, Konstantinos Kephalas assembled these into one
collection, but distributed them according to type and then added several other
collections -- this was somewhat revised c980.
In 1301, Maximus Planudes re-edited Kephalas' anthology, omitting much
and adding some (Paton believes that Planudes' source was missing a book). The Planudean version replaced Kephalas's
version and was printed in 1484.
However a copy of Kephalas's version of c980 was discovered at
Heidelberg in the Palatine Library (hence the anthology is sometimes described
as Palatine) in 1606 and modern versions now use this version as the first 15
books and put all of Planudes' additions as Book 16, the Planudean
Appendix. The Anthology comprises some
4000 poems. Modern scholars view Paton
as obsolete, but I don't know of any later versions of the Anthology which
include the mathematical problems -- e.g. they are not even mentioned in Peter
Jay; The Greek Anthology; Allen Lane, 1973; Penguin, 1981.
See
also: David Singmaster; Puzzles from
the Greek Anthology; Math. Spectrum
17:1 (1984/85) 11-15 for a survey of
these problems.
Meyer. Big Fun Book. 1940.
Jerome
S. Meyer. The Big Fun Book. Greenberg : Publisher, NY, 1940.
MG. Mathematical Gazette.
Mikami. 1913. Yoshio
Mikami. The Development of Mathematics
in China and Japan. Teubner, Leipzig,
1913; reprinted, Chelsea, 1961? See
also: Smith & Mikami.
Minguet. 1733.
Pablo
Minguet (or Minguét) è (or é or e or y) Yról (or Irol) ( -1801?).
Engaños à Ojos Vistas, y Diversion de Trabajos Mundanos, Fundada en
Licitos Juegos de Manos, que contiene todas las diferencias de los Cubiletes, y
otras habilidades muy curiosas, demostradas con diferentes Láminas, para que
los pueda hacer facilmente qualquier entretenido. Pedro Joseph Alonso y Padilla, Madrid, nd [1733]. Frontispiece + 12 + 218 pp. Imprimaturs or licenses dated 3 Nov 1733, 10
Nov 1733 & 12 Dec 1733. [NUC. Christopher 672 for a 18 + 110 pp
version. C&B gives versions with 18
+ 110 pp and with 12 + 218 pp. HPL has
4 versions -- the one I examined was 12 + 218 pp. The BM has a version, but it is not clear which.]
The
early history of this book is confused.
The first edition may have only had 18 + 110 pp (or 105 pp ??), then was
apparently frequently reprinted, without changing the dates, sometimes with
additions. Or it may be that both
versions appeared in 1733. However, my
copy is identical to one in HPL which is catalogued as 1733 and Palau (a list
of Spanish book sales, c1955) lists six sales of the 12 + 218 pp version, dated
1733, and only one sale of the 18 + 110 pp version, dated 1733. Christopher dates the 12 + 218 pp versions
to c1760.
I
now have discovered 26 editions of this work, and 2 and 9 editions of two
derivative works, but I have only seen a few versions. If anyone has access to a copy, I would like
a photocopy of the TP and other publishing details and of the Indice.
3rd
ed, Domingo Fernandez de Arrojo, Madrid, 1755, 18 + 157 + 3 pp. [This includes the same material as 1733,
but it is reset with smaller type and much rearranged and includes some new
material. It seems to be the same as
the 1766. Palau gives this as 18 + 150
+ 10 pp and says there was a cheap edition of 14 + 171 + 10 pp. NUC gives 14 + 171 +11 pp, with publisher
Domingo Fernandez. HPL has two
'entirely different' editions of 1755 -- the one I examined was 18 + 157 + 3
pp. BM has a 1755, but it is not clear
which.]
3rd
ed, Dionisio Hernández, Madrid, 1755?
[Palau, with nd.]
Antonio
del Valle, Madrid, 1756. 171 + 20 (or
40??) pp. [Palau.]
Revised. Pedro J. Alfonso y Padilla, Madrid, c1760,
14 + 218 pp. [Christopher 673 &
674.]
Idem. Añadido en esta quarta impresion, 18
enigmas, 6 quisicosas muy curiosas. 4th
ptg, D. Gabriel Ramirez, Madrid, 1766, 16 + 150 + 10 pp. [BM; Palau; Christopher 675.]
Palau
says he has seen a catalogue with a 1793 ed, but thinks this is a printing
error for either 1733 or 1893.
Palau
lists an 1804 reprint of the 1733, 14 + 218 pp.
Sierra
y Martí, Barcelona, 1820, 1 + 224 pp.
[Identical to 1733, 12 + 218 pp, except the text has been reset,
spelling modernized, the front matter updated and the text starts on p. 10,
with everything the same as the 1733, except page numbers are increased by 9,
and the index is reduced in size. HPL;
Palau; Christopher 676.]
Neuva
Edicion corregida y aumentada por D. J. M. de L. Juan Francisco Piferrer, Barcelona, 1822, 1 + 224 pp. [This is quite differently arranged than the
1733, 12 + 218 pp, and 1820 eds, and contains some extra material. Palau, with
1 + 240 pp; NUC; Shortz; Christopher 677.].
Retitled:
Juegos de Manos ó sea Arte de Hacer Diabluras, y Juegos de Prendas. Que contiene varias demonstraciones de
mágia, fantasmagoria, sombras y otros entretenimientos de diversion para
tertulias y sociedades caseras.
ilustrado con láminas Por D.
Pablo Minguet, y aumentado considerablemente en esta nueva edicion con infinidad
de juegos nuevos, y con laminas intercaladas en el texto. D. Manuel Saurí, Barcelona, 1847, 189 +
10 pp. [Palau; Christopher 678.]
Juegos
de manos; ó sea, Arte de hacer diabluras ...ilustrado con 60 grabados.... New ed.
Simon Blanquet, Mexico, nd [1856].
7 + 498 pp. [NUC.]
Palau
cites an 1857 reprint of the 1847, presumably the 2nd Saurí ed..
Title
varied: Juegos de Manos ó sea Arte de Hacer Diabluras. Contiene: juegos de prendas, de naipes,
varias demonstraciones de majia, fantasmagoria, sombras y otros
entretenimientos de diversion, para tertulias y sociedades caseras. Por D. Pablo Minguet. Tercera Edicion Aumentada con gran número de Juegos nuevos, y grabados
intercalados en el texto. Manuel Saurí,
Barcelona, 1864. 1 + 213 pp. [From TP of 1993 facsimile. This omits one problem and some discussion
that was in 1733 and adds 22 new problems, but I see some of these already
appeared in 1755 and 1822. Palau,
noting that this is the 3rd ed from Saurí and mentioning a recent facsimile; HPL.]
Palau
cites an 1875 Barcelona reprint as 185pp, but has no publisher's name, probably
the 4th Saurí ed.
Title
varied: Juegos de manos; ó sea, El arte de hacer diabluras, contiene 150 clases
de juegos, de prendas, de naipes, varias demonstraciones de mágia,
fantasmagoria, sombras y otros entretenimientos de diversion, para tertulias y
sociedades caseras. 5. ed., aumentada
con gran numero de juegos nuevos y 70 grabados intercalados en el texto. Manuel Saurí, Barcelona, 1876. 192pp.
[NUC.]
Palau
confusingly cites Sauri reprints of 192pp:
8th ed, 1885; 8th ed, 1888; 9th ed, 1888. [Christopher 679 & 680 are the latter two. Perhaps there is an error in the dates, e.g.
the 1885 might really be the 7th ed.]
Title
varied: Juegos de Manos o sea Arte de hacer Diabluras, contiene 150 clases de
juegos, de prendas de naipes, varias demonstraciones de magia, fantasmagorias,
sombras y otros entretenimientos de mucha diversion, para tertulias y
sociedades caseras. Décima edición
aumentada con juegos nuevos, y 70 grabados intercalados en el texto. Juan Tarroll y Cla., Barcelona, 1893. 192pp.
[Palau.]
Palau
lists 11th ed [from Sauri]. Sauri y
Sabater, Barcelona, 1896. 192pp. [NUC lists 2 copies of the 11th ed as Sauri
y Sabater, 1897, 192pp & 189pp.]
12th
ed [from Sauri]. Sauri, Barcelona,
1906. 190pp. [Christopher II 2102. HPL
with nd.]
Facsimile
of the 1864 ed, with Presentación por Joan Brossa: De la brujería blanca. Editorial Alta Fulla, Barcelona, 1981. 11pp new material + II-VIII + 9-213 pp of
facsimile (II is FP; III is TP; V-VIII is Al Lector; 9-12 is a poetic
introductory Relacion; 203-213 is Indice).
2nd
ed (i.e. printing) of the 1981 facsimile of the 1864 ed, with a new cover,
1993. [DBS.]
Pp.
1-25 is a fairly direct translation of the 1723/1725 Ozanam, vol. IV,
pp. 393‑406. A number of
other pictures and texts also are taken from Ozanam. I will give the date as 1733, though the expansion may not have
occurred until c1755. I will cite the
1755, 1822 and 1864 pages in parentheses, e.g.
Pp. 158-159 (1755: 114-115; 1822: 175-176; 1864: 151).
MiS. Mathematics in School.
Mittenzwey. 1880.
Louis
Mittenzwey. Mathematische
Kurzweil. Julius Klinkhardt,
Leipzig. 1880; 2nd ed., 1883; 3rd ed., 1895; 4th ed.,
1904; 5th ed., 1907; 6th ed., 1912; 7th ed., 1918. I will
give the date as 1880 or 1895? I now
have copies of the 1st, 4th, 5th and 7th eds.
In working through the other editions, I have seen many more items than
I had previously recorded and I now think this is one of the most important 19C
puzzle books.
[Ahrens,
MUS #363 lists the first ed. being 1879, apparently taking the date of the
Foreword. He also has 3rd ed., 1903,
but V&T (and another reference) cite 3rd ed., 1895. I think this is a misinterpretation of the
last Vorrede in the 4th ed., which is for the 3rd and 4th ed. and dated
1903. Trey Kazee has obtained a copy of
the 2nd ed. The first two editions have
the author's given name and have 300 problems; the 4th, 5th and 7th have
333. The Vorrede to the 2nd ed in the
2nd ed. says it has no major changes.
The Vorrede to the 3rd & 4th eds in the 4th ed says the 3rd ed was
extended and the 4th ed. has replaced some problems, so it seems that the 3rd
probably had 333 problems. Compared to
the 1st ed, the 4th drops 9 problems and adds 42. The Vorrede to the 3rd, 4th and 5th eds in the 5th ed also says
some problems have been replaced, but I have not discovered any differences
between the 4th and 5th eds except one minor amendment, some resetting which changes
the page breaks on pp. 1-10 and 36-37 and a few line breaks. I suspect that the 4th and 5th eds are very
similar to the 3rd. The 5th and 7th
eds. are very similar, but the 7th is reset with smaller type and occupies five
fewer pages. There are a few amendments
and reorderings and four problems (36, 92, 122, 137) have been replaced, but a
number of simple misprints persist through all editions. I will give pages in 1st, 3rd?, and 7th eds,
e.g.
Prob. 193 & 194, pp. 36 & 89; 1895?: 218 & 219, pp. 41 & 91; 1917: 218 & 219, pp. 37 & 87.]
MM. Mathematics Magazine.
Montucla, Jean Étienne
(1725-1799). See: Ozanam;
Ozanam‑Montucla.
MP. 1926. H.
E. Dudeney. Modern Puzzles. C. Arthur Pearson, London, 1926; new ed., nd
[1936?]. (Almost all of this is in
536.)
MPSL1. 1959 & MPSL2.
1960.
Mathematical
Puzzles of Sam Loyd, vol. 1 & 2, ed. by Martin Gardner, Dover, 1959,
1960. (These contain about 2/3 of the
mathematical problems in the Cyclopedia, often with additional material by
Gardner.)
MRE. W. W. Rouse Ball (1850-1925). Mathematical Recreations and Essays. (First three editions titled Mathematical
Recreations and Problems of Past and
Present Times.) I have compiled an 8
page detailed comparison of the contents of all editions as part of my The Bibliography of Some Recreational
Mathematics Books.
1st
ed., Feb 1892, 240pp.
2nd
ed., May 1892 ('No material changes').
3rd
ed., 1896, 276pp.
4th
ed., 1905, 388pp. [In the 4th ed., it
says 'First Edition, Feb. 1892. Reprinted, May 1892. Second Edition, 1896. Reprinted, 1905.' However, it calls itself the 4th ed. and the 3rd ed. calls itself
3rd and there are substantial changes in the 4th ed.]
5th
ed., 1911, 492pp.
6th
ed., 1914, 492pp.
7th
ed., 1917, 492pp.
8th
ed., 1919, 492pp.
9th
ed., 1920, 492pp.
10th
ed., 1922, 366pp.
11th
ed., 1939, revised by H. S. M. Coxeter, 418pp.
Macmillan
(for 1st to 11th ed.).
12th
ed., 1974, revised by H. S. M. Coxeter, 428pp, U. of Toronto Press.
13th
ed., 1987, revised by H. S. M. Coxeter, 428pp, Dover.
A
few of the editions are actually reprintings of the previous edition: 2 is essentially the
same
as 1, 6 as 5, 9 as 8, 13 as 12. Consequently I will rarely, if ever, cite
editions
2, 6, 9, 13. For each topic occurring
in Ball, I have examined all the
editions,
so the absence of a reference indicates the topic does not occur in that
edition
-- unless it is buried in some highly unlikely place.
See
also: Ball‑FitzPatrick.
MS. Mathematical Spectrum (Sheffield, UK).
MTg. Mathematics Teaching (UK).
MTr. Mathematics Teacher (US).
Munich 14684. 14C.
Munich
Codex Lat. 14684. 14C. Ff. 30‑33. Published by M. Curtze; Mathematisch‑historische Miscellan:
6 -- Arithmetische Scherzaufgaben aus dem 14 Jahrhundert; Bibliotheca Math. (2)
9 (1895) 77‑88. 34 problems. Curtze gives brief notes in German. Curtze says these problems also appear in
Codex Amplonianus Qu. 345, ff. 16‑16', c1325, ??NYS. Munich 14684 comes from the same monastery
(St. Emmeran) as AR and AR incorporates much of it. In a later paper, Curtze says this is 13C -- ??NYR. Cf Folkerts, Aufgabensammlungen.
Murray. 1913. Harold
James Ruthven Murray. A History of
Chess. (OUP, 1913); reprinted by
Benjamin Press, Northampton, Massachusetts, nd [c1986].
MUS. 1910 & 1918.
Wilhelm
Ernst Martin Georg Ahrens (1872-1927).
Mathematische Unterhaltungen und Spiele. 2nd ed., 2 vols., 1910, 1918, Teubner, Leipzig. [The first ed. was 1901 in one volume. There is a 3rd ed. of Vol 1, 1921, but it is
a reprint of the 2nd ed. with just 2 pages of extra notes and the typographical
corrections made.] I will tend to cite
this, as e.g. MUS 1 153-155. MUS #n denotes item n in the substantial
Literarischer Index, MUS 2 375-431.
[MUS
II vii lists 22 items which he had been unable to see and which he suspected
might not exist. I have seen the
following items from the list: 86 (=
Les Amusemens, above), 102 (Hooper,
listed in Some Other Recurring References, below; cf Section 4.A.1), 145 (Jackson, see above), 212 (author's name is Horatio Nelson
Robinson); item 32 exists as an English
edition of van Etten, but the citation gives the publisher as though he was the
author; I have seen a copy of item 82
advertised for sale. A number of items
(135, 152, 164, 221, 223, 277, 297, 317) are cited by Wölffing - op. cit. in
Section 3.B.]
Muscarello. 1478.
Pietro
Paolo Muscarello. Algorismus. MS of 1478.
Published in 2 vols.: I -- facsimile; II -- transcription with notes and
commentaries; Banca Commerciale Italiana, Milan, 1972. I will cite the folios (given in vol. I) and
the pages of the transcribed version in vol. II. Van Egmond's Catalog 275-276.
M500. This is the actual name of the journal of the M500 Society,
the Open University student mathematics society.
NCTM. National Council of Teachers of Mathematics.
nd. no date. Estimated dates may follow in
[ ].
Needham. 1958. Joseph
Needham (1900-1995). Science and
Civilization in China, Vol. 3.
CUP, 1958. (Occasional references
may be made to other volumes: Vol. 2, 1956; Vol. 5, Part IV, 1980.)
The New Sphinx. c1840.
Anonymous. The New Sphinx An elegant Collection of upwards of 500 Enigmas Charades Rebusses
Logogriphes Anagrams Conundrums
&c. &c. To which are
added, a Number of Ingenious Problems.
T. Tegg & Son, London, nd, HB with folding frontispiece. [Vendor suggests it is 1840s. Shortz has 4th ed, by Gye & Baine and
says it is c1840. He also has 'a new(?)
edition', by T. Tegg, and says it is c1843.
He says the chapter of geometrical problems and brainteasers was new in
the 4th ed. Heyl 238 is a 7th ed.,
London, 18??, referring to HPL, where I find it in the Supplement.] The chapter of problems has 27 problems, of
which 21 are copied from Endless Amusement II, 1837 ed., 20 of which come from
the 1826? ed.
E. P. Northrop. Riddles in Mathematics. 1944.
Eugene
P. Northrop. Riddles in
Mathematics. Van Nostrand, 1944; English Universities Press, 1945; revised ed., Penguin, 1961. The Van Nostrand ed has the main text on
pp. 1-262. The EUP ed. has it on
pp. 1-242. The Penguin ed. has it on
pp. 11-240. The revision seems to
consist of only a few additional notes.
I will cite the dates and pages, e.g.
1944: 209-211 & 239;
1945: 195‑197 & 222;
1961: 197‑198 & 222.
H. D. Northrop. Popular Pastimes. 1901.
Henry
Davenport Northrop. Popular
Pastimes for Amusement and Instruction
being a Standard Work on Games,
Plays, Magic and Natural Phenomena Suitable for All Occasions containing
Parlor Games; Charming Tableaux; Tricks of Magic; Charades and Conundrums; Curious Puzzles; Phrenology
and Mind Reading; Palmistry, or How to
Read the Hand; Humorous and Pathetic
Recitations, Dialogues, Etc., Etc. including
The Delightful Art of Entertaining
The Whole Forming a Charming
Treasury of Pastimes for the Home, Public Schools and Academies, Lodges, Social Gatherings, Sunday Schools, Etc., Etc. Frank S. Brant, Philadelphia, 1901. [Not in any of my bibliographies. Vendor says only two copies in NUC, none in
BL.]
NUC. National Union Catalogue Pre‑1956 Imprints. Library of Congress, USA. c1960.
??check details
Nuts to Crack. Nuts to Crack, Part nn. Or, Enigmatical Repository; containing near
mmm Hieroglyphics, Enigmas, Conundrums, Curious Puzzles, and Other Ingenious
Devices. R. Macdonald, 30 Great Sutton
Street, Clerkenwell, London. These are
single broadsheets. The publisher's
details are often trimmed from the bottom of the sheet. At least 25 annual parts appeared, from Part
I of 1832, but the year is not always given.
Answer books -- The Nutcrackers -- also appeared and the publishers kept
the old sheets available for some years.
This series is very rare -- Will Shortz and James Dalgety have the only
examples known to me. I have photocopy
of almost all from Dalgety and Shortz, ??NYR.
mmm is either 250 or 200 and the problems are individually numbered in
each part. I will cite problems as,
e.g. Nuts to Crack I (1832), no. 200.
NX. No copy. Usually prefixed by
?? as a flag for further action.
NYR. Not yet read -- i.e. I have a copy which I
have not yet studied. Usually prefixed
by ??
as a flag for further action.
NYS. Not yet seen. Usually prefixed by
?? as a flag for further action.
OCB. See: Hall, OCB.
OED. Oxford English Dictionary. (As: New English Dictionary, OUP, 1884‑1928),
reprinted with supplements, OUP, 1933 and in various formats since.
o/o. On order.
OPM. 1907-1908. Our Puzzle
Magazine. Produced by Sam Loyd. The pages were unnumbered. The Magazine was reprinted as the Cyclopedia
as follows, but some pages of the Magazine were omitted and the answers to each
magazine were normally in the next one.
Vol. 1, No. 1 (Jun 1907) =
Cyclopedia pp. 7‑70.
Vol. 1, No. 2 (Oct 1907) =
Cyclopedia pp. 71‑121.
Vol. 1, No. 3 (Jan 1908) =
Cyclopedia pp. 122‑178.
Vol. 1, No. 4 (Apr 1908) =
Cyclopedia pp. 179‑234.
Since the Cyclopedia goes
to p. 339, there appear to have been two further issues which have not been
seen by anyone??
(The above data were provided by
Jerry Slocum.)
OUP. Oxford University Press.
Ozanam. 1694.
The
bibliography of this book is a little complicated. I have prepared a more detailed 7 pp. version covering the 19 (or
20) French and 10 English editions, from 1694 to 1854, as well as 15 related
versions -- as part of my The
Bibliography of Some Recreational Mathematics Books.
Jacques
Ozanam (1640-1717). Recreations
Mathematiques et Physiques, qui contiennent Plusieurs Problémes [sic] utiles &
agreables, d'Arithmetique, de Geometrie, d'Optique, de Gnomonique, de
Cosmographie, de Mecanique, de Pyrotechnie, & de Physique. Avec un Traité nouveau des Horloges
Elementaires. 2 vols., Jombert,
Paris, 1694, ??NYS. [Title taken from
my 1696 ed.]
[BNC. NUC -- but NUC lists a 1693 3rd ed. from
Amsterdam, which appears to be a misreading for 1698. MUS II 380 says 1st ed. was Récréations Mathématiques, Paris,
1694, 2 vols. Serge Plantureux's 1993
catalogue describes a 1694 edition in 2 vols., by Jombert and notes that
it is the original edition, that the privilege is dated 11 Jan 1692, but
that it was not printed until 30 Apr 1694.
The dating of the privilege may account for some references to the first
edition being 1692 -- e.g. the Preface to the 1778 Ozanam-Montucla ed. says the
first ed. was 1692, but Hutton changes this to 1694. MRE, 1st ed, 1892, pp. 3‑4, says the the 1st ed. was 2
volumes, Amsterdam, 1696, but this was amended in his 4th ed., 1905.
This
first appeared in two volumes, but later versions were sometimes in one
volume. I have references to versions
in Paris: 1694, 1696, 1697, 1698, possibly 1700? and apparently 1720; and in
Amsterdam: 1696, 1697, 1698, 1700.
There is no indication of any textual changes in these, except that the
pages are numbered consecutively in later versions. Plantureaux describes a 1696 edition as: Paris, Jombert (mais
Hollande), so there seems to have been some piracy going on. I have the 1696 ed. I will assume that the 1696 is essentially
identical in content to the 1694, though in the second volume, the page numbers
may be different and there is some confusion of plate numbering.
The
Traité des Horologes élémentaires, which appears in the 1694 ed., is a
translation of Domenico Martinelli's Horologi elementari. NUC says this was separately paginated in
1694, but it occupies pp. 473-583 of the 1696 ed and pp. 301-482 of vol. 3 of
the 1725 ed.]
About
1723, the work was revised into 4 vols., sometimes described as 3 vols. and a
supplement. MUS #52 gives 1720, 1723,
1724, 1725 and says the dates vary in the literature. The 1725 has privilege dated 1720, but I haven't found any
catalogue entry for a 1720 ed. of this revision, so it may be a spurious date
based on the privilege.
Nouv. ed.
Recreations
Mathematiques et Physiques, qui contiennent Plusieurs Problêmes [sic]
d'Arithmétique, de Géométrie, de Musique, d'Optique, de Gnomonique, de
Cosmographie, de Mécanique, de Pyrotechnie, & de Physique. Avec un Traité des Horloges
Elementaires. Par feu [misprinted
Parfeu in vol. 1] M. Ozanam, de l'Académie Royale des Sciences, &
Professeur en Mathematique. Nouvelle
edition, Revûë, corrigée & augmentée.
Vol. 4 has different title page.
Recreations
Mathematiques et Physiques, ou l'on traite Des Phosphores Naturels &
Artificiels, & des Lampes Perpetuelles.
Dissertation Physique & Chimique.
Avec l'Explication des Tours de Gibeciere, de Gobelets, & autres
récréatifs & divertissans. Nouvelle
edition, Revûë, corrigée & augmentée.
Claude
Jombert, Paris, 1723. [Taken from my
1725 ed.]
[Ball
and Glaisher [op. cit. in 7.P.5, p. 119] both cite a 1723 ed. as though they
had seen it, but there is no BMC entry for this date -- perhaps there is a copy
at Cambridge??. I have seen one volume
in an exhibition which was 1723. MUS
#52 says it was edited by Grandin. NUC
-- "The editor is said to be one Grandin." I have a brief 1899 reference to this ed.
I
have 1725, which is apparently a reprint of the 1723. The privilege/approbation is dated 16 May 1722 in Vol. 3 and Vol.
4 and also 28 Apr and 15 May 1720 in Vol. 3.
NUC says this is 1723 with new title pages. BNC has: Nouvelle édition ... augmentée [par Grandin], 4 vols, Paris,
1725. It was reprinted in 1735, 1737?,
1741, 1750/1749, 1770.
The
text and plates of the 1725 and 1735 eds. seem identical, though some of the
accessory material -- lists of corrections and of plates -- has been omitted
and other has been rearranged. I have
seen two versions of the 1735 -- one has the plates inserted in the text, the
other has them at the end as ordinary pages, while my 1725 has them at the end
on folding pages. Most of the 1725
plates are identical to the 1696 plates, but there were a number of additions
and reorderings. The 1725 plates have
their 1696 plate numbers and 1725 page references at the top with new, more
sequential, plate numbers at the bottom.
The 1725 text sidenotes refer to the plate numbers at the top, while the
1735 and later sidenotes refer to the bottom numbers. (However some of the new illustrations in vol. 4 are not
described in the text and this makes me wonder if there was an earlier version
with these new plates??) I will give
the 1725 top plate numbers, followed by the bottom numbers in ( )
-- e.g. plate 12 (14). The 1741
and the 1750/1749 eds. are essentially identical to the 1735 ed.
Ball,
MRE, and MUS #52 say the 1750 and/or the 1770 ed. were revised by Montucla, but
all other sources say his revision was 1778.
Indeed Montucla was only 25 in 1750.
Inspection of 1750 copies in the Turner Collection and at the Institute
für Geschichte der Naturwissenschaft shows the 1750 is identical to my 1725 ed.
except for some accents and a new publisher.]
English versions
Recreations
Mathematical and Physical; Laying down, and Solving Many Profitable and
Delightful Problems of Arithmetick, Geometry, Opticks, Gnomonicks, Cosmography,
Mechanicks, Physicks, and Pyrotechny.
By Monsieur Ozanam, Professor of the Mathematicks at Paris. Done into English, and illustrated with very
Many cuts. R. Bonwick, et al., London,
1708.
[Pp.
130-191 are omitted, but there is no gap in the text and the Contents also
shows these pages are lacking. Ball,
MRE. BMC. NUC. Hall, OCB, p.
165. Hall, BCB 216. UCL.
Toole Stott 520, noting the gap.
Bodleian. This is a pretty
direct translation of the 1696 French ed. or an early simple revision. Prob. 18-20 of Cosmographie have been
omitted. C&B list this and say
there were three later editions, though they then list the 1756 and 1790
editions.]
2nd ed.
Recreations
for gentlemen and ladies: or, Ingenious Amusements. Being Curious and diverting sports and pastimes, natural and
artificial. With Many Inventions,
pleasant Tricks on the Cards and Dice, Experiments, artificial Fireworks, and other
Curiosities, affording variety of entertainments. James Hoey, Dublin, 1756.
[Taken from Toole Stott entry.]
[NUC. Hall, BCB 212. Hall, OCB, p. 165, but giving the 1790 title. Toole Stott 518, but he has Hoez, which
seems to be either a misreading or a miswriting??]
3rd ed.
Recreations
for Gentlemen and Ladies; being Ingenious Sports and Pastimes. Containing Many curious Inventions, pleasant
Tricks on Cards and Dice; Arithmetical Sports; new Games; Rules for Assuredly
winning at all Games, whether of Cards or Dice; Recreative Fire-works; Tricks
to promote Diversion in Company, and other Curiosities.... James Hoey, Dublin, 1759. [Taken from Hall, BCB 213.]
[Ball,
MRE. Hall, BCB 213. Hall, OCB, p. 165. Bodleian.]
4th ed.
Recreations
for gentlemen and ladies; being ingenious sports and pastimes: containing Many
curious Inventions, pleasant Tricks on Cards and Dice; Arithmetical Sports; new
Games; Rules for Assuredly winning at all Games, whether of Cards or Dice;
Recreative Fire-works; Tricks to promote Diversion in Company, and other
Curiosities.... The fourth
edition. Peter Hoey, Dublin, 1790. [Taken from Toole Stott entry.]
[BMC
calls it the 4th ed. [abridged].
NUC. Hall, BCB 214. Hall, OCB, p. 165. Toole Stott 519.
Bodleian.]
I
will cite these and the following editions by date, though the varying problem
numbers and volume numbers will make this a bit unwieldy. All references to the 4 volume versions are
to vol. I unless specified otherwise.
See the entry in 4.A.1 as an example.
Note that I give the problem number first because this is usually the
same in the 1790, 1803 and 1814 editions, and often in the 1840. Figure numbers will also been given with the
problem number, although the 1840 has the figures in the text with different
numbers. Additions will be entered at
the relevant point.
Ozanam‑Hutton. 1803.
Recreations
in Mathematics and Natural Philosophy: ...
first composed by M. Ozanam, of the Royal Academy of Sciences, &c.
Lately recomposed, and greatly enlarged, in a new Edition, by the celebrated M.
Montucla. And now translated into
English, and improved with many Additions and Observations, by Charles Hutton,
.... 4 vols., T. Davison for G.
Kearsley, London, 1803.
[This
is a pretty direct translation of the 1790 Ozanam-Montucla ed. with a few
changes, and some notes and extra material, generally in sections which do not
interest us. Only one or two plates
have been changed. Erroneous problem
numbers have been retained.
There
was an 1814 ed. by Longman, Hurst, Rees, Orme and Brown, London. The texts are basically identical, but the
1814 has been reset to occupy about 15% fewer pages, the problem numbers have
been corrected, a few corrections/additions have been made and the plates do
not fold out.]
Ozanam‑Montucla. 1778.
J.
E. Montucla (1725-1799)'s revision of Ozanam.
Récréations
Mathématiques et Physiques, Qui contiennent les Problêmes & les Questions
les plus remarquables, & les plus propres à piquer la curiosité, tant des
Mathématiques que de la Physique; le tout traité d'une maniere à la portée des
Lecteurs qui ont seulement quelques connoissances légeres de ces Sciences. Par feu M. Ozanam, de l'Académie royale des
Sciences, &c. Nouvelle Edition,
totalement refondue & considérablement augmentée par M. de C. G. F. Claude Antoine Jombert, fils aîné, Paris,
1778, 4 vols. Approbation &
privilege dated 5 Aug 1775 & 4 Sep 1775.
[BMC
says this is "par M. de C. G. F. [i.e. M. de Chanla, géomètre forézien,
pseudonym of J. E. Montucla.]"
BMC, under Montucla, says M. de Chanla is a pseudonym of Montucla. BNC, under Montucla, describes Montucla as
Éd. Lucas, RM1, p. 242 lists this under
Chaula "attribué à Montucla".
MUS #52 has Chaula. NUC has
Chanla. Montucla's connection with the
book was so little known that the 1778 version was sent to him in his role as
Mathematical Censor and he made some additions to it before approving it. Hutton says the 'last edition' (presumably
1790) bears Montucla's initials.
'Forézien' means from Feurs or the Forez region. Reprinted in 1785-1786 and 1790 (see below).
This
is a considerable reworking of the earlier versions. In particular, the interesting material on conjuring and
mechanical puzzles in Vol. IV has been omitted. There are occasional misnumberings of problems. I recall the plates are folding at the end
of each volume, but I didn't note this specifically.]
Nouv. ed.
"Nouvelle
édition, totalement refondue et considérablement augmentée par M. de M***"
[BMC adds: [i.e. Jean Étienne Montucla]].
Firmin Didot, Paris, 1790.
[NUC
lists this as a reissue of 1778 with new TPs.
I have this ed.]
Ozanam‑Riddle. 1840.
(Edward
Riddle (1788-1854)'s revision of Ozanam‑Hutton.)
Recreations
in mathematics and natural philosophy: translated from Montucla's edition of
Ozanam, by Charles Hutton, LL.D. F.R.S. &c. A new and revised edition, with numerous additions, and
illustrated with upwards of four hundred woodcuts, By Edward Riddle, Master of the Mathematical School, Royal
Hospital, Greenwich. Thomas Tegg,
London, 1840. [Taken from my copy. MUS #130 asserts this is by a C. Biddle, but
this must be due to be a misreading or misprint.]
Another ed.
Recreations
in science and natural philosophy: Dr. Hutton's translation of Montucla's
edition of Ozanam. The present new
edition of this celebrated work is revised By Edward Riddle, Master of the
Mathematical School, Royal Hospital, Greenwich, who has corrected it to the
present era, and made numerous additions.
This Edition is also Illustrated by upwards of Four Hundred
Woodcuts. Thomas Tegg, London, 1844.
Another ed.
Recreations
in Science and Natural Philosophy. Dr.
Hutton's translation of Montucla's edition of Ozanam. New Edition, revised and corrected, with numerous additions, by
Edward Riddle, Master of the Mathematical School, Royal Hospital,
Greenwich. Illustrated by upwards of
Four Hundred Woodcuts. William Tegg,
London, 1851.
[Reprinted by William Tegg, 1854.]
All
the Riddle printings seem to be from the same plates -- the date of the
Prefatory Note and the Erratum given on p. xiv are the same in 1840 and
1851. Marcia Ascher cites an 1844
edition from Nuttall & Hodgson, but this is the printer.
These
have the figures in the text, but otherwise seem to be little different than
Ozanam-Hutton. I will generally cite it
just as 1840.
P. M. Calandri. See:
Benedetto da Firenze.
Pacioli. De Viribus.
c1500.
Luca
Pacioli (or Paciuolo) (c1445-1517). De
Viribus Quantitatis. c1500. Italian MS in Codex 250, Biblioteca
Universitaria di Bologna. Santi 3 dates
it as 1498 and describes the later parts which deal with riddles, etc., but
include some magic, etc. However,
Pacioli petitioned for a privilege to print this in 1508 and Part 2, Chap.
CXXIX, ff 228r-228v, has a date of 1509, so he may have been working on the MS
for many years. Van Egmond's Catalog
55-56 is not as helpful as usual.
Part
1: Delle forze numerali cioe di Arithmetica is described in: A. Agostini;
Il "De viribus quantitatis" di Luca Pacioli; Periodico di Matematiche
(4) 4 (1924) 165‑192 (also separately published with pp. 1-28). Agostini's descriptions are sometimes quite
brief -- unless one knows the problem already, it is often difficult to figure
out what is intended. Further, he
sometimes gives only one case from Pacioli, while Pacioli does the general
situation and all the cases. All
references are to the problem numbers in this part, unless specified
otherwise. I will use problem numbers
and names as in the MS at the problem -- these often differ considerably from
the numbers and names given in the index at the beginning of the MS. There are 81 problems in part 1, but the
Index lists 120 -- in a few cases, the Index name clearly indicates a problem
similar to an actual problem and I will mention this. There are microfilms at the Warburg Institute (currently
misplaced) and at Munich and Siena. I
printed Part 1 and some other relevant material from the Warburg microfilm,
some 125pp. When I read this, I saw
that the material near the last pages I had copied would be of interest, but
when I went back to copy this material, the microfilm had been misplaced.
Dario
Uri has photographed the entire MS and enhanced the images and put them all on
a CD. This has 614 images, including
the insides of the covers. This is
often more legible than the microfilm, but the folio numbers are often faint,
sometimes illegible. It was not until
the transcription (below) became available that I could read the material just
after this point and see that there were more interesting problems, but I found
it difficult to read the Italian (many words are run together and/or archaic)
and the diagrams referred to are lacking.
Dario was able to carry on and found the Chinese Rings and about a dozen
other interesting items. He has put
some material up on his website:
http://digilander.libero.it/maior2000/.
This includes the indexes and a number of the most interesting items,
with his comments and diagrams of later examples of the puzzles. I have now gone through all of the text and
found a number of problems of interest for this bibliography. Nonetheless, quite a number of problems,
some clearly of interest, remain obscure.
Transcription
by Maria Garlaschi Peirani, with Preface and editing by Augusto Marinoni. Ente Raccolta Vinciana, Milano, 1997. (The publisher did not reply to a letter,
but Bill Kalush kindly obtained a copy for me.
Dario Uri says it can be bought from:
Libreria Pecorini, 48 foro Buonaparte, Milano; tel: 02 8646 0660;
fax: 02 7200 1462; web: www.pecorini.com.) I will cite the text as Peirani and any
references to Marinoni's work as Marinoni.
The transcription is not exactly literal in that Peirani has expanded
abbreviations and inserted punctuation, etc.
Also, Peirani seems to have worked from the microfilm or a poor copy as
she sometimes says the manuscript has an incorrect form which she corrects, but
Dario Uri's version clearly shows the MS has the correct form. Peirani uses the problem numbers and names
in the MS (see comment above about these differing from those in the Index),
but with some amendments. I will
probably give problem names as in the MS, with some of Peirani's amendments. I will give translations of the names, but
some of them are pretty uncertain and some have defeated me completely. Marinoni, pp. VIII‑IX indicates
the MS was written in about 1496-1509.
The
title is a bit cryptic, but I think the best English version is: On the Powers
of Numbers, but Pacioli [f. Ir] has 'forze numerali' and R. E. Taylor [op. cit.
in Section 1, pp. 307 & 339] follows Agostini and uses: On the Forces of
Quantity.
Pacioli. Summa.
1494.
Luca
Pacioli (or Paciuolo or Paccioli) (??-c1509).
Sūma de Arithmetica Geometria Proportioni &
Proportionalita. Paganino de Paganini,
Venice, 1494 -- cf Van Egmond's Catalog 325-326; facsimile printed by Istituto Poligrafico e Zecca dello Stato,
Rome, for Fondazione Piero della Francesca, Comune di Sansepolcro, 1994, with
descriptive booklet edited by Enrico Giusti.
There was a second ed., Paganino de Paganinis, Toscolano, 1523, but the
main text seems identical, except for corrections (and errors) and somewhat
different usage of initial letters and colour.
For extensive studies of this book, see the works by Narducci, Taylor,
Davis and Rankin given in Section 1.
Davis identifies material taken from Piero della Francesca's works. Taylor says 99 copies of the 1494 and 36
copies of the 1523 are known. The text
is in two parts -- the second part is geometry and is separately paged. All page references will be to the first
part unless specified as Part II.
Problems are numbered at the right hand edge of the last line of the
previous problem. See Rara 54-59.
Davis
notes that Pacioli's Summa, Part II, ff. 68v - 73v, prob. 1-56, are essentially
identical to della Francesca's Trattato, ff. 105r - 120r.
Panckoucke, André Joseph
(1700-1753). See: Les Amusemens.
Pardon, George Frederick
(1824-1884). See: Indoor & Outdoor; Parlour Pastime; Parlour Pastimes.
Parlour Pastime. 1857.
Parlour
Pastime for the Young: Consisting of Pantomime and Dialogue Charades, Fire-side
Games, Riddles, Enigmas, Charades, Conundrums, Arithmetical and Mechanical Puzzles,
Parlour Magic, etc. etc. Edited by
Uncle George [NUC says this is George Frederick Pardon]. James Blackwood, London, 1857 [Toole Stott
545; Christopher 724]. This was
combined with Games for All Seasons [Toole Stott 311; Christopher 723] into Indoor
& Outdoor, c1859, qv. There is a
later edition, Parlour Pastimes, 1868, qv.
Hence the problems will be cited as: Parlour Pastime, 1857 = Indoor & Outdoor, c1859, Part 1 = Parlour Pastimes, 1868. Many of the problems are identical to Book
of 500 Puzzles.
Parlour Pastimes. 1868?
Parlour
Pastimes: A Repertoire of Acting Charades, Fire-side Games Enigmas, Riddles,
Charades, Conundrums, Arithmetical and Mechanical Puzzles, Parlour Magic, etc.,
etc. James Blackwood, London, nd. [BMC, NUC and Toole Stott 1136 date this
1868 and say it is by George Frederick Pardon.
Toole Stott 1136 indicates that By G. F. P. is on the TP, but it is not
in my example. Toole Stott 1137 is
1870, a slightly smaller ed.] It is an
expanded version of Parlour Pastime, with the material of interest to us being
directly copied, though the page layout varies slightly. The running head of this is actually Parlour
Pastime. Hence the problems will be
cited as: Parlour Pastime, 1857 =
Indoor & Outdoor, c1859, Part 1 =
Parlour Pastimes, 1868. Many of the
problems are identical to Book of 500 Puzzles.
PCP. 1932. H. E.
Dudeney. Puzzles and Curious
Problems. Nelson, 1932; revised ed., nd
[1936?]. (Almost all of this is in
536.) There are almost no changes in
the revised ed., except that problem 175 and its solution have been corrected
-- it is a cross number puzzle and the text was for a different diagram. See: 7.AM.
Peano. Giochi. 1924.
Giuseppe
Peano (1858‑1932). Giochi di
Aritmetica e Problemi Interessanti. G. B.
Paravia, Torino, nd [1924 and later reprints].
(Thanks to Luigi Pepe for a photocopy of this.)
Pearson. 1907. A.
Cyril Pearson. The Twentieth Century
Standard Puzzle Book. Routledge,
London, nd [1907]. Three parts in one
volume, separately paginated. The parts
were also published separately. Each
part has several numbered sequences of problems.
Peck & Snyder. 1886.
Price
List of Out & Indoor Sports & Pastimes. Peck & Snyder, 126‑130 Nassau Street, N. Y., 1886. Reprinted, with some explanatory material,
in the American Historical Catalogue Collection, Pyne Press, Princeton,
1971. Unpaginated -- I have numbered
the pages starting with 1 as the original cover.
Peirani, Maria Garlaschi. See:
Pacioli. De Viribus. c1500.
Perelman. FFF.
1934.
Yakov
Isidorovich Perelman [Я. И. Перелман]
(1882-1942). Figures for Fun. Живая
Математика [Zhivaya
Matematika],
Наука [Nauka], Moscow. The books give no indication of the original
dates, but Tatiana Matveeva has kindly searched the Russian State Library and
found it was originally published by Гос.
техн.-теор.
издат., Leningrad-Moscow, 1934. Schaaf I 9 cites Recreational Arithmetic,
6th ed., Leningrad, 1935 and Sphinx 5 (1935) 96 reviews the 5th ed. of
L'Arithmétique Récréative, Leningrad, 1934 -- both presumably the same book ??
Translated
by G. Ivanov‑Mumjiev.
Foreign Languages Publishing House, Moscow, 1957, 120 sections.
(2nd
ed., 1973 -- used for MCBF, below, apparently the same as the 3rd ed.)
Translated
by G. Ivanov‑Mumjiev. 3rd ed.,
MIR, 1979, 123 sections. The 3rd ed.
drops 3 sections and adds 6 sections and has some amended English.
Perelman. FMP.
1984.
Yakov
Isidorovich Perelman [Я. И. Перелман]. Fun with Maths and Physics. MIR, Moscow, 1984. [This is a translation of Занимательныи
Задачи и Оыты
[Zanimatel'nye Zadachi i Oryty],
Детская
Литературы [Detskaya
Literatura], Moscow.) (There was a
Занимательныи
Задачи, with 4th ed. in 1935 – no
earlier version in Russian State Library.
This title originally published by
Деттиз, 1959.) Compiled
by I. I. Pruskov. Translated by
Alexander Repyev. This is a compilation
from several of his books from 1913 to c1942 (when he died). I have not yet seen the earlier books
(??NYS), but if this is mainly based on the earlier book of the similar title,
this would date the material to c1935? and I will use this date.
Perelman. MCBF.
c1980?
Yakov
Isidorovich Perelman [Я. И. Перелман]. Mathematics Can Be Fun. 3rd ed., MIR, 1985. This consists of the following, originally
separate, works, but with the second part having its page and problem numbers
continued from the first part. Both
these works exist in many other editions and translations.
Figures
for Fun, translated from Живая
Математика
[Zhivaya Matematika], Наука [Nauka], Moscow,
translated 1973 -- with 123 sections.
Translator not specified, but presumably G. Ivanov-Mumjiev, as in FFF
above.
Algebra
Can be Fun, translated from Занимательная
Алгебра
[Zanimatel'naya Algebra] (3rd ed. was published by ОНТИ,
Leningrad-Moscow, 1937 -- The Russian State Library apparently has no earlier
edition), edited and supplemented by V. Boltyansky, Наука [Nauka], Moscow, 1976,
translated by G. Yankovsky, 1976.
References
to this will be to material not in FFF, so will be dated 1937.
Phillips, Hubert
(1891-1964). See: Brush;
Week‑End.
Pike. Arithmetic. 1788.
Nicolas
Pike (1743-1819). A New and Complete
System of Arithmetic, composed for the Use of the Citizens of the United
States. John Mycall, Newbury-Port,
Massachusetts, 1788. (I have a 2nd ed.,
1797, [Halwas 318], ??NYR. This went
through at least 5 editions and then at least six variants, often abridged for
schools [Halwas 318-326].)
Poggendorff. J. C. Poggendorff. Biographisch-Literarisches Handwörterbuch zur Geschichte der
Exacten Wissenschaften enthaltend Nachweisungen über Lebensverhältnisse und
Leistungen von Mathematikern, Astronomen, Physikern, Chemikern, Mineralogen,
Geologen usw aller Völker und Zeiten.
Johann Ambrosius Barth, Leipzig.
Facsimile by Maurizio Martino, Storrs-Mansfield (later Mansfield
Center), Connecticut.
Vols.
I (A - L) and II (M - Z). (1863), nd
[bought in 1996].
Vol.
III (1858-1883), edited by B. W. Feddersen & A. J. von Oettingen;
Parts
I (A - L) and II (M - Z). (1898), nd
[bought in 1998].
Vol.
IV (1883-1904), edited by Arthur von Oettingen;
Parts
I (A - L) and II (M - Z). (1904), nd
[bought in 1998].
[Vol.
V covers 1904-1922. Vol. VI covers
1922-1949. One can get I-VI on
microfiche. Vol. VIIa covers 1932-1953
and apparently comprises 5 volumes.
There is also a VIIa Supplement which gives material supplementary to
vols. I-VI.]
Prévost. Clever and Pleasant Inventions. (1584), 1998.
J.
Prévost. (La Première Partie des
Subtiles et Plaisantes Inventions, Contenant Plusieurs Jeux de Récréation. Antoine Bastide, Lyons, 1584. ??NYS.)
Translated by Sharon King as: Clever
and Pleasant Inventions Part One Containing Numerous Games of Recreations and
Feats of Agility, by Which One May Discover the Trickery of Jugglers and Charlatans. Hermetic Press, Seattle, 1998. [No Second Part ever appeared. Hall, OCB, pp. 43, 100 & 113.] This is apparently the first book primarily
devoted to conjuring. Only five copies
of the original are known. There was a
facsimile in 1987. My thanks to Bill
Kalush for bringing this work to my attention.
Price, Harry (1881-1948). See: HPL.
Problemes. 1612.
Claude‑Gaspar
Bachet (c1587-1638). Problèmes plaisans
& délectables qui se font par les nombres.
(1st ed., 1612); 2nd ed., 1624, P.
Rigaud, Lyon (for 1 & 2 ed.);
revised by A. Labosne, (3rd ed., 1874; 4th ed., 1879); 5th ed., 1884,
Gauthier‑Villars, Paris (for 3, 4, 5 ed.).) 5th ed. reprinted by Blanchard, Paris, 1959 et al., with a
Frontispiece portrait and an introduction by J. Itard, based on the article by
Collet and Itard cited in 1 below.
I
have now obtained a photocopy of the 2nd ed. and have examined a 1st ed. I had believed that Bachet added 10 problems
in the 2nd ed., but the additional section of 10 problems, beginning "S'Ensuivent quelques autres ..."
is already in the 1612 1st ed. In the
1612, there are two problems V, but in
1624, these are made into two parts of prob. V. However, he does extend the initial section of 22 problems to 25
problems, inserting the new material as problems 3, 16 and 21. Prob. 16 (1612) = 18 (1624) has additional
material. Also, Bachet greatly expands
his preliminary material on the properties of numbers from 14 to 52 pages, but
Labosne drops this. Otherwise, the
material seems identical and the main text seems pretty much identical with the
fifth edition except that orthography is modernised -- e.g. plaisans becomes
plaisants, mesme becomes même, luy becomes lui, etc. I have now compared the 3rd ed with the 5th ed and I could find
no differences between them -- though I didn't check every word. Labosne adds a Supplement of 15 problems,
four Notes and a table of contents.
Labosne's Préface given in the 5th ed. is for the 3rd ed. I will cite problem numbers and pages from
the 1st ed., 1612; 2nd ed., 1624 and the 5th ed., 1884 (1959 reprint),
e.g. Prob. XIX, 1612, 99-103. Prob. XXII, 1624: 170-173; 1884: 115‑117. I will generally not give problem titles as
they usually run to several lines. I
will denote Labosne's supplementary problems as Bachet-Labosne, 1874.
I
have seen a 4th ed. by Gauthier‑Villars, 1905, no editor named,
containing only 37 of the 50 problems in the 5th ed. A contemporary review by E. Lampe (Fortschritte der Math. 36
(1905) 300‑301) was also mystified by this edition. C&B list this ed.
Pŗthudakasvâmî or
Pŗthūdaka. See: Chaturveda.
Pseudo-dell'Abbaco. c1440.
This
is attributed to Paolo dell'Abbaco (sometimes called Dagomari)
(c1281-1367). Trattato
d'Aritmetica. (c1370, according to
Arrighi, but see below). Codex
Magliabechiano XI, 86 at Biblioteca Nazionale di Firenze. Edited by Gino Arrighi, Domus Galilaeana,
Pisa, 1964. Arrighi gives some black
& white reproductions of illustrations.
I examined this MS in Sep 1994 and found the illustrations are often
lightly coloured and that Arrighi's illustrations were probably made from
poorish photocopies -- the writing on the opposite side shows through much more
in several of his illustrations than it does in the originals. I have colour slides of 11 pages.
Warren
Van Egmond [New light on Paolo dell'Abbaco; Annali dell'Istituto e Museo di
Storia della Scienza di Firenze 2:2 (1977) 3‑21 and Van Egmond's Catalog
114-115] asserts this MS is a c1440 compilation, based on watermark evidence,
and doubts that it is due to dell'Abbaco, giving the author as
pseudo-dell'Abbaco.
Smith,
Rara, 435‑440 describes a different MS at Columbia, headed 'Trattato
d'Abbaco, d'Astronomia e di segreti naturali e medicinali', which he dates
c1339. Van Egmond, above, gives the
title of this as 'Trattato di tutta l'arte dell'abacho', but Van Egmond's
Catalog 254-255 describes it as Plimpton 167, a codex containing two
works. The first is the dell'Abbaco:
Trattato di tutta l'arte dell'abacho; the second is Rinaldo da Villanova:
Medichamento Generale, which has the title Trattato d'Abbaco, d'Astronomia e di
segreti naturali e medicinali added in a later hand. The first includes the Regoluzze which is sometimes cited
separately. This is quite a different
book than the c1440 Trattato. There is
a c1513 version at Bologna, MS B 2433, which is dated 1339 -- Dario Uri has
sent a CD of images of it. See the
entry for dell'Abbaco in 7.E.
Putnam. Puzzle Fun.
1978.
Graham
R. Putnam, ed. Puzzle Fun. Fun Incorporated, np [Chicago?], 1978.
Rara. 1970. David
Eugene Smith. Rara Arithmetica. (1908; with some addenda, 1910; Addenda,
1939, published both separately and with the 1910 ed.); 4th ed., combining the
original with both Addenda and with De Morgan's Arithmetical Books of 1847 and
a new combined index, Chelsea, 1970.
References are to the main entry of this. Check the index for references to the Addenda and to De Morgan.
Rational Recreations. 1824.
Rational
Recreations. Midsummer MDCCCXXIV. Knight and Lacey, London. This is a six part work, but is bound
together -- perhaps the parts were issued monthly. The parts are consecutively paginated. [Toole Stott 590. Toole
Stott 591 is 2nd ed, 1825 and 592 is 3rd ed, 1825, copublished in Dublin. Hall BCB 235, 236 are 1824 and 1825. C&B.
HPL. Not in Christopher. I have examined the BL copy.]
Recorde. First Part.
1543.
Recorde. Second Part. 1552.
Recorde-Mellis. Third Part.
1582.
Recorde
(or Record), Robert (1510?-1558). The
Grounde of Artes Teaching the worke and practice of Arithmetike. ....
The dating of this book is uncertain.
Smith, Rara, p. 526 records a 1540 edition. An edition by Reynold
Wolff, London, at the Bodleian (Douce R.301) has generally been dated as 1542
and there is a facsimile by Theatrum
Orbis Terrarum, Amsterdam & Da Capo Press, NY, 1969, with the date 1542.
There were reprints in 1543 and 1549.
However, the DSB entry for Recorde ignores the Smith 1540 edition
(presumably because it has not been confirmed) and says the 1542 is now dated
as 1550?, making the 1543 the first edition. This edition only contained material on whole numbers.
In
1552, Recorde added a Second Part dealing with fractions, so the earlier
material will be called the First Part.
At
some stage, John Dee augmented it, but it appears he simply made some revisions
and additions to the existing text without adding new topics. The Dee material was added in 1590 (Smith)
or 1573 (De Morgan).
In
1582, John Mellis added a Third Part, mostly on rules of calculation,
published by J. Harison & H. Bynneman, London.
By
1640, the title was changed to:
Record's Arithmetick, or, The Ground of Arts; Teaching The perfect Work
and Practice of ARITHMETICK, both in whole Numbers and Fractions, after a more
easie and exact form then in former time hath been set forth. Afterwards augmented by Mr. JOHN DEE. And since enlarged with a third part of
RULES of PRACTICE, abridged into a briefer method then hitherto ..., by JOHN
MELLIS.
By
1648, more material was added by Robert Hartwell. I have a 1668 edition which has a little more material by Thomas
Willsford, but these latter two extensions are of no interest to us. It is clear that the text was increased by
accretion, with only minor revisions of Recorde's text, which is generally
preserved in gothic (= black-letter) type, and this is indicated by Smith. So the presence of a problem in Part One or
Part Two or Part Three of the 1668 ed almost certainly indicates its presence in
the first version of these parts. This
is certainly true for Part One, as I have the facsimile to compare, and it is
confirmed by brief examination of a 1582 ed, though at the time, I was looking
at Part Three and did not know of the material in Part Two. I will cite the pages from my 1662 ed and
the side notes which are titles of the problems, and pages of any earlier
editions that I have seen.
Riccardi. Pietro Riccardi. Biblioteca Matematica Italiana dalla Origine della Stampa ai
Primi Anni del Secolo XIX.
G. G. Görlich, Milan, 1952, 2 vols. This work appeared in several parts and supplements in the late
19C and early 20C, mostly published by the Società Tipografica Modense, Modena,
1878-1893. For details, see in Section
3.B.
Riddle, Edward (1788-1854). See:
Ozanam‑Riddle.
The Riddler. See under Boy's Own Book.
Riese. Coss. 1524.
Adam
Riese (c1489-1559). Die Coss. German MS of 1524 found at Marienberg in
1855. Described and abstracted in
Programm der Progymnasial‑ und Realschulanstalt zu Annaberg 1860. Reprinted in 1892. My reference to this comes from Johannes Lehmann; Rechnen und
Raten; Volk und Wissen, Berlin (DDR), 1987, pp. 7‑14, esp. p. 13. I have since seen the Glaisher paper, op.
cit. in 7.G.1 under Widman, esp. p. 37.
Glaisher and Lehmann cite: Bruno Berlet; Adam Riese, sein Leben und
seine Art zu rechnen; Die Coss von Adam Riese; Leipzig & Frankfurt,
1892. Glaisher notes that this was a
pamphlet. BLLD provided a copy from
Biblioth. Regia Berolinen. G., but it was lacking a title page. It seems to have the title: Zur Feier des
vierhundertsten Geburtsjahres von Adam Riese.
It was printed by Königl. Universitätsdruckerei von H. Stürtz in
Würzburg. The Vorwort is dated 1892, but
only signed 'Der Verfasser' and his name does not appear anywhere except on the
spine of the library's cover. The
booklet has two parts.
Adam
Riese, sein Leben, seine Rechenbücher und seine Art zu rechnen (from the
Programm for 1855), pp. 1‑26.
This is a discussion of Riese's Rechnung, but it also mentions some
material from his 'grosse Rechenbuch' titled Rechenung nach der lenge, auf den
Linihen und Feder, written in 1525 but not published until 1550. Glaisher, loc. cit., p. 43, says he sees no
authority for the date of 1525 and assumes it was written c1550. (I have recently obtained a 1976 reprint of
this work, ??NYR)
Die
Coss von Adam Riese (with Abdruck der Coss) (from the Programm for 1860), pp.
27‑62. This gives many numbered
problems -- I will cite the problem number and pages from this.
There
is a recent facsimile of the MS which I have just received -- ??NYR.
Riese. Rechnung. 1522.
Adam
Riese (c1489-1559). Rechnung auff der
Linien unnd Federn ... Erfurt,
1522. I have two reprinted editions. See Rara 138-143.
Christian
Egenolph, Frankfurt, 1544. (Riese's
text is dated 1525. There is a
supplement on gauging by Erhard Helm, dated 1544.) Facsimile by Th. Schäfer, Hannover, 1978.
Christian
Egenolff's Erben, Frankfurt, 1574.
(Riese's text is dated 1525 and appears to be the same text as above,
but reset. The Supplement has further
material.) Facsimile by Th. Schäfer,
Hannover, 1987.
Ripley's Puzzles and Games. 1966.
Ripley's
Believe It or Not! Puzzles and
Games. Essandess Special Edition (Simon
& Schuster), New York, 1966. Much
of the material occurred in the various Ripley's Believe it or Not! books. Most of the material is well known, but
there are a number of unusual variations and some interesting incomplete
assertions and mistakes!
RM. (François-) Édouard (-Anatole) Lucas
(1842-1891). Récréations
mathématiques. (Gauthier‑Villars,
Paris, 4 vols, 1882, 1883, 1893, 1894, 2nd eds. of vol. 1, 1891, vol. 2,
1893) = Blanchard, Paris, (1960), 1975‑1977,
using 2nd ed. of vol. 1 and 1st ed. of vol. 2 (however, there seem to be very
few differences in the editions). I
will cite the Blanchard reprint volumes as RM1, etc. (Dates are as given in Harkin, op. cit. in 1 below, and on the
books, but I have seen other dates cited.)
Lucas;
L'Arithmétique Amusante, 1895, Note IV, pp. 210-260 gives various fragments of
material for further volumes which were found after Lucas's untimely
death. His draft Tables of Contents for
volumes 5 and 6 are given on p. 210, but no material exists for most of the
chapters.
RMM. Recreational Mathematics Magazine. Nos. 1 - 14 (Feb 1961 - Jan/Feb 1964). Quite a bit of the material in this was
abstracted and sometimes extended in: Joseph S. Madachy; Mathematics on
Vacation; Scribner's, NY, 1966; somewhat corrected as: Madachy's Mathematical
Recreations; Dover, 1979.
Rocha. Libro Dabaco. 1541. SEE:
Tagliente. Libro de Abaco. (1515).
1541.
Rohrbough. Lynn Rohrbough edited a series of 20
booklets, called Handy Series, Kits A-J and M‑V, for Cooperative
Recreation Service, Delaware, Ohio, during at least 1925-1941. Most of these were reprinted and revised
several times. Two of these are of
especial interest to us and are listed below.
Several others are cited a few times.
Rohrbough. Brain Resters and Testers. c1935.
Lynn
Rohrbough, ed. Brain Resters and Testers. Handy Series, Kit M, Cooperative Recreation
Service, Delaware, Ohio, nd [c1935].
Rohrbough. Puzzle Craft. 1932.
Lynn
Rohrbough, ed. Puzzle Craft Plans for Making and Solving 40 Puzzles in
Wire, Wood and String. Handy Series,
Kit U, Cooperative Recreation Service, Delaware, Ohio, (1930), 1932.
I
have another version of this, unfortunately undated, but apparently later, so I
have dated it as 1940s? Jerry Slocum
located this by tracking down Rohrbough's successors. The 1932 version has 39 puzzles in its index, but 4 more that
were not indexed and some Notes which are listed in the index of this version,
making 44 items in all. 13 items are
omitted and replaced by 5 in the present version, giving 36 indexed items. The outside of the back cover of the 1932
version shows a number of puzzles, including several not described in either
version of the booklet.
Rudin. 1936. Jacob
Philip Rudin. So You Like Puzzles! Frederick A. Stokes Co., NY, 1936.
SA. Scientific American, usually Martin
Gardner's Mathematical Games column.
For years from at least 1950, SA appeared in two volumes per year, each
of six issues. In year 1950 + n,
vol. 182 + 2n covers Jan-Jun
and vol. 183 + 2n covers
Jul-Dec. See also under Gardner.
Sanford. 1930. Vera
Sanford. A Short History of
Mathematics. Houghton Mifflin, Boston,
1930 & 1958. See also: H&S.
Santi. 1952. Aldo Santi. Bibliografia della Enigmistica. Sansoni Antiquariato, Florence, 1952. 2541 entries, often citing several editions
and versions, arranged chronologically from 1479 onward. My thanks to Dario Uri for providing
this. I will site item numbers. I have only entered part of the information
so far.
de Savigny. Livre des Écoliers. 1846.
M.
l'Abbé de Savigny. Le Livre des
Écoliers Illustré de 400
vignettes. Jeux. -- Récréations. Exercises. -- Arts utiles et
d'agrément. Amusements de la
science. Gustave Havard, Paris, nd
[dealer has written in 1846], HB. [This
is almost entirely the same as Boy's
Own Book, 1843 (Paris). Each has a few
sections the other does not.
de Savigny's illustrations seem to have been copied by a new hand,
generally simplifying a little. Many of
the copies have been done in reverse and this leads to one erroneous
chessboard. However, there is one diagram
in Boy's Own Book, 1843 (Paris), which looks like it was badly copied, so it is
possible that both these books are based on an earlier book.]
Schaaf. 1955-1978.
William
L. Schaaf. A Bibliography of
Recreational Mathematics. Vol. 1,
(1955, 1958, 1963); 4th ed., 1970. Vol.
2, 1970. Vol. 3, 1973. Vol. 4, 1978. National Council of Teachers of Mathematics, Washington, DC. See Schaaf & Singmaster in Section 3.B
for a Supplement to these.
Schott. 1674. Gaspare
Schott. Cursus Mathematicus. Joannis Arnoldi Cholini, Frankfurt,
1674. The material of interest is in
Liber II, Caput VI: De Arithmetica Divinatoria, pp. 57-60, and in Liber XXVI:
Algebra, Pars III: De Exercitatione Algebraicam, Caput I, II, IV, VI, pp.
551-563, and Pars V: Exercitationes Algebraicae, pp. 570-571.
Schwenter. 1636.
Daniel
Schwenter (1585-1636). Deliciæ Physico‑Mathematicae. Oder Mathemat‑ und Philosophische
Erquickstunden, Darinnen Sechshundert Drey und Sechsig, Schöne, Liebliche und
Annehmliche Kunststücklein, Auffgaben und Fragen, auf; der Rechenkunst,
Landtmessen, Perspectiv, Naturkündigung und andern Wissenschafften
genomēn, begriffen seindt, Wiesolche uf der andern seiten dieses blats
ordentlich nacheinander verzeichnet worden: Allen Kunstliebenden zu Ehren,
Nutz, Ergössung des Gemüths und sonderbahren Wolgefallen am tag gegeben Durch
M. Danielem Schwenterum. Jeremiæ
Dümlers, Nuremberg, 1636. [Note: the
text is in elaborate Gothic type with additional curlicues so that it is not
always easy to tell what letter is intended!]
Probably edited for the press by Georg Philip Harsdörffer.
Extended
to three volumes by Harsdörffer in 1651 & 1653, with vol. 1 being a reprint
of the 1636 vol. Vol. 2 & 3 have
titles as follows.
Delitiæ
Mathematicæ et Physicæ Der Mathematischen
und Philosophischen Erquickstunden
Zweiter Theil: Bestehend in
Fünffhundert nutzlichen und lustigen Kunstfragen / nachsinnigen Aufgaben / und
derselben grundrichtigen Erklärungen / Auss Athanasio Kirchero Petro Bettino,
Marion Mersennio, Renato des Cartes, Orontio Fineo, Marino Gethaldo, Cornelio
Drebbelio, Alexandron Tassoni, Sanctorio Sanctorii, Marco Marco, und vielen
anderen Mathematicis und Physicis zusammen getragen durch Georg Philip
Harsdörffern. Jeremia Dümlern,
Nürnberg, 1651.
Delitiæ
Philosophicæ et Mathematicæ Der
Philosophischen und Mathematischen
Erquickstunden / Dritter Theil:
Bestehend in Fünffhundert nutzlichen und lustigen Kunstfragen / und
derselben gründlichen Erklärung: Mit vielen nothwendigen Figuren / so wol in
Kupffer als Holz / gezieret. Und Aus allen neuen berühmten Philosophis und
Mathematicis, mit grossem Fleiss zusammen getragen. Durch Georg Philip
Harsdörffern. Wolffgang dess Jüngern
und Joh. Andreas Endtern, Nürnberg, 1653.
This
3 vol. version was reprinted in 1677 and 1692. Modern facsimile of the 3 vol. version edited by Jörg Jochen
Berns, Keip Verlag, Frankfurt Am Main, 1991, HB. See also MUS II 325-326.
Schott described this as a German translation of van Etten/Leurechon,
but this is quite wrong. V&T, p.
152, say it is 'partially derived from' van Etten/Leurechon. C&B list just the 1636, under
Schwenterum. P. 549 of vol. 1 is
misprinted 249 which indicates that it was the first issue of the 1st ed.
I
have not yet entered all the items from this.
The Secret Out. 1859.
The
Secret Out; or, One Thousand Tricks with Cards, And Other Recreations. Illustrated with over three hundred
engravings. And containing Clear and comprehensive explanations how to
perform with ease, all the curious card deceptions, and slight of hand tricks
extant. With an endless variety of
entertaining experiments in drawing room or white magic, including the
celebrated science of second sight.
Together with a choice collection of intricate and puzzling questions,
amusements in chance, natural magic, etc., etc., etc. By the Author of "The Sociable, or, One Thousand and One
Home Amusements," "The Magician's Own Book," etc, etc. Dick & Fitzgerald, NY, 1859.
[Toole
Stott 191, listing all versions under Cremer.
C&B, under Frikell, have New York, 1859. H. A. Smith dates this as 1869.]
The Secret Out (UK). c1860.
The
Secret Out or, One Thousand Tricks in Drawing-room or White Magic, with an
Endless Variety of Entertaining Experiments.
By the author of "The Magician's Own Book." Translated and edited by W. H. Cremer,
Junr. With three hundred
illustrations.
I
have seen several editions. [Toole
Stott lists all versions under Cremer.]
C&W
(based on the John Camden Hotten, London, 1871?) (with ads from Sep 1886 at back
and inscription dated 12 Oct 1887 on flyleaf).
[NUC; Toole Stott 192; C&B, under Cremer, have London & New
York, 1871, and under Frikell, have London, 1870.]
C&W,
nd [1871? -- NUC lists several dates; Toole Stott 1013 is 1870; Christopher 242
is 1878?].
John
Grant, Edinburgh, nd [1872 -- Toole Stott 1014, no ads].
All
these copies have identical green covers with five magic tricks on the cover.
[Toole
Stott 192 discusses the authorship, saying that Wiljalba (or Gustave) Frikell
is named on the TP of some editions, but that most of the tricks are taken from
the US ed of The Magician's Own Book.
In the US ed, The Author acknowledges his indebtedness to The Sociable
and The Magician's Own Book 'and many other works of similar character and value',
but claims 'that the greater portion of it [i.e. the book] is entirely
original.' In the Preliminary to the UK
ed he says he is indebted to '"Le
Magicien des Salons," revised by references to Decremps, Servière, Leopold, Besson, Kircher, Hildebrandt,
Ozanam, &c., &c.' though an
1874 ad by C&W indicates that it is translated from Le Magicien des
Salons. (This may be Le Magicien de
Société, Delarue, Paris, c1860, but see Rulfs, below.) The back of the TP of Bellew's The Art of
Amusing, Hotten, 1866?, says The Secret Out is a companion volume, just issued,
by Hermann Frikell. BMC & Toole
Stott say it is also attributed to Henry L. Williams. Toole Stott 481 cites a 1910 letter from Harris B. Dick, of the
publishers Dick & Fitzgerald, who thinks their version of The Secret Out
"was a reprint of an English book by W. H. Cremer" -- but there seems
to be no record of a UK ed before the US one.
NUC says an 1871 ed. gives author as Gustave Frikell. Christopher 240-242 are two copies from Dick
& Fitzgerald, c1859, and a C&W, 1878?
He repeats most of the above comments from Toole Stott and 242 cites the
Rulfs article mentioned under Magician's Own Book, above. Rulfs says The Secret Out is largely taken,
illustrations and all, from Blismon de Douai's Manuel du Magicien (1849) and
Richard & Delion's Magicien des salons ou le diable couleur de rose (1857
and earlier). H. A. Smith [op. cit.
under Magician's Own Book] says the first US ed is 1869 (this must be a misprint
or misreading -- though the date is a little hard to read in my copy, it is
clearly 1859) and the UK eds are basically a condensed version with a few
additions. He suggests the book is
taken from DeLion. He doubts whether
Cremer ever wrote anything. C&B,
under Gustave Frikell, say it is a translation of Richard & Delion. C&B, under Herrman Frikell, list London,
1870. C&B, under Secret, list New
York, nd. C&B also list it under
Williams, as London, 1871.]
[Toole
Stott 1056 is [Frikell, Wiljalba]; Parlor Tricks with Cards, ...; By the Author
of Book of Riddles and 500 Home Amusements, etc.; Dick & Fitzgerald, 1860?;
which is described as "abridged from The Secret Out. Toole Stott 547 and 1142 are two versions of
1863, but without the description of the author and hence listed anonymously.]
The
US and UK editions are fairly different.
The US ed has 382 sections, of which 157 (41%) are used in the UK
ed. The UK ed has 323 sections, so 51%
of it is taken from the US ed. The US
ed seems like a magic book, with chapters on Scientific Amusements and
Miscellaneous Tricks. The UK ed has
much less magic and tricks, adding other general tricks and a lot more
scientific tricks. The illustrations
for the common sections are not quite identical -- one was probably copied from
the other. The amount taken from The
Magician's Own Book and The Sociable is fairly small, perhaps 10% from each, in
either edition.
Shortz. Will Shortz's library or his catalogues
thereof, called "Puzzleana".
The most recent I have is: May
1992, 88pp with 1175 entries in 26 categories, with indexes of authors and
anonymous titles. Some entries cover
multiple items. In Jan 1995, he
produced a 19pp Supplement extending to a total of 1451 entries.
SIHGM. 1939-1941.
Ivor
Bulmer Thomas. Selections Illustrating
the History of Greek Mathematics. Loeb
Classical Library, 1939‑1941, 2 vols.
I will give volume and pages as in
SIHGM I 308‑309.
Simpson. Algebra.
1745.
Thomas
Simpson (1710-1761). A Treatise of
Algebra; Wherein the Fundamental Principles Are fully and clearly demonstrated,
and applied to the Solution of a great Variety of Problems. To which is added, The Construction of a
great Number of Geometrical Problems; with the Method of resolving the same
Numerically. John Nourse, London,
1745.
I
also have the 7th ed., with the slightly different title: A Treatise of Algebra. Wherein the Principles Are Demonstrated, And
Applied In many useful and interesting Enquiries, and in the Resolution of a
great Variety of Problems of different Kinds.
To which is added, The Geometrical Construction of a great Number of
Linear and Plane Problems; with the Method of resolving the same
Numerically. The Seventh Edition,
carefully Revised. F. Wingrave,
Successor to Mr. Nourse, London, 1800.
I
have now seen a 6th ed., F. Wingrave, 1790 and it appears identical to the 7th
ed. Both have an Author's Preface to
the Second Edition.
I
will give the 1745 details with the 1790/1800 details in parenthesis like
(1790: ...).
SLAHP. 1928. Sam Loyd
Jr. (1873-1934) Sam Loyd and His
Puzzles. An Autobiographical
Review. Barse & Co., NY, 1928. (This contains somewhat more original
material than I expected, but he claims that he devised a lot of his father's
puzzles given in the Cyclopedia, OPM, and even earlier.)
Slocum. Compendium.
1977.
Jerry
(= G. K.) Slocum. Compendium of
Mechanical Puzzles from Catalogues.
Published by the author, Beverly Hills, 1977, 57pp. This is a compendium of illustrations and
descriptions from 30 catalogues. The
earliest ones are Bestelmeier, 1793, 1807 (Jacoby edition); The Youth's
Companion, 1875; Montgomery Ward, 1886, 1889, 1903, 1930; Peck & Snyder,
1886; Joseph Bland, c1890. Others of
some interest are: Gamage's, 1913, c1915, c1928; Johnson Smith, 1919, 1935,
1937, 1938, 1942.
Slocum & Gebhardt -- see
under Catel.
SM. Scripta Mathematica.
Smith, David Eugene (1860‑1944). See:
Rara and the following four
items.
Smith. History. 1923.
David
Eugene Smith. History of
Mathematics. Two vols. (Ginn, NY, 1923) = Dover, 1958.
Smith. Number Stories. 1919.
David
Eugene Smith. Number Stories of Long
Ago. NCTM, (1919), reprinted 1968? Chaps. IX and X.
Smith. Source Book. 1929.
David
Eugene Smith. A Source Book in
Mathematics. Two vols. (1929)
= Dover, 1959.
Smith & Mikami. 1914.
David
Eugene Smith & Yoshio Mikami. A History of Japanese Mathematics. Open Court, Chicago, 1914.
The Sociable. 1858.
The
Sociable; or, One Thousand and One Home Amusements. Containing Acting Proverbs; Dramatic Charades; Acting Charades,
or Drawing-room Pantomimes; Musical Burlesques; Tableaux Vivants; Parlor Games;
Games of Action; Forfeits; Science in Sport, and Parlor Magic; and a Choice
Collection of Curious Mental and Mechanical Puzzles; &c,&c. Illustrated with nearly three hundred
engravings and diagrams, the whole being a fund of never-ending
entertainment. By the author of
"The Magician's Own Book."
(Dick & Fitzgerald, NY, 1858 [Toole Stott 640 lists this as
anonymous; C&B list it under the title, with no author]); G. G. Evans,
Philadelphia, nd, but the back of the title gives ©1858 by Dick & Fitzgerald, NY, so this copy seems to be a
reprint of the 1858 book. Cf. Book of
500 Puzzles for discussion of possible authorship. The Preface here says that most of the Parlor Theatricals are by
Frank Cahill and George Arnold -- Toole Stott opines that this reference led
Harry Price to ascribe this and the related books to these authors. My thanks to Jerry Slocum for providing a
copy of this.
The
entire section Puzzles and Curious Paradoxes, pp. 285-318, is identical to the
same section in Book of 500 Puzzles, pp. 3-36.
Rulfs (see under Status of the Project in the Introduction) says this
draws on the same sources as Magician's Own Book, with more taken from Endless
Amusement and Parlour Magic.
See
also: Book of 500 Puzzles; Boy's Own Conjuring Book; Illustrated Boy's Own Treasury; Indoor and Outdoor; Landells: Boy's Own Toy-Maker; The Secret Out; Hanky Panky.
SP. Prefixed by ?? is a flag to check
spelling.
"The Sphinx". See:
Lemon.
Sridhara. c900. Śrīdharācārya. Patiganita (= Pâţîgaņita
[NOTE: ţ, ņ denote
t, n with underdot.]). c900.
Transcribed and translated by Kripa Shankar Shukla. Lucknow Univ., Lucknow, 1959. The text is divided into verses and
examples, separately numbered by the editor.
I will cite verse (v.) and example (ex.) and the page of the English
text. The editor has appended answers
on pp. 93‑96, some of which were given by an unknown commentator. (I have seen this dated 8C, which would put
it before Mahavira -- ??)
SSM. School Science and Mathematics.
Struik. Source Book. 1969.
D.
J. Struik (1894- ), ed. A Source Book in Mathematics 1200‑1800. Harvard Univ. Press, 1969.
Sullivan. Unusual.
1943 & 1947.
Orville
A. Sullivan. Problems involving unusual
situations. SM 9 (1943) 114‑118
& 13 (1947) 102‑104.
(Previously listed in Section 2 below.)
Suter. 1900-1902. Heinrich
Suter. Die Mathematiker und Astronomen
der Araber und Ihre Werke. (AGM 10
(1900) & 14 (1902)); reprinted by APA -- Academic Publishers Associated,
Amsterdam, 1981. See also: H. P. J. Renaud; Additions et corrections à
Suter "Die Mathematiker und Astronomen der Araber"; Isis 18 (1932)
166-183.
S&B. 1986. Jerry
(= G. K.) Slocum & Jack Botermans.
Puzzles Old & New -- How to Make and Solve Them. Univ. of Washington Press, Seattle,
1986. Slocum had produced a detailed
index for this and has extended it to a joint index with New Book of Puzzles.
Tabari. Miftāh al-mu‘āmalāt. c1075.
Mohammed
ibn Ayyūb Ţabarī [NOTE:
Ţ denotes a T
with a underdot.]. Miftāh
al-mu‘āmalāt. c1075. Ed. by Mohammed [the h
should have an underdot] Amin Riyāhi [the h should have an
underdot], Teheran, 1970. ??NYS --
frequently cited and sometimes quoted by Tropfke and others.
Tagliente. Libro de Abaco. (1515). 1541.
Girolamo
[& Giannantonio] Tagliente. Libro
de abaco che insegnia a fare ogni raxone marcadantile & apertegare le terre
con larte di la giometria & altre nobilissime raxone straordinarie cō
la tarifa come raspondeno li pexi & monete de molte terre del mondo con la
inclita citta de. Venetia. El qual
Libro se chiama Texauro universale ...; Venice, 1515. See Rara 114-116, 495, 511‑512, which seems to confuse this
with another work whose title starts:
Opera che insegna .... Riccardi
lists 27 editions of this and three editions of the other work. He says Boncompagni has studied this work
and found that Giovanni Rocha made some corrections and that several editions
have only his name, so it is sometimes catalogued under Rocha -- cf below. Smith mentions 25 editions under
Tagliente and says Riccardi mentions
11 others. Van Egmond's Catalog 334-344 lists 31
editions to 1586. It is clear that this
was a major book of its time. I have
briefly looked at a few examples and they seem to have the same material,
though the woodcuts were often changed.
I
have examined: Giovanne Rocha; Libro Dabaco che insegna a fare ogni ragione
mercadãtile: & a ptegare le terre cõ larte di la geometria: & altre
nobilissime ragione straordinarie cõ la Tariffa come respõdeno li pesi &
mone de molte terre del mõdo con la inclita di Vinegia. El qual Libro se chiama Thesauro universale. Venturino Rossinello, Venice, 1541. Smith, Rara, p. 529, only records a 1550 ed.
printed by Giovanni Padovano in Venice.
The Crawford Collection has 1544 & 1550. Not under Rocha in Riccardi.
I found this in the Turner Collection at Keele as A4.32, but it is not
under Rocha in Hill's Catalogue (listed in 3.B), but is under Tagliente. It has nice woodcut(?) illustrations in the
text -- see Rara 512 for an example.
Having first found this book on the shelf at the Turner Collection, I
originally thought it was by Rocha and hence an extraordinarily rare book. I am grateful to Bill Kalush for identifying
this as a version of Tagliente and for pointing out its importance. Cf Van Egmond's Catalogue 338, item 15.
Tartaglia, Nicolo (or Niccolò)
(c1506-1559). See: General Trattato.
Thomas, Ivor B. See:
SIHGM.
Tissandier. Récréations Scientifiques. 1880.
Gaston
Tissandier. Les Récréations
Scientifiques ou L'Enseignement par les Jeux.
G. Masson, Paris, (1880); 2nd
ed., 1881; 3rd ed., 1883; (4th ed., 1884); 5th ed., 1888; (6th ed.,
1893; 7th ed., 1894.) I have seen 3rd ed., 1883, and I have 2nd
ed., 1881, & 5th ed., 1888.
Tissandier was editor of La Nature and the articles often have fine
illustrations by L. Poyet and others, frequently copied elsewhere. See Tom Tit for more of Poyet's work. I have seen the date of the first two
editions as 1881 & 1882, but the Avertissement of the 2nd ed. says the 1st
ed. was Nov 1880 and the Avertissement is dated Apr 1881. [C&B only list 1881.]
Tissandier. Popular Scientific Recreations. 1881.
Translation
and enlargement of the above. Ward
Lock, London, ([1881, 1882]); New enlarged ed., nd [1890, 1891]. [Not clear which French edition the
translation is based on. The new
enlarged ed. contains a Supplement on pp. 775‑876 which includes material
which is in the 5th French ed. of 1888, but not in the 3rd French ed. of 1883,
so it seems the the main text is c1885 and the supplement is c1890. C&B list a London edition, nd,
780pp.] The index refers to a puzzle of
knots and cords on p. 775 which is not present. Most of the Supplement appeared as a series of Scientific
Amusements in (Beeton's) Boy's Own Magazine from 1889 -- I have vol. 3 (1889)
which has the first 10 articles, comprising 62 pages and 67 problems.
Some
of the material appeared in: Marvels of Invention and Scientific Puzzles. Being
A Popular Account of Many Useful and Interesting Inventions and
Discoveries. Ward, Lock, & Co., nd
[c1890]. My copy has no authors listed,
but Jerry Slocum has a copy with Tissandier and Firth on the TP, though it is
difficult to see what Firth could have done to warrant his inclusion. This consists of Chapters 56-60, pp.
726-774, and Chap. 32, pp. 448-465, of Popular Scientific Recreations, set on
smaller pages, plus a few extra items:
An economical mouse-trap (pp. 57-58);
Flying bridges (pp. 64-66);
Performing fleas (pp. 77-79);
Knots and cords and A Curious Toy (pp. 83-85). This last must be what was on pp. 775-776 of
the earlier edition of Popular Scientific Recreations and the other material
seems to have come from some edition of that work or from the articles in
(Beeton's) Boy's Own Magazine. I won't
bother to cite this version.
Tom Tit. 1890-1893?
Arthur
Good [= "Tom Tit"]. La
Science Amusante. 3 vols.,
Larousse, Paris, 1890, 1892, 1893? [I
have seen the dates given as 1889, 1891, 1893, but the Introductions are dated
as I have given, the last being Dec 1893.]
The material originally appeared in the magazine L'Illustration with
classic engravings by Poyet which have been often reproduced, e.g. in: Beeton's Boy's Own Magazine; The Boy's Own Paper; Kolumbus‑Eier (1890, 1976) --
translated as: Columbus' Egg (1978).
Arthur Good's name is clearly English and I have wondered if the
articles were written originally in French or if they were translated from his
English.
I
have the three volume set and five one volume selections/adaptations. I will give references to these by the
initials shown below.
C. François Caradec, ed. La Science Amusante -- 100 Experiences de Physique. Les Editions 1900 [no accent], Paris,
1989. [This consists of all of Vol. 1,
reordered, but otherwise little changed.
Caradec's Preface gives a little about the author. The illustrations are a bit dark.]
H. Magic at Home.
A Book of Amusing Science.
Annotated translation of La Science Amusante, vol. 1, by Prof.
Hoffmann. Cassell, 1891. Cf. VBM below.
VBM. The Victorian Book of Magic Illustrated or Professor
Hoffman's [sic] Curious & Innocent Diversions for Parlour & Refined
Gatherings. Selected [from the above]
& with a note to readers by C. Raymond Reynolds. (Stephen Greene Press, Japan, 1969); Hugh Evelyn, London, nd
[c1970]. [This has 26 of the items in
Vol. 1. Illustrations are small but
good.]
K. Tom Tit.
Scientific Amusements. Selected
and translated by Cargill G. Knott.
Nelson, nd [1918]. [This
contains 178 items, mostly from Vols. 2 & 3, but Knott has added a few
others, possibly taken from Tom Tit's later articles? Knott has also extended some items. Sadly, the illustrations were poorly redrawn for this edition.]
R&A. David Roberts & Cliff Andrew, eds. 100 Amazing Magic Tricks. Cape, London, 1977. [Selections from all three volumes. Although this refers to the original books
and L'Illustration, it avoids mentioning the original author's name! Illustrations are very good.]
[I
have also seen an Italian translation.
There was a US ed., trans. by Camden Curwen & Robert Waters; Magical
Experiments or Science in Play; (© Worthington, 1892); David McKay,
Philadelphia, 1894, 329 pp. [Christopher 398.]]
Todhunter. Algebra, 5th ed. 1870.
Isaac
Todhunter (1820-1884). Algebra For the Use of Colleges and Schools. With Numerous Examples. Macmillan, (1858; 5th ed, 1870); new
edition, 1879, HB. [1st ed was 1858;
2nd, 1861; 5th, 1870. The Preface in my
1879 copy is dated 1870 and says the work has been carefully revised, with two
chapters and 300 miscellaneous examples added, so it was quite different than
previous editions and I will date citations as 1870.]
Tonstall. De Arte Supputandi. 1522.
Cuthbert
Tonstall [often spelled Tunstall] (c1474-1559). De Arte Supputandi Libri Quatuor. ([With Quattuor] Richard Pynson, London, 1522 -- the first
arithmetic printed in England, with TP engraved by Holbein.) I have seen: Robert Stephan, Paris, 1538.
Though the TP and pagination are different, Smith, Rara, gives no
indication that the 1538 text is any different than the 1522, so I will cite
this as 1522. Most citations are to
Book III, whose problems are numbered.
See Rara 132-136.
Toole Stott. 1976-1978.
Raymond
Toole Stott. A Bibliography of English
Conjuring 1581‑1876. 2 vols., published by the author, Derby,
1976, 1978; distributed by Harpur & Sons, Derby. 1414 entries. References
are to the item number.
TP Title Page.
Tropfke. 1980. Johannes
Tropfke, revised by Kurt Vogel, Karin Reich and Helmuth Gericke. Geschichte der Elementarmathematik. 4th ed., Vol. 1: Arithmetik und Algebra. De Gruyter, Berlin, 1980. (Prof. Folkerts says (1994) that vol. 2 is
being edited.)
[The
1st ed. was De Gruyter?, Leipzig, 1902, 2 vols. 2nd ed., De Gruyter, Berlin & Leipzig, 1921-1924, 7
vols. 3rd ed., De Gruyter, Berlin &
Leipzig, 1930-1940, vols. 1-4 (the MSS of the remaining volumes were destroyed
in 1945).]
Tunstall, Cuthbert -- see: Tonstall, Cuthbert.
Turner. The Turner Collection, formerly at
University of Keele. Sadly this
collection was secretly sold by the University in 1998 and has now been
dispersed. A useful catalogue was prepared: Susan Hill; Catalogue of the Turner
Collection of the History of Mathematics Held in the Library of the University
of Keele; University Library, Keele, 1982.
UCL. University College London or its Library,
which includes the Graves Collection, cf Graves.
Uncle George. See:
Parlour Pastime.
Unger. Arithmetische Unterhaltungen.
1832.
Ephraim
Salomon Unger. Arithmetische
Unterhaltungen, bestehend in einer systematisch geordneten Sammlung von mehr
als 900 algebraischen Aufgaben, verbunden mit einer Anleitung, diese Aufgaben
mittelst der einfachsten Regeln der Arithmetick zu lösen. (Erfurt, 1832, 4 + 253 pp., ??NYS); 2nd ed., Keysersche Buchhandlung, Erfurt,
1838, 10 + 268pp. MUS 166 only mentions
the 1838 ed. but a copy of the 1st ed. was advertised by Sändig in Jun 1997. I haven't seen any other reference to this
work. In general, this book believes in
beating problems to death -- each type of problem is done several times. E.g. there are 10 problems of the Chinese
Remainder Theorem with two moduli.
Hence I will generally not describe all the problems.
van Etten. See:
Etten, above.
Van Egmond's Catalog.
Warren
Van Egmond. Practical Mathematics in
the Italian Renaissance: A Catalog of Italian Abbacus Manuscripts and Printed
Books to 1600. Supp. to Annali dell'Istituto
e Museo di Storia della Scienza (1980), fasc. 1. = Istituto e Museo di Storia della Scienza, Monografia N. 4. Florence, 1980. I have consulted this for (almost?) all MSS cited in these
Sources and have made a number of changes of dates and even authorship based on
it. Like any such catalog, there are
some omissions and errors, but it is by far the most authoritative listing of
the material and I have adopted his dates and attributions -- cf Benedetto da
Firenze, P. M. Calandri, dell'Abbaco, pseudo-dell'Abbaco.
Vinot. 1860.
Joseph
Vinot. Récréations Mathématiques Nouveau Recueil de Questions Curieuses et
Utiles Extraites des Auteurs Anciens et Modernes. Larousse & Boyer, Paris, 1860, (3rd ed., Larousse, Paris,
1893; 1898; 1902; 6th ed., nd
[1911]). I have found no difference
between the 1st and 6th eds -- indeed I have found a simple typographical error
repeated.
Vogel, Kurt (1888-1985). See:
AR; BR; Chiu Chang Suan Ching; Columbia Algorism; Tropfke; items in 7.K
under al‑Khwârizmî; items in
7.P.7 and 7.R.2 under Fibonacci.
Vyse. Tutor's Guide. 1771?
Charles
Vyse (fl. 1770-1815). The Tutor's
Guide, being a Complete System of Arithmetic; with Various Branches in the
Mathematics. In Six Parts, .... To which is added, An Appendix, Containing
Different Forms of Acquittances, Bills of Exchange, &c. &c. The whole being designed for the Use of
Schools, .... The Eighth Edition,
corrected and improved, with Additions.
G. G. J. and J. Robinson, London, 1793, HB. 12 + 324 pp. [1st ed. was
1770 or 1771; 2nd ed., 1772; 4th ed., 1779; 6th ed., 1785; 1790; 8th ed., 1793; 1799; 12th ed., 1804; 13th ed., 1807; 14th ed., 1810; 15th ed.,
1815; 1817; 16th ed., 1821.] Since
books of this nature rarely had major changes, I will date this as 1771? until
I see further editions.
Charles
Vyse (fl. 1770-1815). The Tutor's
Guide, being a Complete System of Arithmetic; with Various Branches in the
Mathematics. In Six Parts, .... To which is added, An Appendix, Containing
Different Forms of Acquittances, Bills of Exchange, &c. &c. The whole being designed for the Use of
Schools, .... 10th ed., ed. by J.
Warburton. S. Hamilton for G. G. and J.
Robinson, London, 1799. 12 + 335
pp. [Dedication removed; Preface by the
Editor and references to the Key added.
Though the text is reset with one or two more lines per page, the text
seems to be preserved, though the editor has added substantial footnotes in
places.]
The
Key to the Tutor's Guide; or the Arithmetician's Repository: containing
Solutions to the Questions, &c. in the Tutor's Guide, with references to
the pages where they stand. To which
are added Some Useful Rules, &c.
Likewise An Appendix; showing the Combination of Quantities; The
different Ways they may be varied; with the method of filling the magic
squares, &c. .... Eighth edition; Carefully revised,
corrected, and augmented. G. & J.
Robinson, London, 1802. 8 + 370 pp +
2pp publisher's ads. [1st ed.,
1773; 3rd ed., 1779; 4th ed., 1785; 7th ed., 1799; 8th ed.,
1802; 9th ed., 1807; 11th ed., 1818.]
Despite
the claim in the title of the Key, the references to the answers are in the
text of the 10th ed at the beginning of each set of exercises. There is no mention of the Key in the 8th ed
of the book. I have the 8th and 9th
eds. of the Key and have not seen any difference between them -- indeed I have
found a common misprint -- and they give the answers to the 10th ed. of the
book, on the pages cited in the book, so I assume they are essentially
identical to the 7th ed. of the Key.
V&T, 1952. Volkmann, Kurt &
Tummers, Louis. Bibliographie de
la Prestidigitation Tome I Allemagne et Autriche. Cercle Belge d'Illusionnisme, Bruxelles,
1952. With a 2 page list of
Bibliographies of Conjuring.
Walkingame. Tutor's Assistant.
Francis
Walkingame. The Tutor's Assistant;
Being a Compendium of Arithmetic, and a Complete Question-Book. ...
To which are added, A new and very short Method of extracting the Cube
Root, and a General Table for the ready calculating the Interest .... The Fifteenth Edition. Printed for the Author, London (Great
Russel-Street, Bloomsbury), 1777.
[First ed. was 1751. There were
about a hundred editions in all. See
Wallis's article on this book in MG 47 (No. 361) (1963) 199-208.]
The
Tutor's Assistant. Being a Compendium
of Arithmetic, and a Complete Question-Book.
... To which are added, A new
and very short Method of extracting the Cube Root, and a General Table for the
ready calculating the Interest .... The
Twentieth Edition. Printed for the
Author, London (Kensington), 1784.
Appears to be identical to the 15th ed., except for resetting and some
small changes, both corrections and errors, so I won't cite this separately.
The
Tutor's Assistant, being a Compendium of Arithmetic, and Complete
Question-Book; ...; to which is added, A Compendium of Book-keeping, by Single
Entry, by Isaac Fisher. Thomas
Richardson, Derby; Simpkin, Marshall & Co., London, 1835. [The first Derby edition by Fisher was
1826.]
The
Tutor's Companion; or, Complete Practical Arithmetic. To which is added A Complete Course of Mental Arithmetic, ..., by
Isaac Butler. Webb, Millington, and
Co., London, 1860.
The
editions are pretty similar, but the interesting collection of problems at the
end is much shortened in the 1835 ed. and a few are omitted in the 1860
ed. Consequently I will assume the
problems date from 1751, unless they vary in some important way. I also have a Key to Walkingame from 1840,
but it does not correlate with any of the editions I have!
Wallis. The Wallis Collection of early English
mathematics books. Gathered by Peter J.
Wallis and left to the University of Newcastle upon Tyne in 1992. A typical catalogue entry is Wallis 227 CAR and there are special sections for Newtoniana and Record(e).
Week‑End. 1932. Hubert
Phillips. The Week‑End Problems
Book. Nonesuch Press, London, 1932.
Wehman. New Book of 200 Puzzles. 1908.
Wehman
Bros.' New Book of 200 Puzzles. Wehman Bros., 126 Park Row, NY, 1908. Largely copied from various 19C works: Boy's Own Book, Magician's
Own Book, The Sociable, sometimes with typographical omissions. I only count 130 puzzles! There seems to have been a Johnson Smith
reprint at some time.
Wells. 1698. Edward Wells. Elementa Arithmeticæ Numerosæ et
Speciosæ. In Usum Juventutis
Academicæ. At the Sheldonian Theatre
(i.e. OUP), Oxford, 1698. (My copy was
previously in the Turner Collection, Turner D1.1.) All the material cited is in Appendix Posterior: Viz. Problemata
sive Quæstiones ad exercendas Regulas Arithmeticæ.
Western Puzzle Works. 1926.
Western
Puzzle Works, 979 Marshall Avenue, St. Paul, Minnesota. 1926 Puzzle Catalogue. Photocopy provided by Slocum. Unpaginated, 8pp.
Williams. Home Entertainments. 1914.
Archibald
& F. M. Williams. Home
Entertainments. The Hobby Books, ed. by
Archibald Williams. Nelson, London, nd
[1914 -- BMC].
Williams, Henry Llewellyn, Jr.
(1842-??). Books by Frikell?,
particularly Magician's Own Book, are often attributed to him. [C&B, under Williams, Henry Llewellyn ("W. Frikell") lists:
Hanky Panky; Magician's Own
Book, London & New York; (Magic No
Mystery); The Secret Out and says to also see Cremer.]
Williams, J. L. See:
Boy's Own Book, 1843 (Paris) edition, which lists him as author.
Wilson, Robin J. See:
BLW.
Wingate/Kersey. 1678?.
Edmund
Wingate (1596-1656). Mr. Wingate's
Arithmetick; Containing A Plain and Familiar Method For Attaining the Knowledge
and Practice of Common Arithmetick. (1629.) The Seventh(?) Edition, very much
Enlarged. First Composed by Edmund
Wingate, late of Gray's-Inn, Esquire.
Afterwards, upon Mr. Wingate's Request, Enlarged in his Life-time: Also
since his Decease carefully Revised, and much Improved; ... By John Kersey,
late Teacher of the Mathematicks. My
copy is lacking the TP and pp. 345+, but it appears to be identical to The
Tenth Edition; J. Philips, J. Taylor & J. Knapton, London, 1699 --
though reset, it has the same pagination throughout except for the Dedication. A Librarian's note suggests my earlier
version is the 7th ed. of 1678. This
would be the fourth and last of Kersey's versions. Kersey began editing from the second or third (1658) ed. and did
four versions, the last in 1678.
The
material of interest is almost all in Chapter 10 of Kersey's Appendix: A
Collection of subtil Questions to exercise all the parts of Vulgar Arithmetick;
to which also are added various practical Questions about the mensuration of
Superficial Figures and Solids, with the Gauging of Vessels, pp. 475-527, 75
questions. There are a few further
items in Chapter 11: Sports and Pastimes, pp. 528-544, 7 problems. Chapter 11 is not clearly marked as being by
Kersey in these copies, but is so marked in later editions and it is pretty
clear that the entire Appendix is due to Kersey. At the end, he says he has taken the problems of Chap. 11 from
Bachet's Problemes.
I
have seen a 14th ed. of 1720 which has the same text, reset and repaginated, with
some supplementary material by George Shelley.
I have also seen a 19th ed. of 1760 which has been considerably
reorganized by James Dodson. The two
chapters of puzzle problems have become Chapters XLIII and XLIV and the
material has been changed, generally omitting some problems of interest and
only adding two.
Winning Ways. 1982.
Elwyn
R. Berlekamp, John H. Conway & Richard K. Guy. Winning Ways for Your Mathematical Plays: Vol. 1: Games in
General; Vol. 2: Games in Particular. 2
vols, Academic Press, NY, 1982.
Witgeest. Het Natuurlyk Tover-Boek. 1686.
Simon
Witgeest. Het Natuurlyk Tover-Boek,
Of't Nieuw Speel-Toneel Der Konsten.
Verhandelende over de agt hondert natuurlijke Tover-Konsten. so uyt de
de Gogel-tas, als Kaartspelen, Mathematische Konsten, en meer andered
diergelijke aerdigheden, die tot vermaek, en tijtkorting verstrecken. Mitsgaders een Tractaet van alderley
Waterverwen, en verligteryen; Als oock
Een verhandelinge van veelderley Blanketsels.
Om verscheyde wel-ruykende Wateren, Poederen en Balsemen, als ook
kostelijke beeydselen, om het Aensicht, Hals en Handen, wit en sagt te maecken,
door Simon Witgeest. Jan ten Hoorn,
Amsterdam, 1686. ??NYS -- some
photocopies sent by Jerry Slocum.
'Boek' is 'Boeck' on the frontispiece and running heads. This is a much expanded and retitled 3rd
edition of Witgeest's 1679 work. The
new material is stated to already be in the 2nd ed. of 1682.
[There
were many later editions: 1695; 1698;
Ten Hoorn, 1701; G. de Groot Keur, Amsterdam, 1725 (10th ed.),
1739; 1749; 1760; Amsterdam,
1773; 1815 [Christopher 1098-1099, C&B, HPL]. It was translated as: Naturliches Zauber-Buch (or Zauber=Buch)
oder neuer Spiel-Platz der Künste; Hoffmanns sel Wittw. & Engelbert Streck,
Nürnberg, 1702. There were later
editions (all in Nürnberg?) of 1713 (in 1 or 2 vols); Hoffmann, Nürnberg, 1718;
1730; 1739; 1740; 1745; 1753; Johann Adam Stein, Nürnberg, 1755; 1760-1762;
1763; 1766; 1781; 1786; 1798 and a Lindau reprint of 1978 [Christopher 1092-1097, C&B, V&T],
all apparently based on the 1682 Dutch ed.]
Witgeest. Het Nieuw Toneel der Konsten. 1679.
Simon
Witgeest. Het Nieuw Toneel der Konsten,
Bestaande uyt Sesderley Stukken: het eerste, handelt van alderley aardige
Speeltjes en Klugjes: het tweede, van de Verligt-konst in 't Verwen en
Schilderen: het derde, van het Etzen en Plaat-shijden: het vierde, van de
Glas-konst: het vijfde, heest eenige aardige remedien tegen alderley Ziekten:
het sesde, is van de Vuur-werken. Uyt
verscheyde Autheuren by een vergadert, door S. Witgeest, Middel-borger. Jan ten Hoorn, Amsterdam, 1679; facsimile with epilogue by John Landwehr, A.
W. Sijthoff's Uitgeversmaatschappij N. V., Leiden, 1967 (present from Bill
Kalush).
There
were many later editions, but Nanco Bordewijk has examined these and discovered
that the 3rd ed. of 1686 (I can't recall if he saw the 1682 ed.) was so
extensively revised and extended as to constitute a new book, and it has the
different title given in the previous entry.
(Other sources indicate these revisions are already in the 2nd ed. of
1682.) Landwehr has written a
bibliographical article on this book -- ??NYR.
Wood. Oddities. Clement
Wood. A Book of Mathematical
Oddities. Little Blue Book 1210. Haldeman-Julius, Girard, Kansas, nd [1927].
Young Man's Book. 1839.
Anonymous. The Young Man's Book of Amusement. Containing the Most Interesting and
Instructive Experiments in Various Branches of Science. To Which is Added All the Popular Tricks and
Changes in Cards; and the Art of Making Fire Works. William Milner, Halifax, 1844, HB. 2 + 384 pp + folding plate (originally a frontispiece). My copy has a number of annotations as
though in preparation for another edition.
[Hall BCB 322. Toole Stott
751. BCB 320-323 are 1839, 1840, 1844, 1848. Heyl 358-360 are 1846, 1846 (Milner & Sowerby ??), 1850. Toole Stott 749-752, 1216 are 1839, 1840, 1844, 1846, 1850. Christopher 1111-1113 are 1839, 1846, 1859
(Milner & Sowerby). All these are
apparently the same except for the publisher's name change.]
Young World. c1960.
Young
World Productions. Tricks and
Teasers. 303 Gags Games Tongue Twisters Problems Tricks. Young World Productions, London, nd
[inscribed 1965 on first page, so probably c1960; BLC-Ø].
536. H. E. Dudeney. 536 Puzzles and Curious Problems. Ed. by M. Gardner. Scribner's, NY, 1967.
(This consists of almost all the puzzles from Modern Puzzles (MP) and
Puzzles and Curious Problems (PCP).)
[There is also a Fontana, London, 1970, edition in two volumes: Puzzles
and Curious Problems (258 problems); More Puzzles and Curious Problems (261
problems).]
?? indicates uncertainty and points
where further work needs to be done.
‑ BC, e.g. ‑330 is 330 BC
and ‑5C is
5th century BC.
¹ Inequality or
incongruence (mod m). (My word
processor does not have an incongruence sign.
I may change this in Word using an Arial character.)
Ø Nothing, used after catalogues,
etc., to indicate that I have looked in that catalogue and found no entry. E.g.
BLC-Ø.
SOME OTHER RECURRING REFERENCES
Details
of these items are given under the first reference in Sources. Later references often cite the first
reference. I tend to make entries below
when I use the item, but sometimes I have entered the entry long after the
first usage of the item, and I haven't searched the text for other (perhaps
entered long ago) appearances of these items.
For: See Sections:
Anon: Home Book ..., 1941 4.B.1, 4.B.3, 5.U, 6.AO, 6.BF.4,
7.B, 7.AT, 9.E.1
Anon: Treatise, 1850 7.H, 7.P.6, 7.S, 7.X,
10.A, 10.R
Allen, 1991 5.B,
5.I.1, 5.N, 6.L, 7.E, 7.I, 7.P.1, 7.R.3, 7.AC.6, 7.AH, 7.AL, 9.B, 10.A.4
Always: More Puzzles to Puzzle
You, 1967
6.BF.4,
Always: Puzzles for Puzzlers,
1971
5.D.2,
5.D.5, 7.G.1, 7.AC.1, 9.J, 10.I, 10.K
Always: Puzzles to Puzzle You,
1965
5.K.2,
5.W.1, 5.X.2, 7.AC.3, 7.AC.6, 7.AS,
Always: Puzzling You Again, 1969
5.C,
6.BD, 7.AH,
Ananias of Shirak, c640 7.E, 7.H, 10.A
André, 1876 7.H,
7.H.1, 7.S.1, 7.S.2, 7.AF, 7.AF.2
August, 1939 5.X.1,
6.BE, 7.I, 7.X, 7.Z, 7.AL, 7.AN, 7.AT, 7.AV, 10.H
Badcock, 1823 6.BH, 7.H.3,
7.P.5, 7.Q
Bagley: Paradox Pie, 1944 6.BN, 7.Z, 7.AI, 7.AW, 10.F,
10.Q, 10.S
Bagley: Puzzle Pie, 1944 5.D.5, 6.O, 6.P.1, 6.P.2,
6.R.1, 6.R.2, 6.Y, 6.AF, 6.AI, 7.AV, 10.L
Bath, 1959 5.C,
7.G.1, 7.I, 7.P.5, 7.AC.3, 7.AC.6, 7.AM
Bellew, 1866 5.E,
6.AO.1, 6.AQ
Berloquin, 1981 5.N, 7.H.5,
7.N.3, 10.R
Black, 1952 [1946?] 5.T, 6.F.2, 9.D, 9.F
Bourdon, 1834 7.E.1, 7.H,
7.P.1, 7.S, 7.X, 7.AF.1, 7.AK, 10.A, 10.R
Brandreth Puzzle Book [1895] 5.B, 5.B.1, 5.O, 6.AW.1, 6.AY.1,
7.B, 7.G.1
Bullen, 1789 7.G.1,
7.H, 7.H.5, 7.L.2.b, 7.S.1, 7.AF.1
Bullivant, 1910 5.S, 6.T, 6.AK
[Chambers], 1866? 7.H, 7.L.2.a,
7.L.2.b, 7.Y, 7.AF.2
Chang Chhiu‑Chien -- see
Zhang Qiujian
Colenso, Algebra, 1849 7.P.1, 10.G
Colenso, Arithmetic, 1853 7.H, 7.X, 10.G, 10.R
Devi, 1976 5.D.1,
5.X.1, 7.E, 7.P.1, 7.AC.3, 7.AC.6, 7.AE, 10.A.3; 10.K
Dresner, 1962 5.B.1, 5.C,
5.D.4, 5.K.1, 5.W
Dudeney: World's best puzzles,
1908
2,
5.P.1, 5.S, 6.P.1, 6.S, 6.AI, 6.AO, 6.AW
Elliott, 1872 6.V, 6.AQ,
6.AV, 6.AZ, 11.B, 11.C, 11.D
Filipiak, 1942 5.H.1,
6.W.1, 6.W.2, 6.AK
Fisher, 1968 6.E,
6.P.2, 7.M.4, 7.AI
Fisher, 1973 1, 5.E,
7.S.2, 10.L
Fourrey, Cur. Geom., 1907 6.S.1, 8.G (also 6.R.1)
Fourrey, Rec. Arith., 1899 4.A.1, 5.B, 5.P.1, 5.U, 7.N.1
Fuss, 1843 5.F.1
Goldston, nd [1910?] 6.AK, 11.E
Gomme, 1894/98 4.B.1, 5.R.5 (Also 4.A.3)
Greenblatt, 1965 6.U.2, 6.AE,
7.AG
Heald, 1941 7.Z,
10.E.3, 10.G, 10.0
Hooper, 1774 4.A.1, 5.AA,
6.F, 6.P.2, 7.B, 7.AO, 7.AZ
Hutton, 1804 7.G.2, 7.H,
7.X, 7.AF.1, 7.AK, 10.A, 10.R
Kamp, 1877 5.B,
5.D.1, 5.E, 5.R.7, 7.B, 7.L, 7.Q
Kraitchik, MJ, 1930 4.A.2, 5.J, 7.E,
7.G.1, 7.H.3, 7.AR, 10.B, 10.P
Kraitchik, MR, 1943 4.A.2, 5.J, 6.M,
7.H.2, 7.H.3, 9.D, 9.G, 10.P
Laisant, 1906 6.P.1, 7.AR,
10.A.2, 10.B, 10.H, 10.I
Larte de labbacho, 1478 See: Treviso Arithmetic.
Von der Lasa, 1897 5.F.1, 7.B (Also 4.B.1, 4.B.5)
Licks, 1917 5.A,
6.R.4, 6.AG, 7.P.3, 7.S.2, 7.AC, 7.AD
Van der Linde, 1874 5.F.1, 7.N (Also 4.B.1, 4.B.5)
Littlewood, 1953 5.C, 5.W, 6.J,
8.B, 9.C, 9.D
Lucas, Théorie, 1891 5.L, 5.Z.5, 5.AB
Madachy, 1966 5.O, 6.D, 6.X,
7.N.3, 7.N.4, 7.AC.3, 7.BB
Meyer, 1965 7.I,
7.AC.4, 7.AH, 7.AR, 7.AX
Milne, 1881 7.E, 7.H,
7.R, 7.X, 7.Y, 10.A, 10.A.3, 10.G, 10.R
W. O. J. Moser, 1981 6.I, 6.T
Nordmann, 1927 4.A.4, 5.G.1,
6.AR, 7.AC.3, 7.AR, 11.C, 11.E
Papyrus Rhind, c‑1650 7.C, 7.G.1, 7.L, 7.S.1
Phillips: Playtime Omnibus, 1933 6.AF, 7.S.2, 7.AC.1, 7.AD.1, 7.AE, 9.D
Phillips: Question Time, 1937 5.U, 7.E, 7.AG, 9.G
Ransom, 1955 6.M, 7.F,
7.X, 7.AC.2, 8.B, 10.A.1, 10.B, 10.I
Smith: Origin, 1917 3A, 7.G.2, 7.H, 10.A
Steinhaus: 1938,1950,1960,1969 5.C.1, 6.E, 6.G.1, 6.H, 6.AB
Strutt, 1791?, 1801 4.B.1, 5.R.1,
5.R.5 (Also 4.A.3)
Strutt-Cox, 1903 4.B.1, 5.R.1,
5.R.5 (Also 4.A.3)
Trenchant, 1566 7.L.2.a, 7.S.1
(also 5.B, 5.D.1, 7.E, 7.S.1, 7.AF.1)
Treviso Arithmetic, 1478 7.H, 7.K.1, 7.AL, 10.A
Trigg: Quickies, 1967 5.Q, 6.AE, 6.AN, 7.N.3,
7.W
Wagner, Rechenbuch, 1483 7.G.1, 7.G.2, 7.H, 7.AK, 10.A
Wecker, (1582), 1660 7.L.3, 7.AO, 10.P, 11.I,
11.N
A. C. White, 1913 1, 5.I.1, 6.T,
6.AK, 7.X, 11.E
Widman(n), 1489 7.G.1, 7.H, 7.L.2,
7.P.1, 7.P.5, (7.AL)
Williams & Savage, 1940 7.P.5, 7.X, 7.AC.2, 7.AM,
7.AP, 8.I, 10.E.2
Wolff, 1937 7.R.3,
7.S.2, 7.AC.1, 7.AE, 9.E, 9.E.1, 10.O
Workman, 1902 7.H.1, 7.H.4,
7.J, 7.S.2, 7.AC.2, 10.G
Wyatt, 1928 5.H.1,
6.V, 6.W.1, 6.W.2, 6.AI
Mr. X, 1903, 1911 4.A.1, 5.B, 5.P.1,
5.S, 6.AF, 6.AU, 7.H.3, 7.I, 7.J, 7.M.4, 9.E, 9.J, 10.H
Yang Hui, 1275 7.N, 7.P.1,
7.P.2, 10.A
Zhang Qiujian, 468 7.E, 7.L, 7.P.1, 7.P.6, 10.A
1. BIOGRAPHICAL MATERIAL -- in chronological order
ALCUIN (c735‑804)
Phillip Drennon Thomas. Alcuin of York. DSB I, 104‑105.
Robert Adamson. Alcuin, or Albinus. DNB, (I, 239‑240), 20.
Andrew Fleming West. Alcuin and the Rise of the Christian
Schools. (The Great Educators --
III.) Heinemann, 1893. The only book on Alcuin that I found which
deals with the Propositiones.
Stephen Allott. Alcuin of York c. A.D. 732 to 804
-- his life and letters. William Sessions, York, 1974.
FIBONACCI
[LEONARDO PISANO] (c1170->1240)
See
also the entries for Fibonacci in Common References.
Fibonacci. (1202 -- first paragraph); 1228 -- second paragraph, on p. 1. In this paragraph he narrates almost
everything we know about him. [In the
second ed., he inserted a dedication as the first paragraph.]
The
paragraph ends with the notable sentence which I have used as a motto for this
work. "Si quid forte minus aut
plus iusto vel necessario intermisi, mihi deprecor indulgeatur, cum nemo sit
qui vitio careat et in omnibus undique sit circumspectus." (If I have perchance omitted anything more
or less proper or necessary, I beg indulgence, since there is no one who is
blameless and utterly provident in all things.
[Grimm's translation.])
Richard E. Grimm. The autobiography of Leonardo Pisano. Fibonacci Quarterly 11:1 (Feb 1973)
99-104. He has collated six MSS of the
autobiographical paragraph and presents his critical version of it, with
English translation and notes. Sigler,
below, gives another translation. I
give Grimm's translation, omitting his notes.
After
my father's appointment by his homeland as state official in the customs house
of Bugia for the Pisan merchants who thronged to it, he took charge; and, in
view of its future usefulness and convenience, had me in my boyhood come to him
and there wanted me to devote myself to and be instructed in the study of
calculation for some days. There,
following my introduction, as a consequence of marvelous instruction in the
art, to the nine digits of the Hindus, the knowledge of the art very much
appealed to me before all others, and for it I realized that all its aspects
were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying
methods; and at these places thereafter, while on business, I pursued my study
in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well
as the art of Pythagoras I considered as almost a mistake in respect to the
method of the Hindus. Therefore,
embracing more stringently that method of the Hindus, and taking stricter pains
in its study, while adding certain things from my own understanding and
inserting also certain things from the niceties of Euclid's geometric art, I
have striven to compose this book in its entirety as understandably as I could,
dividing it into fifteen chapters.
Almost everything which I have introduced I have displayed with exact
proof, in order that those further seeking this knowledge, with its pre-eminent
method, might be instructed, and further, in order that the Latin people might
not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or
less proper or necessary, I beg indulgence, since there is no one who is
blameless and utterly provident in all things.
F. Bonaini. Memoria unica sincrona di Leonardo Fibonacci
novamente scoperta. Giornale Storico
degli Archivi Toscani 1:4 (Oct-Dec 1857) 239-246. This reports the discovery of a 1241 memorial of the Comune of
Pisa, which I reproduce as it is not well known. This grants Leonardo an annual honorarium of 20 pounds. In 1867, a plaque bearing this inscription
and an appropriate heading was placed in the atrium of the Archivio di Stato in
Pisa.
"Considerantes
nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam
quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli,
in abbacandis estimationibus et rationibus civitatis eiusque officialium et
aliis quoties expedit, conferunter; ut eidem Leonardo, merito dilectionis et
gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem
substinet in audiendis et consolidandis estimationibus et rationibus
supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede
sive salario suo, annis singulis, libre xx denariorum et amisceria consueta
dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero
more solito serviat), presenti constitutione firmamus."
A
translation follows, but it can probably be improved. My thanks to Steph Maury Gannon for many improvements over my
initial version.
Considering
the honour and progress of our city and its citizens that is brought to them
through both the knowledge and the diligent application of the discreet and
wise Maestro Leonardo Bigallo in the art of calculation for valuations and
accounts for the city and its officials and others, as often as necessary; we
declare by this present decree that there shall be given to the same Leonardo,
from the Comune and on behalf of the Comune, by reason of affection and
gratitude, and for his excellence in science, in recompense for the labour
which he has done in auditing and consolidating the above mentioned valuations
and accounts for the Comune and the public bodies, as his wages or salary, 20
pounds in money each year and his usual fees (the same Pisano shall continue to
render his usual services to the Comune and its officials in the art of
calculation etc.).
Bonaini
also quotes a 1506 reference to Lionardo Fibonacci.
Mario Lazzarini. Leonardo Fibonacci Le sue Opere e la sua Famiglia.
Bolletino di Bibliografia e Storia delle Scienze Matematiche 6 (1903) 98‑102 &
7 (1904) 1-7. Traces the family
to late 11C, saying Leonardo's father was Guglielmo and his grandfather was
probably Bonaccio. He estimates the
birth date as c1170. He describes a
contract of 28 Aug 1226 in which Leonardo Bigollo, his father, Guglielmo, and
his brother, Bonaccingo, buy a piece of land from a relative. This land included a tower and other
buildings, outside the city, near S. Pietro in Vincoli. [G. Milanesi; Documento inedito intorno a
Leonardo Fibonacci; Rome, 1867 -- ??NYS].
Says nothing is known of the 1202 ed of Liber Abbaci. Quotes the above memorial.
R. B. McClenon. Leonardo of Pisa and his liber
quadratorum. AMM 26:1 (Jan 1919) 1-8.
Gino Loria. Leonardo Fibonacci. Gli Scienziati Italiana dall'inizio del
medio evo ai nostri giorni. Ed. by Aldo
Mieli. (Dott. Attilio Nardecchia
Editore, Rome, 1921;) Casa Editrice
Leonardo da Vinci, Rome, 1923. Vol. 1,
pp. 4-12. This reproduces much of the
material in Lazzarini and the opening biographical paragraph of Liber
Abaci.
Ettore Bortolotti. Article on Fibonacci in: Enciclopedia Italiana. G. Treccani, Rome, 1949 (reprint of 1932
ed.).
Charles King. Leonardo Fibonacci. Fibonacci Quarterly 1:4 (Dec 1963) 15-19.
Gino Arrighi, ed. Leonardo Fibonacci: La Practica di Geometria
-- Volgarizzata da Cristofano di Gherardo di Dino, cittadino pisano. Dal Codice 2186 della Biblioteca Riccardiana
di Firenze. Domus Galilaeana, Pisa, 1966. The Frontispiece is the mythical portrait of
Fibonacci, taken from I Benefattori dell'Umanità, vol. VI; Ducci, Florence,
1850. (Smith, History II 214 says it is
a "Modern engraving. The portrait
is not based on authentic sources".)
P. 15 shows the plaque erected in the Archivio di Stato di Pisa in 1855
which reproduces the above memorial with an appropriate heading, but Arrighi
has no discussion of it. P. 19 is a
photo of the statue in Pisa and p. 16 describes its commissioning in 1859.
Joseph and Francis Gies. Leonard of Pisa and the New Mathematics of
the Middle Ages. Crowell, NY,
1969. This is a book for school
students and contains a number of dubious statements and several false
statements.
Kurt Vogel. Fibonacci, Leonardo, or Leonardo of
Pisa. DSB IV, 604-613.
A. F. Horadam. Eight hundred years young. Australian Mathematics Teacher 31 (1975) 123‑134. Good survey of Fibonacci's life &
work. Gives English of a few
problems. This is available on
Kimberling's website - see below.
Ettore Picutti. Leonardo Pisano. Le Scienze 164 (Apr 1982) ??NYS.
= Le Scienze, Quaderni; 1984, pp. 30-39. (Le Scienze is a magazine;
the Quaderni are collections of articles into books.) Mostly concerned with the Liber Quadratorum,
but surveys Fibonacci's life and work.
Says he was born around 1170.
Includes photo of the plaque in the Archivo di Stato di Pisa.
Leonardo Pisano Fibonacci. Liber quadratorum, 1225. Translated and edited by L. E. Sigler
as: The Book of Squares; Academic
Press, NY, 1987. Introduction: A brief
biography of Leonardo Pisano (Fibonacci) [1170 - post 1240], pp. xv-xx. This is the best recent biography,
summarizing Picutti's article. Says he
was born in 1170 and his father's name was Guilielmo -- cf Loria above. Gives another translation of the
biographical paragraph of the Liber Abbaci.
A. F. Horadam & J. Lahr. Letter to the Editor. Fibonacci Quarterly 28:1 (Feb 1990) 90. The authors volunteer to act as coordinators
for work on the life and work of Fibonacci.
Addresses: A. F. Horadam,
Mathematics etc., Univ. of New England, Armidale, New South Wales, 2351,
Australia; J. Lahr, 14 rue des Sept
Arpents, L‑1139 Luxembourg, Luxembourg.
Thomas Koshy. Fibonacci and Lucas Numbers with
Applications. Wiley-Interscience,
Wiley, 2001. Claims to be 'the first
attempt to compile a definitive history and authoritative analysis' of the
Fibonacci numbers, but the history is generally second-hand and marred with a
substantial number of errors, The
mathematical work is extensive, covering many topics not organised before, and
is better done, but there are more errors than one would like.
Laurence E. Sigler. Translation of Liber Abaci as: Fibonacci's Liber Abaci A Translation into Modern English of
Leonardo Pisano's Book of Calculation.
Springer, 2002.
Clark Kimberling's site web
includes biographical material on Fibonacci and other similar number
theorists.
http://cedar.evansville.edu/~ck6/bstud/fibo.html .
Ron Knott has a huge website on
Fibonacci numbers and their applications, with material on many related topics,
e.g. continued fractions, π, etc. with some history.
www.ee.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html .
Luca
PACIOLI (c1445-1517)
S. A. Jayawardene. Luca Pacioli. BDM 4, 1897-1900.
Bernardino Baldi (Catagallina)
(1553-1617). Vita di Pacioli. (1589, first published in his Cronica de Mathematici
of 1707.) Reprinted in: Bollettino di
bibliografia e di storia delle scienze matematiche e fisiche 12 (1879)
421-427. ??NYS -- cited by Taylor,
p. 338.
Enrico Narducci. Intorno a due edizioni della "Summa de
arithmetica" di Fra Luca Pacioli.
Rome, 1863. ??NYS -- cited by
Riccardi [Biblioteca Matematica Italiana, 1952]
D. Ivano Ricci. Luca Pacioli, l'uomo e lo scienziato. San Sepolcro, 1940. ??NYS -- cited in BDM.
R. Emmett Taylor. No Royal Road Luca Pacioli and His Times.
Univ. of North Carolina Press, Chapel Hill, 1942. BDM describes this as lively but unreliable.
Ettore Bortolotti. La Storia della Matematica nella Università
di Bologna. Nicola Zanichelli Editore,
Bologna, 1947. Chap. I, § V, pp. 27-33:
Luca Pacioli.
Margaret Daly Davis. Piero della Francesca's Mathematical
Treatises The "Trattato
d'abaco" and "Libellus de quinque corporibus regularibus". Longo Editore, Ravenna, 1977. This discusses Piero's reuse of his own
material and Pacioli's reuse of Piero's material.
Fenella K. C. Rankin. The Arithmetic and Algebra of Luca
Pacioli. PhD thesis, Univ. of London,
1992 (copy at the Warburg Institute), ??NYR.
Enrico Giusti, ed. Descriptive booklet accompanying the 1994
facsimile of the Summa -- qv in Common References.
Edward A. Fennell. Figures in Proportion: Art, Science and the
Business Renaissance. The contribution
of Luca Pacioli to culture and commerce in the High Renaissance. Catalogue for the exhibition, The Institute
of Chartered Accountants in England and Wales, London, 1994.
Claude-Gaspar
BACHET de Méziriac (1581‑1638)
C.‑G. Collet & J.
Itard. Un mathématicien humaniste --
Claude‑Gaspar Bachet de Méziriac (1581‑1638). Revue d'Histoire des Sciences et leurs
Applications 1 (1947) 26‑50.
J. Itard. Avant-propos. IN: Bachet; Problemes;
1959 reprint, pp. v‑viii. Based
on the previous article.
There is a Frontispiece portrait
in the reprint.
Underwood Dudley. The first recreational mathematics
book. JRM 3 (1970) 164‑169. On Bachet's Problemes.
William Schaaf. Bachet de Méziriac, Claude‑Gaspar. DSB I, 367‑368.
Jean
LEURECHON (c1591‑1670)
and Henrik VAN ETTEN
A. Deblaye. Étude sur la récréation mathématique du P.
Jean Leurechon, Jésuite. Mémoires de la
Société Philotechnique de Pont-à-Mousson 1 (1874) 171-183. [MUS #314.
Schaaf. Hall, OCB, pp. 86, 88
& 114, says the only known copy of this journal is at Harvard, which has
kindly supplied me with a photocopy of this article. Hall indicates the article is in vol. II and says it is 12 pages,
but only cites pp. 171 & 174.] This
simply assumes Leurechon is the author and gives a summary of his life. The essential content is described by Hall.
G. Eneström. Girard Desargues und D.A.L.G. Biblioteca Mathematica (3) 14 (1914) 253‑258. D.A.L.G. was an annotator of van Etten's
book in c1630. Although D.A.L.G. was
used by Mydorge on one of his other books, it had been conjectured that this
stood for Des Argues Lyonnais Girard
(or Géomètre). Eneström can find no
real evidence for this and feels that Mydorge is the most likely person.
Trevor H. Hall. Mathematicall Recreations. An Exercise in Seventeenth Century
Bibliography. Leeds Studies in
Bibliography and Textual Criticism, No. 1.
The Bibliography Room, School of English, University of Leeds, 1969,
38pp. Pp. 18‑38 discuss the
question of authorship and Hall feels that van Etten probably was the author
and that there is very little evidence for Leurechon being the author. Much of the mathematical content is in
Bachet's Problemes and may have been copied from it or some common source. [This booklet is reproduced as pp. 83-119 of
Hall, OCB, with the title page of the 1633 first English edition reproduced as
plate 5, opp. p. 112. Some changes have
been made in the form of references since OCB is a big book, but the only other
substantial change is that he spells the name of the dedicatee of the book as
Verreyken rather than Verreycken.]
William Schaaf. Leurechon, Jean. DSB VIII, 271‑272.
Jacques Voignier. Who was the author of "Recreation
Mathematique" (1624)? The
Perennial Mystics #9 (1991) 5-48 (& 1-2 which are the cover and its
reverse). [This journal is edited and
published by James Hagy, 2373 Arbeleda Lane, Northbrook, Illinois, 60062,
USA.] Presents some indirect evidence
for Leurechon's authorship.
Jacques
OZANAM (1640‑1717)
On the flyleaf of J. E.
Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil portrait
labelled Ozanam -- the only one I know of.
This copy is at the Institut für Geschichte der Naturwissenschaft in
Munich. Hofmann published the picture
-- see below.
Charles Hutton. A Mathematical and Philosophical
Dictionary. 1795-1796. Vol. II, pp. 184-185. ??NYS
[Hall, OCB, p. 166.]
Charles Hutton. On the life and writings of Ozanam, the
first author of these Mathematical Recreations. Ozanam-Hutton. Vol.
I. 1803: xiii-xv; 1814: ix-xi.
William L. Schaaf. Jacques Ozanam on mathematics .... MTr 50 (1957) 385-389. Mostly based on Hutton. Includes a sketchy bibliography of Ozanam's
works, generally ignoring the Recreations.
Joseph Ehrenfried Hofmann. Leibniz und Ozanams Problem, drei Zahlen so
zu bestimmen, dass ihre Summe eine Quadratzahl und ihre Quadratsumme eine
Biquadratzahl ergibt. Studia Leibnitiana
1:2 (1969) 103-126. Outlines Ozanam's
life, gives a bibliography of his works and reproduces the above-mentioned
drawing as a plate opp. p. 124. (My
thanks to Menso Folkerts for this information and a copy of Hofmann's article.)
William L. Schaaf. Ozanam, Jacques. DSB X, 263‑265.
Jean
Étienne MONTUCLA (1725-1799)
Charles Hutton. Some account of the life and writings of
Montucla. Ozanam‑Hutton. Vol. I.
1803: viii-xii; 1814: iv-viii.
Charles Hutton. A Philosophical and Mathematical
Dictionary. 2nd ed. of the Dictionary
cited under Ozanam, 1815, Vol. II, pp.
63-64. ??NYS. According to Hall, OCB, p. 167, this is not in the 1795-1796 ed.
and is a reworking of the previous item.
Lewis
CARROLL (1832-1898)
Pseudonym
of Charles Lutwidge Dodgson. There is
so much written on Carroll that I will only give references to his specifically
recreational work and some basic references.
The Diaries of Lewis
Carroll. Edited by Roger Lancelyn
Green. (OUP, 1954); 2 vols, Greenwood Publishers, Westport,
Connecticut, 1971, HB.
Lewis Carroll's Diaries The private journals of Charles Lutwidge
Dodgson (Lewis Carroll) The first
complete version of the nine surviving volumes with notes and annotations by
Edward Wakeling. Introduction by Roger
Lancelyn-Green. The Lewis Carroll
Society, Publications Unit, Luton, Bedfordshire. [There were 13 journals, but 4 are lost.]
Vol.
1. Journal 2, Jan-Sep 1855. 1993, 158pp.
Vol.
2. Journal 4, Jan-Dec 1856. 1994, 158pp.
Vol.
3. Journal 5, Jan 1857 - Apr 1858. 1995, 199pp.
Vol.
4. Journal 8, May 1862 - Sep 1864 and a
reconstruction of the four missing
years,
1858-1862. 1997, 399pp.
Vol.
5. Journal 9, Sep 1864 - Jan 1868,
including the Russian Journal.
1999,
416pp.
Vol.
6. Journal 10, Apr 1868 - Dec
1876. 2001, 552pp.
Vol.
7. Journal 11, Jan 1877 - Jun
1883. 2003, 606pp.
The Letters of Lewis
Carroll. Edited by Morton N. Cohen with the assistance of Roger Lancelyn
Green. Volume One ca.1837 - 1885; Volume Two 1886 -
1898. Macmillan London, 1979.
Stuart Dodgson Collingwood. The Life and Letters of Lewis Carroll. T. Fisher Unwin, London, 1898.
Stuart Dodgson Collingwood,
ed. The Lewis Carroll Picture
Book. T. Fisher Unwin, London,
1899. = Diversions and
Digressions of Lewis Carroll, Dover, 1961.
= The Unknown Lewis Carroll, Dover, 1961(?). Reprint, in reduced format, Collins, c1910. The pagination of the main text is the same
in the 1899 and in both Dover reprints, but is quite different than the
Collins. Cited as: Carroll-Collingwood,
qv in Common References.
R. B. Braithwaite. Lewis Carroll as logician. MG 16 (No. 219) (Jul 1932) 174-178. He notes that Carroll assumed that a
universal statement implied the existence of an object satisfying the
antecedent, e.g. 'all unicorns are blue' would imply the existence of unicorns,
contrary to modern convention.
Derek Hudson. Lewis Carroll -- An Illustrated
Biography. Constable, 1954; new illustrated ed., 1976.
Warren Weaver. Lewis Carroll: Mathematician. SA 194:4 (Apr 1956) 116‑128. +
Letters and response. SA 194:6
(Jun 1956) 19-22.
Martin Gardner. The Annotated Alice. C. N. Potter, NY, 1960. Penguin, 1965; 2nd ed., 1971. Revised as: More Annotated Alice, 1990, qv.
Martin Gardner. The Annotated Snark. Bramhall House, 1962. Penguin, 1967; revised, 1973 & 1974.
John Fisher. The Magic of Lewis Carroll. Nelson, 1973. Penguin, 1975.
Morton N. Cohen, ed. The Selected Letters of Lewis Carroll. Papermac (Macmillan), 1982.
Martin Gardner. More Annotated Alice. [Extension of The Annotated Alice.] Random House, 1990.
Edward Wakeling. Lewis Carroll's Games and Puzzles. Dover and the Lewis Carroll Birthplace
Trust, 1992. Cited as Carroll-Wakeling,
qv in Common References.
Francine F. Abeles, ed. The Pamphlets of Lewis Carroll -- Vol. 2:
The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces. Lewis Carroll Society of North America,
distributed by University Press of Virginia, Charlottesville, 1994.
Edward Wakeling. Rediscovered Lewis Carroll Puzzles. Dover, 1995. Cited as Carroll‑Wakeling II, qv in Common References.
Martin Gardner. The Universe in a Handkerchief. Lewis Carroll's Mathematical Recreations,
Games, Puzzles and Word Plays.
Copernicus (Springer, NY), 1996.
Cited as Carroll‑Gardner, qv in Common References.
Martin Gardner. The Annotated Alice: The Definitive
Edition. 1999. [A combined version of The Annotated Alice
and More Annotated Alice.]
Professor
Louis HOFFMANN (1839‑1919)
Pseudonym
of Angelo John Lewis.
Joseph Foster. Men-at-the-Bar: A biographical Hand-List of
the Members of the Various Inns of Court, including Her Majesty's Judges,
etc. 2nd ed, the author, 1885. P. 277 is the entry for Lewis. Born in London, eldest son of John
Lewis. Graduated from Wadham College,
Oxford. Entered Lincoln's Inn as a
student in 1858, called to the bar there in 1861. Married Mary Ann Avery in 1864.
Author of Manual of Indian Penal Code and Manual of Indian
Civil Procedure. Address: 12
Crescent Place, Mornington Crescent, London, NW. (My thanks to the Library of Lincoln's Inn for this information.)
Anonymous. Professor Hoffmann. Mahatma 4:1 (Jul 1900) 377-378. A brief note, with photograph, stating that
he is Mr. Angelo Lewis, M.A. and Barrister-at-Law.
Will Goldston. Will Goldston's Who's Who in Magic. My version is included in a compendium
called: Tricks that Mystify; Will Goldston, London, nd [1934-NUC]. Pp. 106-107. Says he was a barrister, retired to Hastings about 1903 and died
in 1917.
Who Was Who, 1916-1928, p.
627. This says he attended North London
Collegiate School and that he only practised law until 1876. He was on the staff of the Saturday Review
and a contributor to many journals. Won
the £100 prize offered by Youth's Companion (Boston) for best short story for
boys. Lists 36 books by him and 9 card
games he invented. Address:
Manningford, Upper Bolebrooke Road, Bexhill-on-Sea. (My thanks to the Library of Lincoln's Inn for this information.)
J. B. Findlay & Thomas A.
Sawyer. Professor Hoffmann: A
Study. Published by Thomas A. Sawyer,
Tustin, California, 1977. A short book,
12 + 67 pp, with two portraits (one from Mahatma) and 27pp of
bibliography. He was born at 3 Crescent
Place, Mornington Crescent, London. He
was a barrister and wrote two books on Indian law.
Charles Reynolds. Introduction -- to the reprint of
Hoffmann's Modern Magic, Dover, 1978, pp. v‑xiv. This says Lewis was a barrister, which is
mentioned in another reprint of a Hoffmann book and in S. H. Sharpe's
translation of Ponsin on Conjuring.
Edward Hordern. Foreword to this edition. In:
Hoffmann's Puzzles Old and New (see under Common References), 1988
reprint, pp. v‑vi. This says he
was the Reverend Lewis, but this is corrected in Hoffmann-Hordern to saying he
was a barrister.
Hoffmann-Hordern, p. viii, is a
version of the photograph in Mahatma.
Hall, OCB, p. 189, gives
Hoffmann's address as Ireton Lodge, Cromwell Ave., N. -- presumably the
Cromwell Ave. in Highgate.
Toole Stott 386 gives a little
information about Hoffmann and Modern Magic, including an address in Mornington
Crescent in 1877.
No DNB or DSB entry -- I have
suggested a DNB entry.
Sam
LOYD (1841‑1911) and
Sam LOYD JR. (1873‑1934)
[W. R. Henry.] Samuel Loyd. [Biography.] Dubuque
Chess Journal, No. 66 (Aug-Sep 1875) 361-365.
??NX -- o/o (11 Jul 91).
Loyd. US Design 4793 -- Design for Puzzle-Blocks. 11 April 1871. These are solid pieces, but unfortunately the drawing did not
come with this, so I am not clear what they are. ??Need drawing -- o/o (11 Jul 91).
Anonymous & Sam Loyd. Loyd's puzzles (Introductory column). Brooklyn Daily Eagle (22 Mar 1896) 23. Says he lives at 153 Halsey St., Brooklyn.
L. D. Broughton Jr. Samuel Loyd. [A Biography.] Lasker's
Chess Magazine 1:2 (Dec 1904) 83-85. About
his chess problems with a mention of some of his puzzles.
G. G. Bain. The prince of puzzle‑makers. An interview with Sam Loyd. Strand Magazine 34 (No. 204) (Dec 1907) 771‑777. Solutions of Sam Loyd's puzzles. Ibid. 35 (No. 205) (Jan 1908) 110.
Walter Prichard Eaton. My fifty years in puzzleland -- Sam Loyd and
his ten thousand brain‑teasers.
The Delineator (New York) (April 1911) 274 & 328. Drawn portrait of Loyd, age 69.
Anon. Puzzle inventor dead.
New-York Daily Tribune (12 Apr 1911) 7.
Says he died at his house, 153 Halsey St. "He declared no one had ever succeeded in solving [the
"Disappearing Chinaman"]."
Says he is survived by a son and two daughters (!! -- has anyone ever
tracked the daughters and their descendents??).
Anon. Sam Loyd, puzzle man, dies.
New York Times (12 Apr 1911) 13.
Says he was for some time editor of The Sanitary Engineer and a shrewd
operator on Wall Street.
Anon. Sam Loyd. SA (22 Apr
1911) 40-41?? Says he was for some
years chess editor of SA and was puzzle editor of Woman's Home Companion when
he died.
W. P. Eaton. Sam Loyd.
The American Magazine 72 (May 1911) 50, 51, 53. Abridged version of Eaton's earlier article. Photo of Loyd on p. 50.
P. J. Doyle. Letter to the Chess column. The Sunday Call [Newark, NJ] (21 May 1911),
section III, p. 10.
A. C. White. Sam Loyd and His Chess Problems. Whitehead and Miller, Leeds, UK, 1913; corrected, Dover, 1962.
Alain C. White. Supplement to Sam Loyd and His Chess
Problems. Good Companion Chess Problem
Club, Philadelphia, vol. I, nos. 11-12 (Aug 1914), 12pp. This is mostly corrections of the chess
problems, but adds a few family details with a picture of the Loyd Homestead
and Grist Mill in Moylan, Pennsylvania.
Alain C. White. Reminiscences of Sam Loyd's family. The Problem [Pittsburgh]
(28 Mar 1914) 2, 3, 6, 7.
Louis C. Karpinski. Loyd, Samuel. Dictionary of American Biography, Scribner's, NY, vol. XI,
1933, pp. 479‑480.
Loyd Jr. SLAHP.
1928. Preface gives some details
of his life, making little mention of his father, "who was a famous
mathematician and chess player".
He claims to have created over 10,000 puzzles. There are some vague
biographical details on pp. 1‑22, e.g. 'Father conducted a printing
establishment.' 'My "Missing Chinaman
Puzzle"'. (It may have been some
such assertion that led me to estimate his birthdate as 1865, but I now see it
is well known to be 1873.)
Anonymous. Sam Loyd dead; puzzle creator. New York Times (25 Feb 1934). Obituary of Sam Loyd Jr. Says he resided at 153 Halsey St.,
Brooklyn -- the same address as his father -- see the Brooklyn Daily Eagle
article of 1896, above. He worked from
a studio at 246 Fulton St., Brooklyn.
It says Jr. invented 'How Old is Ann?'.
Clark Kinnaird. Encyclopedia of Puzzles and Pastimes. Grosset & Dunlap, NY, 1946. Pp. 263‑267: Sam Loyd. Asserts that Loyd Jr. invented 'How Old is
Ann?'
Gardner. Sam Loyd: America's greatest puzzlist. SA (Aug 1957) c= First Book, Chap. 9.
Gardner. Advertising premiums. SA (Nov 1971) c= Wheels, chap. 12.
Will Shortz is working on a
biography.
No DSB entry.
François
Anatole Édouard LUCAS (1842‑1891)
Jeux Scientifiques de Ed.
Lucas. Advertisement by Chambon &
Baye (14 rue Etienne-Marcel, Paris) for the 1re Serie of six games. Cosmos.
Revue des Sciences et Leurs Applications 39 (NS No. 254) (7 Dec 1889) no
page number on my photocopy.
B. Bailly [name not given, but
supplied by Hinz]. Article on Lucas's
puzzles. Cosmos. Revue des Sciences et Leurs Applications. NS, 39 (No. 259) (11 Jan 1890) 156-159. NEED 156‑157.
Nécrologie: Édouard Lucas. La Nature 19 (1891) II, 302.
Obituary notice: "La
Nature announces the death of Prof. Edouard Lucas ...." Nature 44 (15 Oct 1891) 574-575.
Duncan Harkin. On the mathematical work of François‑Édouard‑Anatole
Lucas. L'Enseignement Math. (2) 3
(1957) 276‑288. Pp. 282‑288
is a bibliography of 184 items. I have
found many Lucas publication not listed here and have started a new
Bibliography -- see below.
P. J. Campbell. Lucas' solution to the non‑attacking
rooks problem. JRM 9 (1976/77) 195‑200. Gives life of Lucas.
A photo of Lucas is available
from Bibliothèque Nationale, Service Photographique, 58 rue Richelieu, F‑75084
Paris Cedex 02, France. Quote Cote du
Document Ln27 . 43345 and Cote du Cliche 83 A 51772. (??*)
I have obtained a copy, about 55 x 85 mm, with the photo in an oval
surround. It looks like a carte-de-visite,
but has Édouard LUCAS (1842-1891). --
Phot. Zagel. underneath. (Thanks to H. W. Lenstra for the information.)
Norman T. Gridgeman. Lucas, François‑Édouard‑Anatole. DSB VIII, 531‑532.
Susanna S. Epp. Discrete Mathematics with Applications. Wadsworth, Belmont, Calif., 1990, p. 477
gives a small photo of Lucas which looks nothing like the photo from the
BN. I have since received a note from
Epp via Paul Campbell that a wrong photo was used in the first edition, but
this was corrected in later editions.
Alain Zalmanski. Edouard Lucas Quand l'arithmétique devient amusante. Jouer Jeux Mathématiques 3 (Jul/Sep 1991) 5. Brief notice of his life and work.
Andreas M. Hinz. Pascal's triangle and the Tower of
Hanoi. AMM 99 (1992) 538-544. Sketches Lucas' life and work, giving
details that are not in the above items.
David Singmaster. The publications of Édouard Lucas. Draft version, 14pp, 1998. I discovered many items in Dickson's History
of the Theory of Numbers and elsewhere which are not given by Harkin (cf
above). This has 248 items, though many
of these are multiple items so the actual count is perhaps 275. However, Dickson does not give article
titles, and may not give the pages of the entire article, so the same article
may be cited more than once, at different pages. I hope to fill in the missing information at some time.
Hermann
Cäsar Hannibal SCHUBERT
(1848-1911)
Acta Mathematica 1882-1912.
Table Générale des Tomes 1-35.
1913. P. 169. Portrait of Schubert.
Werner Burau. Schubert, Hermann Cäsar Hannibal. DSB XII, 227‑229.
Walter
William Rouse BALL (1850‑1925)
Anon. Obituary: Mr. Rouse
Ball. The Times (6 Apr 1925) 16.
Anon. Funeral notice: Mr. W. W.
R. Ball. The Times (9 Apr 1925) 13.
(Lord) Phillimore. Letter:
Mr. Rouse Ball. The Times (9 Apr
1925) 15.
"An old pupil". The late Mr. Rouse Ball. The Times (13 Apr 1925) 12.
J. J. Thomson. W. W. Rouse Ball. The Cambridge Review (24 Apr 1925) 341-342.
Anon. Obituary of W. W. Rouse Ball.
Nature 115 (23 May 1925) 808‑809.
Anon. The late Mr. W. W. Rouse Ball.
The Trinity Magazine (Jun 1925) 53-54.
Anon. Entry in Who's Who, 1925, p. 127.
Anon. Wills and bequests: Mr.
Walter William Rouse Ball. The Times
(7 Sep 1925) 15.
E. T. Whittaker. Obituary.
W. W. Rouse Ball. Math. Gaz. 12
(No. 178) (Oct 1925) 449-454, with photo opp. p. 449.
F. Cajori. Walter William Rouse Ball. Isis 8 (1926) 321‑324. Photo on plate 15, opp. p. 321. Copy of Ball's 1924 Xmas card on p. 324.
J. A. Venn. Alumni Cantabrigienses. Part II:
From 1752 to 1900. Vol. I, p.
136. CUP, 1940.
David Singmaster. Walter William Rouse Ball (1850-1925). 6pp handout for 1st UK Meeting on the
History of Recreational Mathematics, 24 Oct 1992. Plus extended biographical (6pp) and bibliographical (8pp) notes
which repeat some of the material in the handout.
No DNB or DSB entry -- however I
have offered to write a DNB entry. I
have since seen the proposed list of names for the next edition and Ball is
already on it.
Henry
Ernest DUDENEY (1857‑1930)
Anon. & Dudeney. A chat with the puzzle king. The Captain 2 (Dec? 1899) 314‑320, with
photo. Partly an interview. Includes photos of Littlewick Meadow.
Anon. Solutions to "Sphinx's puzzles". The Captain 2:6 (Mar 1900) 598‑599 &
3:1 (Apr 1900) 89.
Anon. Master of the breakfast table problem. Daily Mail (1 Feb 1905) 7.
An interview with Dudeney in which he gives the better version of his
spider and fly problem.
Fenn Sherie. The Puzzle King: An Interview with Henry E.
Dudeney. Strand Magazine 71 (Apr 1926)
398‑4O4.
Alice Dudeney. Preface to PCP, dated Dec 1931, pp. vii‑x. The date of his death is erroneously given
as 1931.
Gardner. Henry Ernest Dudeney: England's greatest
puzzlist. SA (Jun 1958) c= Second Book, chap. 3.
Angela Newing. The Life and Work of H. E. Dudeney. MS 21 (1988/89) 37‑44.
Angela Newing is working on a biography.
No DNB or DSB entry. I have suggested a DNB entry.
Wilhelm
Ernst Martin Georg AHRENS (1872‑1927)
Wilhelm Lorey. Wilhelm Ahrens zum Gedächtnis. Archiv für Geschichte der Mathematik, der
Naturwissenschaften und der Technik 10 (1927/28) 328‑333. Photo on p. 328.
O. Staude. Dem Andenken an Dr. Wilhelm Ahrens. Jahresbericht DMV 37 (1928) 286-287.
No DSB entry.
Yakov
Isidorovich PERELMAN [Я. И. Перелман] (1882-1942)
Perelman. FMP.
1984. P. 2 (opp. TP) is a sketch
of his life and the history of the book.
There is a small drawing of Perelman at the top of the page.
Patricio Barros. Website -- Yakov I. Perelman [in Spanish]:
www.geocities.com/yakov_perelman/index.html.
This includes a four page biography, in collaboration with Antonio Bravo,
and two photos.
Hubert
PHILLIPS (1891-1964)
Hubert Phillips. Journey to Nowhere. A Discursive Autobiography. Macgibbon & Kee, London, 1960. ??NYR
No DNB entry -- I have suggested
one.
2. GENERAL PUZZLE COLLECTIONS AND SURVEYS
H. E. Dudeney. Great puzzle crazes. London Magazine 13?? (Nov 1904) 478‑482. Fifteen Puzzle. Pigs in Clover, Answers, Pick-me-up (spiral ramp) and other
dexterity puzzles. Get Off the Earth. Conjurer's Medal (ring maze). Chinese Rings. Chinese Cross (six piece burr).
Puzzle rings. Solitaire. The Mathematician's Puzzle (square, circle,
triangle). Imperial Scale. Heart and Balls.
H. E. Dudeney. Puzzles from games. Strand Magazine 35 (No. 207) (Mar 1908)
339‑344. Solutions. Ibid. 35 (No. 208) (Apr 1908) 455‑458.
H. E. Dudeney. Some much‑discussed puzzles. Strand Magazine 35 (No. 209) (May 1908)
580‑584. Solutions. Ibid. 35 (No. 210) (Jun 1908) 696.
H. E. Dudeney. The world's best puzzles. Strand Magazine 36 (No. 216) (Dec 1908) 779‑787. Solutions.
Ibid. 37 (No. 217) (Jan 1909) 113‑116.
H. E. Dudeney. The psychology of puzzle crazes. The Nineteenth Century 100:6 (Dec 1926) 868‑879. Repeats much of his 1904 article.
Sam Loyd Jr. Are you good at solving puzzles? The American Magazine (Sep 1931) 61‑63,
133‑137.
Orville A. Sullivan. Problems involving unusual situations. SM 9 (1943) 114‑118 &
13 (1947) 102‑104.
3. GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL
I have tried to divide this material into historical and
bibliographical parts, but the two overlap considerably.
3.A. GENERAL HISTORICAL MATERIAL
Raffaella Franci. Giochi matematici in trattati d'abaco del
medioevo e del rinascimento. Atti del
Convegno Nazionale sui Giochi Creative, Siena, 11-14 Jun 1981. Tipografia Senese for GIOCREA (Società
Italiana Giochi Creativi), 1981. Pp.
18-43. Describes and quotes many
typical problems. 17 references, several
previously unknown to me.
Heinrich Hermelink. Arabische Unterhaltungsmathematik als
Spiegel Jahrtausendealter Kulturbeziehungen zwischen Ost und West. Janus 65 (1978) 105-117, with English
summary. An English translation
appeared as: Arabic recreational
mathematics as a mirror of age-old cultural relations between Eastern and
Western civilizations; in: Ahmad Y. Al-Hassan,
Ghada Karmi & Nizar Namnum, eds.; Proceedings of the First International
Symposium for the History of Arabic Science, April 1976 -- Vol. Two: Papers in
European Languages; Institute for the History of Arabic Science, Aleppo, 1978,
pp. 44-52. (There are a few translation
and typographical errors, which make it clear that the English version is a
translation of the German.)
D. E. Smith. On the origin of certain typical
problems. AMM 24 (1917) 64‑71. (This is mostly contained in his History,
vol. II, pp. 536‑548.)
Many of the items cited in the Common
References have extensive bibliographies.
In particular: BLC; BMC;
BNC; DNB; DSB;
Halwas; NUC; Schaaf;
Smith & De Morgan: Rara; Suter
are basic bibliographical sources.
Datta & Singh; Dickson; Heath: HGM;
Murray; Sanford: H&S &
Short History; Smith:
History & Source Book; Struik; Tropfke
are histories with extensive bibliographical references. AR;
BR are editions of early texts
with substantial bibliographical material.
Ahrens: MUS; Ball: MRE; Berlekamp, Conway & Guy: Winning Ways; Gardner;
Lucas: RM are recreational books
with some useful bibliographical material.
Of these, the material in Ahrens is by far the most useful. The magic bibliographies of Christopher,
Clarke & Blind, Hall, Heyl, Price (see HPL), Toole Stott and Volkmann &
Tummers have considerable overlap with the present material, particularly for
older books, though Hall, Heyl and Toole Stott restrict themselves to English
material, while Volkmann & Tummers only considers German. Santi is also very useful. Below I give some additional bibliographical
material which may be useful, arranged in author order.
Anonymous. Mathematical bibliography. SSM 48 (1948) 757‑760. Covers recreations.
Wilhelm Ahrens. Mathematische Spiele. Section I G 1 of Encyklopadie der Math.
Wiss., Vol. I, part 2, Teubner, Leipzig, 1900‑1904, pp. 1080‑1093.
Raymond Clare Archibald. Notes on some minor English mathematical serials. MG 14 (1928-29) 379-400.
Elliott M. Avedon &
Brian Sutton‑Smith. The
Study of Games. (Wiley, NY, 1971); Krieger, Huntington, NY, 1979.
Anthony S. M. Dickins. A Catalogue of Fairy Chess Books and
Opuscules Donated to Cambridge University Library, 1972‑1973, by Anthony
Dickins M.A. Third ed., Q Press,
Kew Gardens, UK, 1983.
Underwood Dudley. An annotated list of recreational
mathematics books. JRM 2:1
(Jan 1969) 13-20. 61 titles, in
English and in print at the time.
Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games
and Some Related Material. There have
been several versions with slightly varying titles. The most recent printed version is: 400 items, 28 pp., including 4 pp of text, Sep 1990. Technical Report CS90‑23, Weizmann
Institute of Science, Rehovot, Israel.
= Proc. Symp. Appl. Math. 43 (1991) 191-226. Fraenkel has since produced Update 1 to this which lists 430
items on 31pp, Aug 1992; and Update 2,
480 items on 33pp, with 5 pp of text, accidentally dated Aug 1992 at the top
but produced in Feb 1994. On 22 Nov
1994, it became a dynamic survey on the Electronic J. Combinatorics and can be
accessed from:
http://ejc.math.gatech.edu:8080/journal/surveys/index.html.
It
can also be accessed via anonymous ftp from
ftp.wisdom.weizmann.ac.il. After
logging in, do cd pub/fraenkel and then
get one of the following three
compressed files: games.tex.z; games.dvi.z; games.ps.z.
Martin P. Gaffney &
Lynn Arthur Steen. Annotated
Bibliography of Expository Writing in the Mathematical Sciences. MAA, 1976.
JoAnne S. Growney. Mathematics and the arts -- A
bibliography. Humanistic Mathematics
Network Journal 8 (1993) 22-36. General
references. Aesthetic standards for
mathematics and other arts.
Biographies/autobiographies of mathematicians. Mathematics and display of information (including
mapmaking). Mathematics and humor. Mathematics and literature (fiction and
fantasy). Mathematics and music. Mathematics and poetry. Mathematics and the visual arts.
JoAnne S. Growney. Mathematics in Literature and Poetry. Humanistic Mathematics Network Journal 10
(Aug 1994) 25-30. Short survey. 3 pages of annotated references to 29
authors, some of several books.
R. C. Gupta. A bibliography of selected book [sic] on history
of mathematics. The Mathematics
Education 23 (1989) 21-29.
Trevor H. Hall. Mathematicall Recreations. Op. cit. in 1. This is primarily concerned with the history of the book by van
Etten. [This booklet is revised as pp.
83-119 of Hall, OCB -- see Section 1.]
Catherine Perry Hargrave. A History of Playing Cards and a
Bibliography of Cards and Gaming.
(Houghton Mifflin, Boston, 1930);
Dover, 1966.
Susan Hill. Catalogue of the Turner Collection of the
History of Mathematics Held in the Library of the University of Keele. University Library, Keele, 1982. (Sadly this collection was secretly sold by
Keele University in 1998 and has now been dispersed.)
Honeyman Collection -- see:
Sotheby's.
Horblit Collection -- see:
Sotheby's and H. P. Kraus.
Else Høyrup. Books about Mathematics. Roskilde Univ. Center, PO Box 260, DK‑4000,
Roskilde, Denmark, 1979.
D. O. Koehler. Mathematics and literature. MM 55 (1982) 81-95. 64 references. See Utz for some further material.
H. P. Kraus (16 East 46th Street,
New York, 10017). The History of
Science including Navigation.
Catalogue
168. A First Selection of Books from
the Library of Harrison D. Horblit. Nd
[c1976].
Catalogue
169. A Further Selection of Books, 1641-1700
(Wing Period) from the Library of Harrison D. Horblit. Nd [c1976].
Catalogue
171. Another Selection of Books from
the Library of Harrison D. Horblit. Nd
[c1976].
These are the continuations of the catalogues
issued by Sotheby's, qv.
John S. Lew. Mathematical references in literature. Humanistic Mathematics Network Journal 7
(1992) 26-47.
Antonius van der Linde. Das erst Jartausend [sic] der
Schachlitteratur -- (850‑1880).
(1880); Facsimile reprint by
Caissa Limited Editions, Yorklyn, Delaware, 1979, HB.
Andy Liu. Appendix III: A selected bibliography on popular mathematics. Delta-k 27:3 (Apr 1989) --
Special issue: Mathematics for
Gifted Students, 55-83.
Édouard Lucas. Récréations mathématiques, vol 1 (i.e. RM1),
pp. 237-248 is an Index Bibliographique.
Felix Müller. Führer durch die mathematische Literature
mit besonderer Berücksichtigung der historisch wichtigen Schriften. Abhandlungen zur Geschichte der Mathematik
27 (1903).
Charles W. Newhall. "Recreations" in secondary mathematics. SSM 15 (1915) 277‑293.
Mathematical Association. 259 London Road, Leicester, LE2 3BE.
Catalogue of Books and Pamphlets
in the Library. No details, [c1912],
19pp, bound in at end of Mathematical Gazette, vol. 6 (1911‑1912).
A
First List of Books & Pamphlets in the Library of the Mathematical
Association -- Books and Pamphlets acquired before 1924. Bell, London, 1926.
A
Second List of Books & Pamphlets in the Library of the Mathematical
Association -- Books and Pamphlets acquired during 1924 and 1925. Bell, London, 1929.
A
Third List of Books & Pamphlets in the Library of the Mathematical
Association -- Books and Pamphlets added from 1926 to 1929. Bell, London, 1930.
A
Fourth List of Books & Pamphlets in the Library of the Mathematical
Association -- Books and Pamphlets added from 1930 to 1935. Bell, London, 1936.
Lists 1‑4 edited by E. H.
Neville.
Books
and Periodicals in the Library of the Mathematical Association. Ed. by R. L. Goodstein. MA, 1962.
Includes the four previous lists and additions through 1961.
SEE
ALSO: Riley; Rollett; F. R. Watson.
Stanley Rabinowitz. Index to Mathematical Problems 1980-1984.
MathPro Press, Westford, Massachusetts, 1992.
Cecil B. Read &
James K. Bidwell.
Selected
articles dealing with the history of elementary Mathematics. SSM 76 (1976) 477-483.
Periodical
articles dealing with the history of advanced mathematics -- Parts I &
II. SSM 76 (1976) 581-598 &
687-703.
Rudolf H. Rheinhardt. Bibliography on Whist and Playing
Cards. From: Whist Scores and Card-table Talk, Chicago, 1887. Reprinted by L. & P. Parris, Llandrindod
Wells, nd [1980s].
Pietro Riccardi. Biblioteca Matematica Italiana dalla Origine
della Stampa ai Primi Anni del Secolo XIX.
G. G. Görlich, Milan, 1952, 2 vols. This work appeared in several parts and supplements in the late
19C and early 20C, mostly published by the Società Tipografica Modense, Modena,
1878-1893. Because it appeared in
parts, the contents of early copies are variable and even the reprints may vary. The contents of this set are as follows.
I. 20pp prelims +
Col. 1 - 656 (Abaco - Kirchoffer).
[= original Vol. I.]
Col.
1 - 676 (La Cometa - Zuzzeri) + 2pp correzioni. [= original Vol. II.]
II. 4pp
titles and reverses. Correzioni ed
Aggiunte. [= original Appendice.]
Serie
I.a Col. 1 - 78 +
1½pp Continuazione delle
Correzioni (note that these
have Pag.
when they mean Col.).
Serie
II.a. Col. 81 - 156.
Serie
III.a. Col. 157 - 192 +
Aggiunte al Catalogo delle Opere di sovente citate,
col.
193-194 + 1p Continuazione delle Correzioni (note that these have
Pag. when they mean Col.).
Serie
IV.a. Col. 197 - 208 +
Seconda Aggiunta al Catalogo delle Opere più di
sovente
citate, col. 209 - 212 + Continuazione delle Correzioni in
col.
211-212.
Serie
V.a. Col. 1 - 180.
Serie
VI.a. Col. 179 - 200.
Serie
V & VI must have been published as one volume as Serie V ends
halfway
down a page and then Serie VI begins on the same page.
Serie
VII.a. 2pp introductory note
by Ettore Bortolotti in 1928 saying that this
material
was left as a manuscript by Riccardi and never previously
published +
Col. 1 - 106.
Indice
Alfabetico, of authors, covering the original material and all seven Series
of
Correzioni ed Aggiunte, in 34 unnumbered columns.
Parte
Seconda. Classificazione per materie
delle opere nella Parte I. 18pp
(including
a chronological table) + subject index, pp. 1 - 294.
Catalogo Delle opere più di sovente citate, col. 1 -
54.
[I
have seen an early version which had the following parts: Vol. I, 1893, col. 1‑656; Vol. II, 1873, col. 1-676; Appendice, 1878-1880-1893, col. 1-228. Appendice, nd, col. 1-212. Serie V, col. 1-228. Parte 2, Vol. 1, 1880, pp. 1-294. Renner Katalog 87 describes it as 5 in 2
vols.]
A. W. Riley. School Library Mathematics List --
Supplement No. 1. MA, 1973.
SEE
ALSO: Rollett.
Tom Rodgers. Catalog of his collection of books on
recreational mathematics, etc. The
author, Atlanta, May 1991, 40pp.
Leo F. Rogers. Finding Out in the History of
Mathematics. Produced by the author,
London, c1985, 52pp.
A. P. Rollett. School Library Mathematics List. Bell, London, for MA, 1966.
SEE
ALSO: Riley.
Charles L. Rulfs. Origins of some conjuring works. Magicol 24 (May 1971) 3-5.
José A. Sánchez Pérez. Las Matematicas en la Biblioteca del
Escorial. Imprenta de Estanislao
Maestre, Madrid, 1929.
William L. Schaaf.
List
of works on recreational mathematics.
SM 10 (1944) 193-200.
PLUS: A. Gloden; Additions to Schaaf's "List of works on
mathematical recreations"; SM 13 (1947) 127.
A
Bibliography of Recreational Mathematics.
Op. cit. in Common References, 4 vols., 1955-1978. In these volumes he gives several lists of
relevant books.
Books for the periods 1900-1925 and
1925-c1956 are given as Sections 1.1 (pp. 2-3) and 1.2 (pp. 4-12) in Vol.
1.
Chapter 9, pp. 144-148, of Vol. 1, is
a Supplement, generally covering c1954-c1962, but with some older items.
In Vol. 2, 1970, the Appendix, pp.
181-191, extends to c1969, including some older items and repeating a few from
the Supplement of Vol. 1.
Appendix A of Vol. 3, 1973, pp.
111-113, adds some more items up through 1972.
Appendix A, pp. 134-137, of Vol. 4,
1978, extends up through 1977.
The following VESTPOCKET BIBLIOGRAPHIES are
extensions of the material
in his Bibliographies.
No. 1: Pythagoras and rational triangles; Geoboards and lattices. JRM 16:2 (1983-84) 81-88.
No. 2: Combinatorics; Gambling and sports. JRM
16:3 (1983-84) 170-181.
No. 3: Tessellations and polyominoes; Art and music. JRM 16:4 (1983-84) 268‑280.
No. 4: Recreational miscellany. JRM 17:1 (1984-85) 22-31.
No. 5: Polyhedra;
Topology; Map coloring. JRM 17:2 (1984-85) 95-105.
No. 6: Sundry algebraic notes. JRM 17:3 (1984-85) 195-203.
No. 7: Sundry geometric notes. JRM 18:1 (1985-86) 36-44.
No. 8: Probability;
Gambling. JRM 18:2 (1985-86)
101-109.
No. 9: Games and puzzles. JRM 18:3 (1985-86) 161-167.
No.
10: Recreational
mathematics; Logical puzzles; Expository mathematics. JRM 18:4 (1985-86) 241-246.
No.
11: Logic, Artificial
intelligence, and Mathematical foundations.
JRM 19:1 (1987) 3-9.
No.
12: Magic squares and cubes; Latin squares; Mystic arrays and Number patterns. JRM 19:2 (1987) 81-86.
The
High School Mathematics Library. NCTM,
(1960, 1963, 1967, 1970, 1973); 6th
ed., 1976; 7th ed., 1982; 8th ed., 1987.
SEE ALSO:
Wheeler; Wheeler & Hardgrove.
Early
Books on Magic Squares. JRM 16:1
(1983-84) 1-6.
William L. Schaaf &
David Singmaster. Books on
Recreational Mathematics. A Supplement
to the Lists in William L. Schaaf's A Bibliography of Recreational
Mathematics. Collected by William L.
Schaaf; typed and annotated by David Singmaster. School of Computing, Information Systems and
Mathematics, South Bank University, London, SE1 0AA. 18pp, Dec 1992 and revised several times afterwards.
Peter Schreiber.
Mathematik
und belletristik [1.] & 2. Teil.
Mitteilungen der Mathematischen Gesellschaft der Deutschen
Demokratischer Republik. (1986), no. 4,
57-71 & (1988), no. 1-2, 55-61.
Good on German works relating mathematics and arts.
Mathematiker
als Memoirenschreiber. Alpha (Berlin)
(1991), no. 4, no page numbers on copy received from author. Extends previous work.
S. N. Sen. Scientific works in Sanskrit, translated
into foreign languages and vice‑versa in the 18th and 19th century
A.D. Indian J. History of Science 7
(1972) 44‑70.
Will Shortz. Puzzleana [catalogue of his puzzle
books]. Produced by the author. 14 editions have appeared. The latest is: May 1992, 88pp with 1175 entries in 26 categories, with indexes
of authors and anonymous titles. Some
entries cover multiple items. In Jan
1995, he produced a 19pp Supplement extending to a total of 1451 entries.
David Singmaster.
The
Bibliography of Some Recreational Mathematics Books. School of Computing, Information Systems
and Mathematics, South Bank Univ.
13 Nov 1994, 39pp. Technical Report SBU-CISM-94-09.
2nd ed., Aug 1995, 41pp. Technical Report SBU-CISM-95-08.
3rd ed., Jun 1996, 42pp. Technical Report SBU-CISM-96-12.
4th ed., Jun 1998, 44pp. Technical Report SBU-CISM-98-02.
(Current version is 61pp.)
Books
on Recreational Mathematics. School of Computing,
Information Systems and
Mathematics, South Bank Univ., until
1996.
21 Jan 1991.
Approx. 2951 items on 120pp, ringbound.
30 Jan 1992.
Approx. 3314 items on 138pp, ringbound.
10 Jan 1993.
Approx. 3606 items on 95pp, ringbound.
10 Dec 1994.
Approx. 4303 items plus 67 Old Books on 110pp. Technical
Report SBU‑CISM-94-11.
10 Oct 1996.
Approx. 4842 items plus 84 Old Books on 127pp. Technical
Report SBU-CISM-96-17.
24 May 1999.
Approx. 6015 items plus 133 Old Books on 166pp. Technical
Report SBU-CISM-99-14.
26 Feb 2002.
Approx. 7185 items plus 192 Old Books plus Supplement of
Calculating Devices, on 220pp.
thermal bound.
22 Nov 2003.
Approx. 7811 items plus 202 Old Books plus Supplement of
Calculating Devices, on 244pp.
thermal bound.
Index
to Martin Gardner's Columns and Cross Reference to His Books. (Oct 1993.)
Slightly revised as: Technical
Report SBU-CISM-95-09; School of Computing, Information Systems, and
Mathematics; South Bank University, London, Aug 1995, 22pp. (Current version is 23pp and Don Knuth has
sent 9pp of additional material and I will combine these at some time.)
Harold Adrian Smith. Dick and Fitzgerald Publishers. Books at Brown 34 (1987) 108-114.
Sotheby's [Sotheby Parke
Bernet].
Catalogue
of the J. B. Findlay Collection Books
and Periodicals on Conjuring and the Allied Arts. Part I: A-O 5-6 Jul 1979. Part II: P-Z plus: Mimeographed Books
and Instructions; Flick Books
Catalogues of Apparatus and Tricks
Autograph Letters, Manuscripts, and Typescripts 4-5 Oct 1979. Part III: Posters and Playbills
3-4 Jul 1980. Each with estimates
and results lists.
The
Celebrated Library of Harrison D. Horblit Esq.
Early Science Navigation &
Travel Including Americana with a few medical books. Part I
A - C 10/11 Jun 1974. Part II
D - G 11 Nov 1974. HB.
The sale was then cancelled and the library was sold to E. P. Kraus, qv,
who issued three further catalogues, c1976.
The
Honeyman Collection of Scientific Books and Manuscripts. Seven volumes, each
with estimates and results booklets.
Part I: Printed Books A-B, 30-31 Oct 1978.
Part II:
Printed Books C-E, 30 Apr - 1 May 1979.
Part III: Manuscripts and Autograph Letters of
the 12th to the 20th Centuries.
Part IV: Printed Books F-J, 5-6 Nov 1979.
Part V: Printed Books K-M, 12-13 May 1980.
Part VI: Printed Books N-Sa, 10-11 Nov 1980.
Part VII:
Printed Books Sc-Z and Addenda, 19-20 May 1981.
Lynn A. Steen, ed.
Library
Recommendations for Undergraduate Mathematics.
MAA Reports No. 4, 1992.
Two-Year
College Mathematics Library Recommendations.
MAA Reports No. 5, 1992.
Strens/Guy Collection. Author/Title Listing. Univ. of Calgary. Preliminary Catalogue, 319 pp., July 1986. [The original has a lot of blank space. I have a computer version which is reduced
to 67pp.]
Eva Germaine Rimington
Taylor. The Mathematical Practitioners
of Tudor & Stuart England
1485-1714. CUP for the Institute
of Navigation, 1970.
Eva Germaine Rimington
Taylor. The Mathematical Practitioners
of Hanoverian England 1714‑1840. CUP for the Institute of Navigation,
1966.
PLUS: Kate Bostock, Susan Hurt & Michael Hart;
An Index to the Mathematical Practitioners of Hanoverian England 1714-1840; Harriet Wynter Ltd., London,
1980.
W. R. Utz. Letter:
Mathematics in literature. MM 55
(1982) 249‑250. Utz has sent his
3pp original more detailed version along with 4pp of further citations. This extends Koehler's article.
George Walker. The Art of Chess-Play: A New Treatise on the
Game of Chess. 4th ed., Sherwood,
Gilbert & Piper, London, 1846.
Appendix: Bibliographical
Catalogue of the chief printed books, writers, and miscellaneous articles on
chess, up to the present time, pp. 339-375.
Frank R. [Joe] Watson, ed. Booklists.
MA.
Puzzles,
Problems, Games and Mathematical Recreations.
16pp, 1980.
Selections
from the Recommended Books. 18pp, 1980.
Full
List of Recommended Books. 105pp, 1984.
Margariete Montague
Wheeler. Mathematics Library --
Elementary and Junior High School. 5th
ed., NCTM, 1986.
SEE
ALSO: Schaaf; Wheeler & Hardgrove.
Margariete Montague Wheeler &
Clarence Ethel Hardgrove.
Mathematics Library -- Elementary and Junior High School. NCTM, (1960; 1968; 1973); 4th ed., 1978.
SEE
ALSO: Schaaf; Wheeler.
Ernst Wölffing. Mathematischer Bücherschatz. Systematisches Verzeichnis der wichtigsten
deutschen und ausländischen Lehrbücher und Monographien des 19. Jahrhunderts
auf dem Gebiete der mathematischen Wissenschaften. I: Reine Mathematik; (II:
Angewandte Mathematik never appeared).
AGM 16, part I (1903).
Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games
and Some Related Material. Op. cit. in
3.B.
4.A. GENERAL THEORY AND NIM‑LIKE GAMES
Conway's
extension of this theory is well described in Winning Ways and later work is
listed in Fraenkel's Bibliography -- see section 3.B & 4 -- so I will not
cover such material here.
See
MUS I 145-147.
(a,
b) denotes the game where one can
take 1, 2, ..., or
a away from one pile, starting
with b
in the pile, with the last player winning. The version (10,
100) is sometimes called Piquet des
Cavaliers or Piquet à Cheval, a name which initially perplexed me. Piquet is one of the older card games, being
well known to Rabelais (1534) and was known in the 16C as Cent (or Saunt or
Saint) because of its goal of 100 points.
See: David Parlett; (Oxford Guide
to Card Games, 1990 =) A History of Card Games; OUP, 1991, pp. 24
& 175-181. The connection with
horses undoubtedly indicates that (10,
100) was viewed as a game which could
be played without cards, while riding -- see Les Amusemens, Decremps.
( 3, 13) Dudeney, Stong
( 3, 15) Mittenzwey, Hoffmann, Mr. X,
Dudeney, Blyth,
( 3, 17) Fourrey,
( 3, 21) Blyth, Hummerston,
( 4, 15) Mittenzwey,
( 6, 30) Pacioli, Leske, Mittenzwey,
Ducret,
( 6, 31) Baker,
( 6, 50) Ball-FitzPatrick,
( 6, 52) Rational Recreations
( 6, 57) Hummerston,
( 7, 40) Mittenzwey,
( 7, 41) Sprague,
( 7, 45) Mittenzwey,
( 7, 50) Decremps,
( 7, 60) Fourrey,
( 8, 100) Bachet,
Carroll,
( 9, 100) Bachet,
Ozanam, Alberti
(10, 100) Bachet, Henrion, Ozanam, Alberti, Les Amusemens,
Hooper, Decremps,
Badcock,
Jackson, Rational Recreations, Manuel des Sorciers,
Boy's Own
Book, Nuts to Crack, Young Man's Book, Carroll,
Magician's Own
Book, Book of 500 Puzzles, Secret Out,
Boy's Own Conjuring
Book, Vinot, Riecke, Fourrey, Ducret, Devant,
(10, 120) Bachet,
(12, 134) Decremps,
General case: Bachet, Ozanam, Alberti, Decremps, Boy's Own
Book, Young Man's Book, Vinot, Mittenzwey, (others ?? check)
Versions
with limited numbers of each value or using a die -- see 4.A.1.a.
Version
where an odd number in total has to be taken:
Dudeney, Grossman & Kramer, Sprague.
Versions
with last player losing: Mittenzwey,
Pacioli. De Viribus.
c1500. Ff. 73v - 76v. XXXIIII effecto afinire qualunch' numero
na'ze al compagno anon prendere piu de un termi(n)ato .n. (34th effect to
finish whatever number is before the company, not taking more than a limiting
number) = Peirani 109‑112.
Phrases it as an addition problem.
Considers (6, 30) and the general problem.
David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games.
Penguin, 1991, pp. 174-175.
"Early references to 'les luettes', said to have been played by
Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in
1534, seem to suggest a game of the Nim family (removing numbers of objects
from rows and columns)."
Cardan. Practica Arithmetice. 1539.
Chap. 61, section 18, ff. T.iiii.v - T.v.r (p. 113). "Ludi mentales". One has
1, 3, 6 and the other has 2, 4, 5;
or one has
1, 3, 5, 8, 9 and
the other has 2, 4, 6, 7, 10; one one wants to make 100.
"Sunt magnæ inventionis, & ego inveni æquitando & sine
aliquo auxilio cum socio potes ludere & memorium exercere ...."
Baker. Well Spring of Sciences.
1562? Prob. 5: To play at 31
with Numbers, 1670: pp. 353‑354.
??NX. (6, 31).
Bachet. Problemes.
1612. Prob. XIX: 1612,
99-103. Prob. XXII, 1624: 170-173; 1884: 115‑117. Phrases it as an addition problem. First considers (10, 100), then (10, 120), (8, 100), (9, 100), and the general case. Labosne omits the demonstration.
Dennis Henrion. Nottes to van Etten. 1630.
Pp. 19-20. (10, 100) as an addition problem, citing Bachet.
Ozanam. 1694.
Prob. 21, 1696: 71-72; 1708: 63‑64. Prob. 25, 1725: 182‑184. Prob. 14, 1778: 162-164; 1803: 163-164; 1814: 143-145. Prob. 13,
1840: 73-74. Phrases it as an addition
problem. Considers (10, 100)
and (9, 100) and remarks on the general case.
Alberti. 1747.
Due persone essendo convenuto ..., pp. 105‑108 (66‑67). This is a slight recasting of Ozanam.
Les Amusemens. 1749.
Prob. 10, p. 130: Le Piquet des Cavaliers. (10, 100) in additive
form. "Deux amis voyagent à
cheval, l'un propose à l'autre un cent de Piquet sans carte."
William Hooper. Rational Recreations, In which the
Principles of Numbers and Natural Philosophy Are clearly and copiously
elucidated, by a series of Easy, Entertaining, Interesting Experiments. Among which are All those commonly performed
with the cards. [Taken from my 2nd ed.]
4 vols., L. Davis et al., London, 1774;
2nd ed., corrected, L. Davis et al., London, 1783-1782 (vol. 1 says
1783, the others say 1782; BMC gives 1783-82);
3rd ed., corrected, 1787; 4th
ed., corrected, B. Law et al., London, 1794.
[Hall, BCB 180-184 & Toole Stott 389-392. Hall says the first four eds. have identical
pagination. I have not seen any
difference in the first four editions, except as noted in Section 6.P.2. Hall, OCB, p. 155. Heyl 177 notes the different datings of the 2nd ed, Hall, BCB 184 and Toole Stott 393 is a 2
vol. 4th ed., corrected, London, 1802.
Toole Stott 394 is a 2 vol. ed. from Perth, 1801. I have a note that there was an 1816 ed, but
I have no details. Since all relevant
material seems the same in all volumes, I will cite this as 1774.] Vol. 1, recreation VIII: The magical
century. (10, 100) in additive form. Mentions other versions and the general rule.
I
don't see any connection between this and Rational Recreations, 1824.
Henri Decremps. Codicile de Jérôme Sharp, Professeur de
Physique amusante; Où l'on trouve parmi
plusieurs Tours dont il n'est point parlé dans son Testament, diverses
récréations relatives aux Sciences & Beaux-Arts; Pour servir de troisième suite
À La Magie Blanche Dévoilée. Lesclapart, Paris, 1788.
Chap. XXVII, pp. 177-184: Principes mathématiques sur le piquet à
cheval, ou l'art de gagner son diner en se promenant. Does (10, 100) in
additive form, then discusses the general method, illustrating with (7, 50) and (12, 134).
Badcock. Philosophical Recreations, or, Winter
Amusements. [1820]. Pp. 33-34, no. 48: A curious recreation with
a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50
counters. Discusses general
case, but doesn't notice that the limitation to 50 counters each
considerably changes the game!
Jackson. Rational Amusement. 1821.
Arithmetical Puzzles, no. 47, pp. 11 & 64. Additive form of (10,
100).
Rational Recreations. 1824. Exercise 12(?), pp. 57-58.
As in Badcock. Then says it can
be generalised and gives (6, 52).
Manuel des Sorciers. 1825.
Pp. 57-58, art. 30: Le piquet sans cartes. ??NX (10, 100) done subtractively.
The Boy's Own Book.
The
certain game. 1828: 177; 1828-2: 236;
1829 (US): 104; 1855: 386‑387; 1868: 427.
The
magical century. 1828: 180; 1828-2: 236‑237; 1829 (US): 104-105; 1855: 391‑392.
Both are additive phrasings of (10, 100).
The latter mentions using other numbers and how to win then.
Nuts to Crack V (1836), no.
70. An arithmetical problem. (10, 100).
Young Man's Book. 1839.
Pp. 294-295. A curious
Recreation with a Hundred Numbers, usually called the Magical Century. Almost identical to Boy's Own Book.
Lewis Carroll.
Diary
entry for 5 Feb 1856. In
Carroll-Gardner, pp. 42-43. (10,
100). Wakeling's note in the Diaries
indicates he is not familiar with this game.
Diary
entry for 24 Oct 1872. Says he has
written out the rules for Arithmetical Croquet, a game he recently invented. Roger Lancelyn Green's abridged version of
the Diaries, 1954, prints a MS version dated 22 Apr 1889. Carroll-Wakeling, prob. 38, pp. 52-53 and
Carroll-Gardner, pp. 39 & 42 reprint this, but Gardner has a misprinted
date of 1899. Basically (8, 100),
but passing the values 10, 20,
..., requires special moves and one may
have to go backward. Also, when a move
is made, some moves are then barred for the next player. Overall, the rules are typically
Carrollian-baroque.
Magician's Own Book. 1857.
The certain
game, p. 243. As in Boy's Own Book.
The
magical century, pp. 244-245. As in
Boy's Own Book.
Book of 500 Puzzles. 1859.
The
certain game, p. 57. As in Boy's Own
Book.
The
magical century, pp. 58-59. As in Boy's
Own Book.
The Secret Out. 1859.
Piquet on horseback, pp. 397-398 (UK: 130‑131) -- additive (10, 100)
unclearly explained.
Boy's Own Conjuring Book. 1860.
The
certain game, pp. 213‑214. As in
Boy's Own Book.
Magical
century, pp. 215. As in Boy's Own Book.
Vinot. 1860. Art. XI: Un cent de
piquet sans cartes, pp. 19-20. (10.
100). Says the idea can be generalised,
giving (7, 52) as an example.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 563-III, pp.
247: Wer von 30 Rechenpfennigen den letzen wegnimmt, hat gewonnen. (6, 30).
F. J. P. Riecke. Mathematische Unterhaltungen. 3 vols., Karl Aue, Stuttgart, 1867, 1868
& 1873; reprint in one vol.,
Sändig, Wiesbaden, 1973. Vol. 3, art
22.2, p. 44. Additive form of (10, 100).
Mittenzwey. 1880.
Probs. 286-287, pp. 52 & 101-102;
1895?: 315-317, pp. 56 & 103-104;
1917: 315-317, pp. 51 & 98.
(6,
30), last player wins.
(4,
15), last player loses, the solution discusses other cases: (7, 40), (7, 45)
and indicates the general solution.
(added
in 1895?) (3, 15), last player loses.
Hoffmann. 1893.
Chap VII, no. 19: The fifteen matches puzzle, pp. 292 & 300‑301
= Hoffmann-Hordern, p. 197. (3,
15). c= Benson, 1904, The fifteen match
puzzle, pp. 241‑242.
Ball-FitzPatrick. 1st ed., 1898. Deuxième exemple, pp. 29-30.
(6, 50).
E. Fourrey. Récréations Arithmétiques. (Nony, Paris, 1899; 2nd ed., 1901); 3rd ed., Vuibert & Nony, Paris, 1904; (4th ed., 1907); 8th ed., Librairie Vuibert, Paris, 1947. [The 3rd and 8th eds are identical except
for the title page, so presumably are identical to the 1st ed.] Sections 65‑66: Le jeu du piquet à
cheval, pp. 48‑49. Additive forms
of (10, 100) and (7, 60). Then gives subtractive form for a pile of
matches for (3, 17).
Étienne Ducret. Récréations Mathématiques. Garnier Frères, Paris, nd [not in BN, but a
similar book, nouv. ed., is 1892]. Pp.
102‑104: Le piquet à cheval.
Additive version of (10,
100) with some explanation of the use
of the term piquet. Discusses (6, 30).
Mr. X [possibly J. K. Benson --
see entry for Benson in Abbreviations].
His Pages. The Royal Magazine
9:3 (Jan 1903) 298-299. A good game for
two. (3, 15) as a subtraction game.
David Devant. Tricks for Everyone. Clever Conjuring with Everyday Objects. C. Arthur Pearson, London, 1910. A counting race, pp. 52-53. (10, 100).
Dudeney. AM.
1917. Prob. 392: The pebble
game, pp. 117 & 240. (3, 15) &
(3, 13) with the object being to
take an odd number in total. For 15,
first player wins; for 13, second
player wins. (Barnard (50 Telegraph
..., 1985) gives the case (3, 13).)
Blyth. Match-Stick Magic.
1921.
Fifteen
matchstick game, pp. 87-88. (3, 15).
Majority
matchstick game, p. 88. (3, 21).
Hummerston. Fun, Mirth & Mystery. 1924.
Two
second-sight tricks (no. 2), p. 84. (6,
57), last player losing.
A
match mystery, p. 99. (3, 21), last player losing.
H. D. Grossman & David
Kramer. A new match-game. AMM 52 (1945) 441‑443. Cites Dudeney and says Games Digest (April
1938) also gave a version, but without solution. Gives a general solution whether one wants to take an odd total
or an even total.
C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. How to design a "Pircuit" or Puzzle circuit, pp.
388-394. On pp. 388-391, Harry Rudloe
describes a relay circuit for playing the subtractive form of (3, 13), which he calls the "battle of
numbers" game.
Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London,
1963. Problem 24: "Ungerade"
gewinnt, pp. 16 & 44‑45. (=
'Odd' is the winner, pp. 18 & 53‑55.) (7, 41) with the winner
being the one who takes an odd number in total. Solves (7, b) and states the structure for (a, b).
I
also have some other recent references to this problem. Lewis (1983) gives a general solution which
seems to be wrong.
Numerical
variations: Badcock, Gibson, McKay.
Die
versions: Secret Out (UK), Loyd,
Mott-Smith, Murphy.
Baker. Well Spring of Sciences.
1562? Prob. 5: To play at 31
with Numbers, 1670: pp. 353‑354.
??NX. (6, 31). ??CHECK if this has the limited use of
numbers.
John Fisher. Never Give a Sucker an Even Break. (1976);
Sphere Books, London, 1978.
Thirty-one, pp. 102-104. (6,
31) additively, but played with just 4
of each value, the 24 cards of ranks 1
-- 6, and the first to exceed 31
loses. He says it is played extensively
in Australia and often referred to as "The Australian Gambling Game of
31". Cites the 19C gambling expert
Jonathan Harrington Green who says it was invented by Charles James Fox (1749‑1806). Gives some analysis.
Badcock. Philosophical Recreations, or, Winter
Amusements. [1820]. Pp. 33-34, no. 48: A curious recreation with
a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50
counters. Discusses general
case, but doesn't notice that the limitation to 50 counters each
considerably changes the game!
Nuts to Crack V (1836), no.
71. (6, 31) additively, with four of
each value. "Set down on a slate,
four rows of figures, thus:-- ... You agree to rub out one figure alternately,
to see who shall first make the number thirty-one."
Magician's Own Book. 1857.
Art. 31: The trick of thirty‑one, pp. 70‑71. (6, 31)
additively, but played with just 4 of each value -- e.g. the 24 cards of
ranks 1 -- 6. The author advises you not to play it for money with
"sporting men" and says it it due to Mr. Fox. Cf Fisher.
= Boy's Own Conjuring Book; 1860; Art. 29: The trick of thirty‑one,
pp. 78‑79. = The Secret Out;
1859, pp. 65-66, which adds a footnote that the trick is taken from the book
One Hundred Gambler Tricks with Cards by J. H. Green, reformed gambler,
published by Dick & Fitzgerald.
The Secret Out (UK), c1860. To throw thirty‑one with a die before
your antagonist, p. 7. This is
incomprehensible, but is probably the version discussed by Mott-Smith.
Edward S. Sackett. US Patent 275,526 -- Game. Filed: 9 Dec 1882; patented: 10 Apr 1883. 1p
+ 1p diagrams. Frame of six rows
holding four blocks which can be slid from one side to the other to play the 31
game, though other numbers of rows, blocks and goal may be used. Gives an example of a play, but doesn't go
into the strategy at all.
Larry Freeman. Yesterday's Games. Taken from "an 1880 text" of games. (American edition by H. Chadwick.) Century House, Watkins Glen, NY, 1970. P. 107: Thirty-one. (6, 31) with 4 of each value -- as in Magician's Own Book.
Algernon Bray. Letter:
"31" game. Knowledge 3
(4 May 1883) 268, item 806. "...
has lately made its appearance in New York, ...." Seems to have no idea as how to win.
Loyd. Problem 38: The twenty‑five up puzzle. Tit‑Bits 32 (12 Jun &
3 Jul 1897) 193 & 258.
= Cyclopedia. 1914. The dice game, pp. 243 & 372. = SLAHP: How games originate, pp. 73 &
114. The first play is arbitrary. The second play is by throwing a die. Further values are obtained by rolling the
die by a quarter turn.
Ball-FitzPatrick. 1st ed., 1898. Généralization récente de cette question, pp. 30-31. (6, 50)
with each number usable at most 3 times. Some analysis.
Ball. MRE, 4th ed., 1905, p. 20.
Some analysis of (6, 50) where each player can play a value at most 3
times -- as in Ball-FitzPatrick, but with the additional sentence: "I have never seen this extension
described in print ...." He also
mentions playing with values limited to two times. In the 5th ed., 1911, pp. 19-21, he elaborates his analysis.
Dudeney. CP.
1907. Prob. 79: The thirty-one
game, pp. 125-127 & 224. Says it
used to be popular with card-sharpers at racecourses, etc. States the first player can win if he starts
with 1, 2 or 5, but the analysis of cases 1 and 2 is complicated. This occurs as No. 459: The thirty-one
puzzle, Weekly Dispatch (17 Aug 1902) 13 & (31 Aug 1902) 13, but he leaves
the case of opening move 2 to the reader, but I don't see the answer given in
the next few columns.
Devant. Tricks for Everyone. Op. cit. in 4.A.1. 1910. The thirty-one
trick, pp. 53-54. Says to get to 3, 10, 17, 24.
Hummerston. Fun, Mirth & Mystery. 1924.
Thirty-one -- a game of skill, pp. 95-96. This uses a layout of four copies of the numbers 1, 2, 3, 4, 5, 6 with one copy of 20 in a
5 x 5 square with the 20
in the centre. Says to get
to 3, 10, 17, 24, but that this will lose to an experienced
player.
Loyd Jr. SLAHP.
1928. The "31 Puzzle
Game", pp. 3 & 87. Loyd Jr
says that as a boy, he often had to play it against all comers with a $50 prize
to anyone who could beat 'Loyd's boy'.
This is the game that Loyd Sr called 'Blind Luck', but I haven't found
it in the Cyclopedia. States the first
player wins with 1, 2 or 5, but only sketches the case for opening with 5. I have seen an example of Blind Luck -- it
has four each of the numbers 1 - 6 arranged around a frame containing a
horseshoe with 13 in it.
McKay. Party Night. 1940. The 21 race, pp. 166. Using the numbers 1, 2, 3, 4, at most four
times, achieve 21. Says to get 1, 6, 11, 16. He doesn't realise that the sucker can be mislead into playing
first with a 1 and losing! Says that
with 1, ..., 5 at most four times, one wants to achieve 26
and that with 1, ..., 6 at most four times, one wants to achieve 31. Gives just the key numbers each time.
Geoffrey Mott-Smith. Mathematical Puzzles for Beginners and
Enthusiasts. (Blakiston, 1946); revised 2nd ed., Dover, 1954.
Prob.
179: The thirty-one game, pp. 117-119
& 231-232. As in Dudeney.
Prob.
180: Thirty-one with dice, p. 119
& 232-233. Throw a die, then make quarter turns to
produce a total of 31. Analysis based
on digital roots (i.e. remainders (mod 9)).
First player wins if the die comes up 4, otherwise the second player can
win. He doesn't treat any other totals.
"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and
Sciences, Croydon, 1947. "Trente
et un", pp. 56-57. Says he doesn't
know any name for this. Get 31 using 4
each of the cards A, 2, ..., 6. Says first player loses easily if he starts
with 4, 5, 6 (not true according to Dudeney) and that gamblers dupe the sucker
by starting with 3 and winning enough that the sucker thinks he can win by
starting with 3. But if he starts with
a 1 or 2, then the second player must play low and hope for a break.
Walter B. Gibson. Fell's Guide to Papercraft Tricks, Games and
Puzzles. Frederick Fell, NY, 1963. Pp. 54-55: First to fifty. First describes (50, 6), but then adds a
version with slips of paper: eight
marked 1 and seven marked with 2, 3, 4,
5, 6 and you secretly extract a 6 slip
when the other player starts.
Harold Newman. The 31 Game. JRM 23:3 (1991) 205-209.
Extended analysis. Confirms
Dudeney. Only cites Dudeney &
Mott-Smith.
Bernard Murphy. The rotating die game. Plus 27 (Summer 1994) 14-16. Analyses the die version as described by
Mott-Smith and finds the set,
S(n), of winning moves for
achieving a count of n by the first player, is periodic with period
9 from n = 8, i.e. S(n+9) =
S(n) for n ³ 8. There is no first player winning move if and only if n is
a multiple of 9. [I have confirmed this
independently.]
Ken de Courcy. The Australian Gambling Game of 31. Supreme Magic Publication, Bideford, Devon,
nd [1980s?]. Brief description of the game
and some indications of how to win. He
then plays the game with face-down cards!
However, he insures that the cards by him are one of of each rank and he
knows where they are.
Loyd?? Problem 43: The daisy game.
Tit‑Bits 32 (17 Jul
& 7 Aug 1897) 291 &
349. (= Cyclopedia. 1914.
A daisy puzzle game, pp. 85 & 350.
c= MPSL2, prob. 57, pp. 40‑41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of Kayles with 13
objects. Solution uses a symmetry
argument -- but the Tit‑Bits solution was written by Dudeney.
Dudeney. Problem 500: The cigar puzzle. Weekly Dispatch (7 Jun, 21 Jun, 5 Jul,
1903) all p. 16. (= AM, prob. 398,
pp. 119, 242.) Symmetry in placement
game, using cigars on a table.
Loyd. Cyclopedia. 1914. The great Columbus problem, pp. 169 &
361. (= MPSL1, prob. 65, pp. 62 &
144. = SLAHP: When men laid eggs,
pp. 75 & 115.) Placing eggs on
a table.
Maurice Kraitchik. La Mathématique des Jeux. Stevens, Bruxelles, 1930. Section XII, prob. 1, p. 296. (= Mathematical Recreations; Allen &
Unwin, London, 1943; Problem 1, pp. 13‑14.) Child plays black and white against two
chess players and guarantees to win one game.
[MJ cites L'Echiquier (1925) 84, 151.]
CAUTION. The 2nd edition of Math. des Jeux, 1953, is
a translation of Mathematical Recreations and hence omits much of the earlier
edition.
Leopold. At Ease!
1943. Chess wizardry in two
minutes, pp. 105‑106. Same as
Kraitchik.
This
has objects in a line or a circle and one can remove one object or two adjacent
objects (or more adjacent objects in a generalized version of the game). This derives from earlier games with an
array of pins at which one throws a ball or stick.
Murray 442 cites Act 17 of Edward IV,
c.3 (1477): "Diversez novelx
ymagines jeuez appellez Cloishe Kayles ..." This outlawed such games.
A 14C picture is given in [J. A. R. Pimlott; Recreations; Studio Vista,
1968, plate 9, from BM Royal MS 10 E IV f.99] showing a 3 x 3
array of pins. A version is
shown in Pieter Bruegel's painting "Children's Games" of 1560 with
balls being thrown at a row of pins by a wall, in the back right of the
scene. Versions of the game are given
in the works of Strutt and Gomme cited in 4.B.1. Gomme II 115‑116 discusses it under Roly‑poly, citing
Strutt and some other sources. Strutt
270‑271 (= Strutt-Cox 219-220) calls it "Kayles, written also cayles
and keiles, derived from the French word quilles". He has redrawings of two 14C engravings
(neither that in Pimlott) showing lines of pins at which one throws a stick (=
plate opp. 220 in Strutt-Cox). He also
says Closh or Cloish seems to be the same game and cites prohibitions of it in
c1478 et seq. Loggats was analogous and
was prohibited under Henry VIII and is mentioned in Hamlet.
14C MS in the British Museum,
Royal Library, No. 2, B. vii.
Reproduced in Strutt, p. 271.
Shows a monk(?) standing by a line of eight conical pins and another
monk(?) throwing a stick at the pins.
Anonymous. Games of the 16th Century. The Rockliff New Project Series. Devised by Arthur B. Allen. The Spacious Days of Queen Elizabeth. Background Book No. 5. Rockliff Publishing, London, ©1950, 4th
ptg. The Background Books seem to be
consecutively paginated as this booklet is paginated 129-152. Pp. 133-134 describes loggats, quoting
Hamlet and an unknown poet of 1611. P.
137 is a photograph of the above 14C illustration. The caption is "Skittles, or "Kayals", and
Throwing a Whirling Stick".
van Etten. 1624.
Prob. 72 (misnumbered 58) (65), pp 68‑69 (97‑98): Du jeu des
quilles (Of the play at Keyles or Nine-Pins).
Describes the game as a kind of ninepins.
Loyd. Problem 43: The daisy game.
Tit‑Bits 32 (17 Jul
& 7 Aug 1897) 291 &
349. (= Cyclopedia. 1914.
A daisy puzzle game, pp. 85 & 350.
c= MPSL2, prob. 57, pp. 40‑41 & 140. c= SLAHP: The daisy game, pp. 42 &
99.) Circular version of Kayles with 13
objects. See also 4.A.2.
Dudeney. Sharpshooters puzzle. Problem 430. Weekly Dispatch (26 Jan, 9 Feb, 1902) both p. 13. Simple version of Kayles.
Ball. MRE, 4th ed., 1905, pp. 19-20.
Cites Loyd in Tit‑Bits.
Gives the general version:
place p counters in a circle and one can take not
more than m adjacent ones.
Dudeney. CP.
1907. Prob. 73: The game of
Kayles, pp. 118‑119 & 220.
Kayles with 13 objects.
Loyd. Cyclopedia. 1914. Rip van Winkle puzzle, pp. 232 & 369‑370. (c= MPSL2, prob. 6, pp. 5 & 122.) Linear version with 13 pins and the second
knocked down. Gardner asserts that Dudeney
invented Kayles, but it seems to be an abstraction from the old form of the
game.
Rohrbough. Puzzle Craft, later version, 1940s?. Daisy Game, p. 22. Kayles with 13 petals of a daisy.
Philip Kaplan. More Posers. (Harper & Row, 1964);
Macfadden-Bartell Books, 1965.
Prob. 45, pp. 48 & 95.
Circular kayles with five objects.
Doubleday - 2. 1971.
Take your pick, pp. 63-65. This
is Kayles with a row of 10, but he says the first player can only take one.
Nim is the game with a number of piles
and a player can take any number from one of the piles. Normally the last one to play wins.
David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games.
Penguin, 1991. Pp. 174-175. "Early references to 'les luettes',
said to have been played by Anne de Bretagne and Archduke Philip the Fair in
1503, and by Gargantua in 1534, seem to suggest a game of the Nim family
(removing numbers of objects from rows and columns)."
Charles L. Bouton. Nim: a game with a complete mathematical
theory. Annals of Math. (2) 3 (1901/02)
35‑39. He says Nim is played at
American colleges and "has been called Fan‑Tan, but as it is not the
Chinese game of that name, the name in the title is proposed for it." He says Paul E. More showed him the misère
(= last player loses) version in 1899,
so it seems that Bouton did not actually invent the game himself.
Ahrens. "Nim", ein amerikanisches Spiel
mit mathematischer Theorie.
Naturwissenschaftliche Wochenschrift
17:22 (2 Mar 1902) 258‑260.
He says that Bouton has admitted that he had confused Nim and Fan‑Tan. Fan‑Tan is a Chinese game where you
bet on the number of counters (mod 4) in someone's hand. Parker, Ancient Ceylon, op. cit. in 4.B.1,
pp. 570-571, describes a similar game, based on odd and even, as popular in
Ceylon and "certainly one of the earliest of all games".
For
more about Fan-Tan, see the following.
Stewart Culin. Chess and playing cards. Catalogue of games and implements for
divination exhibited by the United States National Museum in connection with
the Department of Archæology and Paleontology of the University of Pennsylvania
at the Cotton States and International Exposition, Atlanta, Georgia, 1895. IN: Report of the U. S. National Museum,
year ending June 30, 1896. Government
Printing Office, Washington, 1898, HB, pp. 665-942. [There is a reprint by Ayer Co., Salem, Mass., c1990.] Fan-Tan (= Fán t‘án = repeatedly
spreading out) is described on pp. 891 & 896, with discussion of related
games on pp. 889-902.
Alan S. C. Ross. Note 2334:
The name of the game of Nim. MG
37 (No. 320) (May 1953) 119‑120.
Conjectures Bouton formed the word 'nim' from the German 'nimm'. Gives some discussion of Fan‑Tan and
quotes MUS I 72.
J. L. Walsh. Letter:
The name of the game of Nim. MG
37 (No. 322) (Dec 1953) 290.
Relates that Bouton said that he had chosen the word from the German
'nimm' and dropped one 'm'.
W. A. Wythoff. A modification of the game of Nim. Nieuw Archief voor Wiskunde (Groningen) (2)
7 (1907) 199‑202. He considers a
Nim game with two piles allows the extra move of taking the same amount from
both piles. [Is there a version with
more piles where one can take any number from one pile or equal amounts from
two piles?? See Barnard, below for a
three pile version.]
Ahrens. MUS I.
1910. III.3.VII: Nim, pp. 72‑88. Notes that Nim is not the same as Fan‑Tan,
has been known in Germany for decades and is played in China. Gives a thorough discussion of the theory of
Nim and of an equivalent game and of Wythoff's game.
E. H. Moore. A generalization of the game called
Nim. Annals of Math. (2) 11 (1910) 93‑94. He considers a Nim game with n
piles and one is allowed to take any number from at most k
piles.
Ball. MRE, 5th ed., 1911, p. 21.
Sketches the game of Nim and its theory.
A. B. Nordmann. One Hundred More Parlour Tricks and
Problems. Wells, Gardner, Darton &
Co., London, nd [1927 -- BMC]. No. 13:
The last match, pp. 10-11. Thirty
matches divided at random into three heaps.
Last player loses. Explanation
of how to win is rather cryptic:
"you must try and take away ... sufficient ... to leave the matches
in the two or three heaps remaining, paired in ones, twos, fours, etc., in
respect of each other."
Loyd Jr. SLAHP.
1928. A tricky game, pp. 47
& 102. Nim (3, 4, 8).
Emanuel Lasker. Brettspiele der Völker. 1931.
See comments in 4.A.5. Jörg
Bewersdorff [email of 6 Jun 1999] says that Lasker considered a three person
Nim and found an equilibrium for it -- see: Jörg Bewersdorff; Glück, Logik und
Bluff Mathematik im Spiel -- Methoden,
Ergebnisse und Grenzen; Vieweg, 1998, Section 2.3 Ein Spiel zu dritt, pp. 110-115.
Lynn Rohrbough, ed. Fun in Small Spaces. Handy Series, Kit Q, Cooperative Recreation
Service, Delaware, Ohio, nd [c1935].
Take Last, p. 10. Last player
loses Nim (3, 5, 7).
Rohrbough. Puzzle Craft. 1932.
Japanese
Corn Game, p. 6 (= p. 6 of 1940s?).
Last player loses Nim (1, 2, 3, 4, 5).
Japanese
Corn Game, p. 23. Last player loses
Nim (3, 5, 7).
René de Possel. Sur la Théorie Mathématique des Jeux de
Hasard et de Réflexion. Actualités
Scientifiques et Industrielles 436.
Hermann, Paris, 1936. Gives the
theory of Nim and also the misère version.
Depew. Cokesbury Game Book.
1939. Make him take it, pp.
187-188. Nim (3, 4, 5), last player loses.
Edward U. Condon, Gereld L.
Tawney & Willard A. Derr. US Patent
2,215,544 -- Machine to Play Game of Nim.
Filed: 26 Apr 1940; patented: 24
Sep 1940. 10pp + 11pp diagrams.
E. U. Condon. The Nimatron. AMM 49 (1942) 330‑332.
Has photo of the machine.
Benedict Nixon & Len
Johnson. Letters to the Notes &
Queries Column. The Guardian
(4 Dec 1989) 27. Reprinted in: Notes & Queries, Vol. 1; Fourth Estate,
London, 1990, pp. 14-15. These describe
the Ferranti Nimrod machine for playing Nim at the Festival of Britain,
1951. Johnson says it played Nim (3, 5, 6) with a maximum move of
3. The Catalogue of the
Exhibition of Science shows this as taking place in the Science Museum.
H. S. M. Coxeter. The golden section, phyllotaxis, and
Wythoff's game. SM 19 (1953) 135‑143. Sketches history and interconnections.
H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Chap. 11: The golden section and phyllotaxis, pp. 160-172. Extends his 1953 material.
A. P. Domoryad. Mathematical Games and Pastimes. (Moscow, 1961). Translated by Halina Moss.
Pergamon, Oxford, 1963. Chap.
10: Games with piles of objects, pp. 61‑70. On p. 62, he asserts that Wythoff's game is 'the Chinese national
game tsyanshidzi ("picking stones")'. However M.‑K. Siu cannot recognise such a Chinese game,
unless it refers to a form of jacks, which has no obvious connection with
Wythoff's game or other Nim games. He
says there is a Chinese character, 'nian', which is pronounced 'nim' in
Cantonese and means to pick up or take things.
N. L. Haddock. Note 2973:
A note on the game of Nim. MG 45
(No. 353) (Oct 1961) 245‑246.
Wonders if the game of Nim is related to Mancala games.
T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965.
Section on misère version of Wythoff's game, p. 133. Richard Guy (letter of 27 Feb 1985) says
this is one of O'Beirne's few mistakes -- cf next entry.
Winning Ways. 1982.
P. 407 says Wythoff's game is also called Chinese Nim or Tsyan‑shizi. No reference given. See comment under Domoryad above. This says many authors have done this
incorrectly.
D. St. P. Barnard. 50 Daily Telegraph Brain‑Twisters. Javelin Books, Poole, Dorset, 1985. Prob. 30: All buttoned up, pp. 49‑50,
91 & 115. He suggests three pile
game where one can take any number from one pile or an equal number from any
two or all three piles. [See my note to
Wythoff, above.]
Matthias Mala. Schnelle Spiele. Hugendubel, Munich, 1988.
San Shan, p. 66. This describes
a nim-like game named San Shan and says it was played in ancient China.
Jagannath V. Badami. Musings on Arithmetical Numbers Plus Delightful Magic Squares. Published by the author, Bangalore, India,
nd [Preface dated 9 Sep 1999]. Section
4.16: The game of Nim, pp. 124-125.
This is a rather confused description of one pile games (21, 5) and (41,
5), but he refers to solving them by (mentally) dividing the pile into
piles. This makes me think of combining
the two games, i.e. playing Nim with several piles but with a limit on the
number one can take in a move.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. Ff. 134-144 are: Essay 10 Part 5. See 4.B.1 for more details.
At the top of f. 134.r, he has added a note: "This is probably my earliest Note on
Games of Skill. I do not recollect the
date. 3 March 1865". He then describes Tit Tat To and makes some
simple analysis, but he never uses a name for it.
Charles Babbage. Notebooks -- unpublished collection of MSS
in the BM as Add. MS 37205. ??NX. See 4.B.1 for more details. On f. 304, he starts on analysis of
games. Ff. 310‑383 are
almost entirely devoted to Tit-Tat-To, with some general discussions. F. 321.r, 10 Sep 1860, is the beginning
of a summary of his work on games of skill in general. F. 324-333, Oct 1844, studies "General
laws for all games of Skill between two players" and draws flow charts
showing the basic recursive analysis of a game tree (ff. 325.v &
325.r). On f. 332, he counts the
number of positions in Tit Tat To as
9! + 8! + ... + 1! = 409,113.
F. 333 has an idea of the tree structure of a game.
John M. Dubbey. The Mathematical Work of Charles
Babbage. CUP, 1978, pp. 96‑97
& 125‑130. See 4.B.1 for more
details. He discusses the above Babbage
material. On p. 127, Dubbey
has: "The basic problem is one
that appears not to have been previously considered in the history of
mathematics." Dubbey, on p. 129,
says: "This analysis ... must
count as the first recorded stochastic process in the history of
mathematics." However, it is
really a deterministic two-person game.
E. Zermelo. Über eine Anwendung der Mengenlehre auf die
Theorie des Schachspiels. Proc. 5th ICM
(1912), CUP, 1913, vol. II, 501‑504. Gives general idea of first and second
person games.
Ahrens. A&N.
1918. P. 154, note. Says that each particular Dots and Boxes
board, with rational play, has a definite outcome.
W. Rivier. Archives des Sciences Physiques et
Naturelles (Nov/Dec 1921). ??NYS --
cited by Rivier (1935) who says that the later article is a new and simpler
version of this one.
H. Steinhaus. Difinicje potrzebne do teorji gry i
pościgu (Definitions for a theory of games and pursuit). Myśl Akademicka (Lwów) 1:1 (Dec 1925)
13‑14 (in Polish). Translated,
with an introduction by Kuhn and a letter from Steinhaus in: Naval Research Logistics Quarterly 7 (1960)
105‑108.
Dénès König. Über eine Schlussweise aus dem Endlichen ins
Unendliche. Mitteilungen der Universitä
Szeged 3 (1927) 121-130. ??NYS -- cited
by Rivier (1935). Kalmár cites it to
the same Acta as his article.
László Kalmár. Zur Theorie der abstracten Spiele. Acta Litt. Sci. Regia Univ. Hungaricae
Francisco‑Josephine (Szeged) 4 (1927) 62‑85. Says there is a gap in Zermelo which has
been mended by König. Lengthy approach,
but clearly gets the idea of first and second person games.
Max Euwe. Proc. Koninklijke Akadamie van Wetenschappen
te Amsterdam 32:5 (1929). ??NYS --
cited by Rivier (1935).
Emanuel Lasker. Brettspiele der Völker. Rätsel‑ und mathematische Spiele. A. Scherl, Berlin, 1931, pp. 170‑203. Studies the one pile game (100, 5)
and the sum of two one‑pile games: (100, 5) + (50, 3). Discusses Nimm, "an old Chinese game according to Ahrens"
and says the solver is unknown. Gives
Lasker's Nim -- one can take any amount from a pile or split it in two -- and
several other variants. Notes that 2nd person + 2nd person is
2nd person while 2nd person + 1st person is
1st person. Gives the idea of
equivalent positions. Studies three
(and more) person games, assuming the pay‑offs are all different. Studies some probabilistic games. Jörg Bewersdorff [email of 6 Jun 1999]
observes that Lasker's analysis of his Nim got very close to the idea of the
Sprague-Grundy number. See: Jörg
Bewersdorff; Glück, Logik und Bluff
Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998,
Section 2.5 Lasker-Nim: Gewinn auf
verborgenem Weg, pp. 118-124.
W. Rivier. Une theorie mathématique des jeux de
combinaisions. Comptes-Rendus du
Premier Congrès International de Récréation Mathématique, Bruxelles, 1935. Sphinx, Bruxelles, 1935, pp. 106‑113. A revised and simplified version of his 1921
article. He cites and briefly discusses
Zermelo, König and Euwe. He seems to be
classifying games as first player or second player.
René de Possel. Sur la Théorie Mathématique des Jeux de
Hasard et de Réflexion. Actualités
Scientifiques et Industrielles 436.
Hermann, Paris, 1936. Gives the
theory of Nim and also the misère version.
Shows that any combinatorial game is a win, loss or draw and describes
the nature of first and second person positions. He then goes on to consider games with chance and/or bluffing,
based on von Neumann's 1927 paper.
R. Sprague. Über mathematische Kampfspiele. Tôhoku Math. J. 41 (1935/36) 438‑444.
P. M. Grundy. Mathematics and games. Eureka 2 (1939) 6‑8. Reprinted, ibid. 27 (1964) 9‑11. These two papers develop the Sprague-Grundy
Number of a game.
D. W. Davies. A theory of chess and noughts and
crosses. Penguin Science News 16 (Jun
1950) 40-64. Sketches general ideas of
tree structure, Sprague-Grundy number, rational play, etc.
H. Steinhaus. Games, an informal talk. AMM 72 (1965) 457‑468. Discusses Zermelo and says he wasn't aware
of Zermelo in 1925. Gives Mycielski's
formulation and proof via de Morgan's laws.
Goes into pursuit and infinite games and their relation to the Axiom of
Choice.
H. Steinhaus. (Proof that a game without ties has a
strategy.) In: M. Kac; Hugo Steinhaus -- a
reminiscence and a tribute; AMM 81 (1974) 572‑581. Repeats idea of his 1965 talk.
See
5.M for Sim and 5.R.5 for Fox and Geese, etc.
Most
of the board games described here are classic and have been extensively
described and illustrated in the various standard books on board games,
particularly the works of Robert C. Bell, especially his Board and
Table Games from Many Civilizations;
OUP, vol. I, 1960, vol. II, 1969;
combined and revised ed., Dover, 1979 and the older work of Edward G.
Falkener; Games Ancient and Oriental and How to Play Them; Longmans, Green, 1892; Dover, 1961. The works by Culin (see 4.A.4, 4.B.5 and 4.B.9) are often
useful. Several general works on games
are cited in 4.B.1 and 4.B.5 -- I have read Murray's History of Board Games
Other than Chess, but not yet entered the material. Note that many of these works are more concerned with the game
than with its history and have a tendency to exaggerate the ages of games by
assuming, e.g. that a 3 x 3 board must have been used for
Tic-Tac-Toe. I will not try to
duplicate the descriptions by Bell, Falkener and others, but will try to
outline the earliest history, especially when it is at variance with common
belief. The most detailed mathematical
analyses are generally in Winning Ways.
4.B.1. TIC‑TAC‑TOE = NOUGHTS AND CROSSES
Popular
belief is that the game is ancient and universal -- e.g. see Brandreth,
1976. However the game appears to have
evolved from earlier three‑in‑a‑row games, e.g. Nine Holes or
Three Men's Morris, in the early 19C.
See also the historical material in 4.B.5. The game is not mentioned in Strutt nor most other 19C books on games,
not even in Kate Greenaway's Book of Games (1889), nor in Halliwell's section
on slate games (op. cit. in 7.L.1, 1849, pp. 103-104), but there may be an
1875 description in Strutt-Cox of 1903.
Babbage refers to it in his unpublished MSS of c1820 as a children's
game, but without giving it a name. In
1842, he calls it Tit Tat To and he uses slight variations on this name in his
extended studies of the game -- see below.
The OED's earliest references are:
1849 for Tip‑tap‑toe;
1855 for Tit‑tat‑toe;
1861 for Oughts and Crosses.
However, the first two entries may be referring to some other game --
e.g. the entries for Tick‑tack‑toe for 1884 & 1899 are clearly
to the game that Gomme calls Tit‑tat‑toe. Von der Lasa cites a 1838-39 Swedish book for Tripp, Trapp,
Trull. Van der Linde (1874, op. cit. in
5.F.1) gives Tik, Tak, Tol as the Dutch name.
Using the works of Strutt, Gomme, Strutt-Cox, Fiske, Murray, the OED and
some personal communications, I have compiled a separate index of 121
variant names which refer to
5 basic games, with a few
variants and a few unknown games. The
Murray and Parker material is given first, as it deals generally with the
ancient history. Then I list several
standard sources and then summarize their content. Other material follows that.
Fiske says that van der Linde and von der Lasa (see 5.F.1) mention early
appearances of Morris games, but rather briefly and I don't always have that
material.
The
usual # shape board will be so indicated. If one is setting down pieces, then the board is often drawn as a
'crossed square', i.e. a square with its horizontal and vertical midlines
drawn, and one plays on the intersections.
Fiske 127 says this form is common in Germany, but unknown in England
and the US. In addition, the diagonals
are often drawn, producing a 'doubly crossed square'. The squares are sometime drawn as circles giving a 'crossed
circle' and a 'doubly crossed circle', though it is hard to identify the
corners in a crossed circle. The 3 x 3
array of dots sometimes occurs.
The standard # pattern is sometimes surrounded by a square
producing a '3 x 3 chessboard'.
Fiske
129 says the English play with O and
+, while the Swedes play with O
and 1. My experience is that English and Americans play with O
and X. One English friend said that where she grew up, it was called 'Exeter's
Nose' as a deliberate corruption of 'Xs and Os'.
The
first clear references to the standard game of Noughts and Crosses are Babbage
(1820) and the items discussed under Tic-tac-toe below. Further clear references are: Cassell's, Berg, A wrangler ..., Dudeney,
White and everything entered below after White.
Misère
version: Gardner (1957); Scotts (1975);
Murray mentions Morris, which he
generally calls Merels, many times.
Besides the many specific references mentioned below and in 4.B.5, he shows,
on p. 614, under Nine Holes and Three Men's Morris, a number of 3 x 3
diagrams.
Kurna,
Egypt, (-14C) -- a double crossed square and a double crossed circle -- see
Parker below.
Ptolemaic
Egypt (in the BM, no. 14315) -- a square with
# drawn inside. See below where I describe this, from a
recent exhibition, as just a # board.
Ceylon
-- a doubly crossed square -- see Parker below.
Rome
and Pompeii -- doubly crossed circles.
Under
Nine Holes, he says a piece can be moved to any vacant point; under Three Men's Morris, he says a man can
only be moved along a marked line to an adjacent point, i.e. horizontally,
vertically or along a main diagonal.
Under
Nine Holes, he shows the # board for English Noughts and Crosses. He specifically notes that the pieces do not
move. His only other mention of this
board is for a Swedish game called Tripp, Trapp, Trull, but he does not state
that the pieces do not move. He gives
no other examples of the # board nor of non‑moving pieces.
He
also mentions Five (or Six) Men's Morris, of which little is known. On p. 133, he mentions a 3 x 3
"board of nine points used for a game essentially identical with
the 'three men's merels', which has existed in China from at least the time of
the Liang dynasty (A.D. 502‑557).
The 'Swei shu' (first half of the 7th c.) gives the names of twenty
books on this game."
H. Parker. Ancient Ceylon. ??, London, 1909; Asian
Educational Services, New Delhi, 1981.
Nerenchi keliya, pp. 577‑580 & 644. There is a crossed square with small holes at the intersections
at the Temple of Kurna, Upper Egypt, ‑14C. [Rohrbough, loc. cit. in 4.B.5, says this temple was started by
Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc.
(1783) 17.] On p. 644, he shows 34
mason's diagrams from Kurna, which include
#, # in a circle, crossed square with small holes at the
intersections, doubly crossed square, doubly crossed circle. He cites Bell, Arch. Survey of Ceylon, Third
Progress Report, p. 5 note, for for a doubly crossed square in Ceylon, c1C, but
Noughts and Crosses is not found in the interior of Ceylon. The doubly crossed square was used in 18C
Ireland. On pp. 643-665, he discusses
appearances of the crossed square and doubly crossed circle as designs or
characters and claims they have mystic significance. On p. 662, he lists many early appearances of the #
pattern.
Murray 440, note 63, includes a
reference to Soutendam; Keurboek van Delft; Delft, c1425, f. 78 (or p. 78?);
who says games of subtlety are allowed, e.g. ... ticktacken. There is no indication if this may be our
game and the OED indicates that such names were used for backgammon back to
1558. The OED doesn't cite: W. Shakespeare; Measure for Measure,
c1604. Act I, scene ii, line 180 (or
196): "foolishly lost at a game of
ticktack". Later it was more
common as Tric-trac.
Murray 746 notes a Welsh game
Gwyddbwyll mentioned in the Mabinogion (14C).
The name is cognate with the Irish Fidchell and may be a Three Men's
Morris, but the game was already forgotten by the 15C.
STANDARD
SOURCES ON GAMES
Joseph Strutt. The Sports and Pastimes of the People of
England. (With title starting: Glig‑Gamena Angel-Ðeod., or the Sports
...; J. White, London, 1791, 1801, 1810).
A new edition, with a copious index, by William Hone. Tegg, London, 1830, 1831, 1833, 1834, 1838,
1841, 1850, 1855, 1875, 1876, 1891.
[The 1830 ed. has a preface, omitted in 1833, stating that the 1810 ed.
is the same as the 1801 ed. and that Hone has only changed it by adding the
Index and incorporating some footnotes into the text.] [Hall, BCB 263-266 are: 1801,
1810, 1830, 1831.
Toole Stott 647-656 are:
1791; 1801; 1810;
1828-1830 in 10 monthly parts with Index by Hone; 1830; 1830;
1833; 1838; 1841;
1876, an expanded ed, ed by Hone.
Heyl 300-302 gives 1830; 1838;
1850. Toole Stott 653 says the
sheets were remaindered to Hone, who omitted the first 8pp and issued it
in 1833, 1834, 1838, 1841.
I have seen an 1855 ed. C&B
list 1801, 1810, 1830, 1903. BMC has
1801, 1810, 1830, 1833, 1834, 1838, 1841, 1875, 1876, 1898.]
Strutt-Cox. The Sports and Pastimes of the People of
England. By Joseph Strutt. 1801.
A new edition, much enlarged and corrected by J. Charles Cox. Methuen, 1903. The Preface sketches Strutt's life and says this is based on the
'original' 1801 in quarto, with separate plates which were often hand coloured,
but not consistently, while the 1810 reissue had them all done in a terra‑cotta
shade. Hone reissued it in octavo in 1830
with the plates replaced by woodcuts in the text and this was reissued in 1837,
1841 and 1875. (From above we see that
there were other reissues.) "Mr.
Strutt has been left for the most part to speak in his own characteristic
fashion .... A few obvious mistakes and
rash conclusions have been corrected, ... certain unimportant omissions have
been made. ... Nearly a third of the book is new." Reprinted in 1969 and in the 1960s?
J. T. Micklethwaite. On the indoor games of school boys in the
middle ages. Archaeological Journal 49
(Dec 1892) 319-328. Describes
various 3 x 3 boards and games on them, including Nine Holes and "tick,
tack, toe; or oughts and crosses, which I suppose still survives wherever
slate and pencil are used as implements of education", Three Men's Morris
and also Nine Men's Morris, Fox and Geese, etc.
Alice B. Gomme. The Traditional Games of England, Scotland,
and Ireland. 2 vols., David Nutt,
London, 1894 & 1898. Reprinted in
one vol., Thames & Hudson, London, 1984.
Willard Fiske. Chess in Iceland and in Icelandic Literature
with Historical Notes on Other Table-Games.
The Florentine Typographical Society, Florence, 1905. Esp. pp. 97-156 of the Stray Notes. P. 122 lists a number of works on ancient
games.
These
and the OED have several entries on Noughts and Crosses and Tic‑tac‑toe
and many on related games, which are summarised below. Gomme often cites or quotes Strutt. The OED often gives the same quotes as
Gomme. Gomme's references are highly
abbreviated but full details of the sources can usually be found in the OED.
(Nine
Men's) Morris,
where Morris is spelled about 30 different ways, e.g. Marl, Merelles, Mill,
Miracles, Morals, and Nine Men's may be given as, e.g. Nine‑peg, Nine
Penny, Nine Pin. Also known as Peg
Morris and Shepherd's Mill. Gomme I 80
& 414‑419 and Strutt 317‑318 (c= Strutt-Cox 256-258 & plate
opp. 246, which adds reference to Micklethwaite) are the main entries. See 4.B.5 for material more specifically on
this game.
Nine
Holes, also
known as Bubble‑justice, Bumble‑puppy, Crates, and possibly Troll‑madam,
Troule‑in‑Madame. Gomme I
413‑414 and Strutt 274‑275 & 384 (c= Strutt‑Cox
222-223 & 304) are the main entries.
Twelve Holes is similar [Gomme II 321 gives a quote from 1611]. There seem to be cases where Nine Men's
Morris was used in referring to Nine Holes [Gomme I 414‑419]. There are two forms of the game: one form has holes in an upright board that
one must roll a ball or marble through;
the other form has holes in the ground, usually in a 3 x 3 array, that one must roll balls into. Unfortunately, none of the references
implies that one has to get three in a row -- see Every Little Boys Book for a
version where this is certainly not the case.
There are references going back to 1572 for Crates (but mentioning
eleven holes) [Gomme I 81 & II 309] and 1573 [OED] for Nine Holes. Botermans et al.; The World of Games; op.
cit. in 4.B.5; 1989; p. 213, shows a 17C engraving by Ménian showing Le Jeu de
Troumadame as having a board with holes in it, held vertically on a table and
one must roll marbles through the holes.
They say it is nowadays known as 'bridge'.
Three
Men's Morris. This is less common, but occurs in several
variant spellings corresponding to the variants of Nine Men's Morris,
including, e.g. Three‑penny Morris, Tremerel. The game is played on a 3
x 3 board and each player has three men. After making three plays each, consisting of setting men on the
cells, further play consists of picking up one of your own men and placing it
on a vacant cell, with the object of getting three in a row. There are several versions of this game,
depending on which cells one may play to, but the descriptions given rarely
make this clear. [Gomme I 414‑419]
quotes from F. Douce; Illustrations of Shakespeare and of Ancient Manners;
1807, i.184. "In the French
merelles each party had three counters only, which were to be placed in a line
to win the game. It appears to have
been the tremerel mentioned in an old fabliau.
See Le Grand, Fabliaux et Contes, ii.208. Dr. Hyde thinks the morris, or merrils, was known during the time
that the Normans continued in possession of England, and that the name was
afterwards corrupted into three men's morals, or nine men's morals." [Hyde.
Hist. Nederluddi [sic], p. 202.]
In practice, the board is often or usually drawn as a crossed
square. If one can move along all
winning lines, then it would be natural to draw a doubly crossed square. See under Alfonso MS (1283) in 4.B.5 for
versions called marro, tres en raya and riga di tre. Again, much of the material on this game is in 4.B.5.
Five‑penny
Morris. None of the references make it clear, but
this seems to be (a form of) Three Men's Morris. Gomme I 122 and the OED [under Morrell] quote: W. Hawkins; Apollo Shroving (a play of
1627), act III, scene iv, pp. 48-49.
"...,
Ovid hath honour'd my exercises.
He describes in verse our boyes play.
Twise
three stones, set in a crossed square where he wins the game
That
can set his three along in a row,
And
that is fippeny morrell I trow."
Most of the references (and
myself) are perplexed by the reference to five, though the fact that one has at
most five moves in Tic‑tac‑toe might have something to do with
it?? Since Three Men's Morris is less
well known, some writers have assumed Five‑penny Morris was Nine Men's
Morris and others have called all such games by the same name. A few lines later, Hawkins has: "I
challenge him at all games from blowpoint upward to football, and so on to
mumchance, and ticketacke. ... rather
than sit out, I will give Apollo three of the nine at Ticketacke,
..."
Corsicrown [Gomme I 80] seems to be a
version of Three Men's Morris, but using seven of the nine cells, omitting two
opposite side cells. Gomme quotes from
J. Mactaggart; The Scottish Gallovidian Encyclopedia; (1871 or possibly
1824?): "each has three men ....
there are seven points for these men to move about on, six on the edges of the
square and one at the centre."
Tic‑tac‑toe. The earliest clearly described versions are given in Babbage
(with no name given), c1820, and Gomme I 311, under Kit‑cat‑cannio,
where she quotes from: Edward Moor;
Suffolk Words and Phrases; 1823 (This word does not occur in the OED). Gomme also gives entries for Noughts and
Crosses [I 420‑421] and Tip‑tap‑toe [II 295‑296] with
variants Tick‑tack‑toe and Tit‑tat‑toe. In 1842-1865, Babbage uses Tit Tat To and
slight variants. Under Tip‑tap‑toe,
Gomme says the players make squares and crosses and that a tie game is a score
for Old Nick or Old Tom. (When I was
young, we called it Cat's Game, and this is an old Scottish term [James T. R.
Ritchie; The Singing Street Scottish
Children's Games, Rhymes and Sayings; (O&B, 1964); Mercat Press,
Edinburgh, 2000, p. 61].) She quotes
regional glossaries for Tip‑tap‑toe (1877), Tit‑tat‑toe
(1866 & 1888), Tick‑tack‑toe (1892). The OED entry for Oughts and Crosses seems to be this game and
gives an 1861 quote. Von der Lasa cites
a 1838-39 Swedish book for Tripp, Trapp, Trull. Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as
the Dutch name.
Tit‑tat‑toe [Gomme II 296‑298]. This is a game using a slate marked with a
circle and numbered sectors. The player
closes his eyes and taps three times with a pencil and tries to land on a good
sector. Gomme gives the verse:
Tit,
tat, toe, my first go,
Three
jolly butcher boys all in a row
Stick
one up, stick one down,
Stick
one in the old man's ground.
But cf Games and Sports for
Young Boys, 1859, below.
The
OED entries under Tick‑tack, Tip‑tap and Tit give a number of
variant spellings and several quotations, which are often clearly to this game,
but are sometimes unclear. Also some
forms seem to refer to backgammon.
In
her 'Memoir on the study of children's games' [Gomme II 472‑473], Gomme
gives a somewhat Victorian explanation of the origin of Old Nick as the winner
of a tie game as stemming from "the primitive custom of assigning a
certain proportion of the crops or pieces of land to the devil, or other earth
spirit."
Franco Agostini & Nicola
Alberto De Carlo. Intelligence
Games. (As: Giochi della Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. P. 81 says examples of boards were
discovered in the lowest level of Troy and in the Bronze Age tombs in Co.
Wicklow, Ireland. Their description is
a bit vague but indicates that the Italian version of Tic-tac-toe is actually
Three Men's Morris.
Anonymous. Play the game. Guardian Education section (21 Sep 1993) 18-19. Shows a stone board with the #
incised on it 'from Bet Shamesh, Israel, 2000 BC'. This might be the same as the first board
below??
A small exhibition of board
games organized by Irving Finkel at the British Museum, 1991, displayed the
following.
Stone
slab with the usual # Tick-Tac-Toe board incised on it, but
really a 4 x 3 board. With nine stone men. From Giza, >-850. BM items EA 14315 & 14309, donated by W.
M. Flinders Petrie. Now on display in
Room 63, Case C.
Stone
Nine Holes board from the Temple of Artemis, Ephesus, 2C-4C. Item BM GR 1873.5.5.150. This is a
3 x 3 array of
depressions. Now on display in Room 69,
Case 9.
Robbie Bell & Michael
Cornelius. Board Games Round the
World. CUP, 1988. P. 6 states that the crossed square board
has been found at Kurna (c-1400) and at the Ptolemaic temple at Komombo
(c-300). They state that Three Men's
Morris is the game mentioned by Ovid in Ars Amatoria. They say that it was known to the Chinese at the time of
Confucius (c-500) under the name of Yih, but is now known as Luk tsut k'i. They also say the game is also known as Nine Holes -- which seems
wrong to me.
The Spanish Treatise on
Chess-Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of
Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototypic
Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913. (There was also an edition by Arnald
Steiger, Geneva, 1941.) See 4.B.5 for
more details of this work. Vol. 2, f.
93v, p. CLXXXVI, shows a doubly crossed square board. ??NX -- need to study text.
Pieter Bruegel (the Elder). Children's Games. Painting dated 1560 at the Kunsthistorisches Museum, Vienna. In the right background, children are
playing a game involving throwing balls into holes in the ground, but the holes
appear to be in a straight line.
Anonymous. Games of the 16th Century. 1950.
Op. cit. in 4.A.3. P. 134
describes nine-holes, quoting an unknown poet of 1611: "To play at
loggats, Nine-holes, or Ten-pinnes".
The author doesn't specify what positions the balls are to be rolled
into. P. 152 describes Troll-my-dames
or Troule-in-madame: "they may have in the end of a bench eleven holes
made, into which to troll pummets, or bowls of lead, ...."
William Wordsworth. The Prelude, Book 1. Completed 1805, published 1850. Lines 509‑513.
At
evening, when with pencil, and smooth slate
In
square divisions parcelled out and all
With
crosses and with cyphers scribbled o'er,
We
schemed and puzzled, head opposed to head
In
strife too humble to be named in verse.
It
is not clear if this is referring to Noughts and Crosses.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. F. 4r is part of
the Table of Contents. It shows Noughts
and Crosses games played on the # board and on a 4 x 4 board adjacent to
entry 4: The Mill. Ff. 124-146
are: Essay 10 -- Of questions requiring
the invention of new modes of analysis.
On f. 128.r, he refers to a game in which "the relative
positions of three of the marks is the object of inquiry." Though the reference is incomplete, a Noughts
and Crosses game is drawn on the facing page, f. 127.v. Ff. 134-144 are: Essay 10 Part 5. At the
top of f. 134.r, he has added a note:
"This is probably my earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". The Essay begins: "Amongst the simplest of those games requiring any degree of
skill which amuse our early years is one which is played at in the following
manner." He then describes the
game in detail and makes some simple analysis, but he never uses a name for it.
Charles Babbage. Notebooks -- unpublished collection of MSS
in the BM as Add. MS 37205. ??NX. On f. 304, he starts on analysis of games. Ff. 310-383 are almost entirely devoted to
Tit-Tat-To, with some general discussions.
Most of this material comprises a few sheets of working, carefully
dated, sometimes amended and with the date of the amendment. A number of sheets describe parts of the
automaton that he was planning to build which would play the game, but no such
machine was built until 1949. The
sheets are not always in strict chronological order.
F.
310.r is the first discussion of the game, called Tit Tat To, dated 17 Sep
1842. On F. 312.r, 20 Sep 1843, he says
he has "Reduced the 3024 cases D to 199 which include many Duplicates by
Symmetry." F. 321.r, 10 Sep 1860,
is the beginning of a summary of his work on games of skill in general. He refers to Tit-tat-too. F. 322.r continues and he says: "I have found no game of skill more
simple that that which children often play and which they call Tit‑tat-to." F. 324-333, Oct 1844, studies "General
laws for all games of Skill between two players" and draws flow charts
showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he counts the number of
positions as 9! + 8! + ... + 1!
= 409,113. F. 333 has an idea of the tree structure of a game. On ff. 337-338, 8 Sep 1848, he has Tit-tat
too. On ff. 347.r-347.v, he suggests
Nine Men's Morris boards in triangular and pentagonal shapes and does various
counting on the different shapes. On
ff. 348-349, 26 Oct 1859, he uses Tit-Tat-To.
John M. Dubbey. The Mathematical Work of Charles
Babbage. CUP, 1978, pp. 96‑97
& 125‑130. He discusses the
above Babbage material. On p. 127,
Dubbey has: "After a surprisingly
lengthy explanation of the rules, he attempts a mathematical formulation. The basic problem is one that appears not to
have been previously considered in the history of mathematics." Babbage represents the game using roots of
unity. Dubbey, on p. 129, says: "This analysis ... must count as the
first recorded stochastic process in the history of mathematics." However, it is really a deterministic
two-person game.
Games and Sports for Young
Boys. Routledge, nd [1859 - BLC]. P. 70, under Rhymes and Calls: "In the game of Tit-tat-toe, which is
played by very young boys with slate and pencil, this jingle is used:--
Tit,
tat, toe, my first go:
Three
jolly butcher boys all in a row;
Stick
one up, stick one down.
Stick
one on the old man's crown."
Baron Tassilo von Heydebrand und
von der Lasa. Ueber die griechischen
und römischen Spiele, welche einige ähnlichkeit mit dem Schach hatten. Deutsche Schachzeitung (1863) 162-172, 198-199,
225-234, 257-264. ??NYS -- described on
Fiske 121-122 & 137, who says van der Linde I 40-47 copies much of it. Von der Lasa asserts that the Parva Tabella
of Ovid is Kleine Mühle (Three Men's Morris).
He says the game is called Tripp, Trapp, Trull in the Swedish book
Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57) --
??NYS. Van der Linde says that the
Dutch name is Tik, Tak, Tol. Fiske
notes that both of these refer to Noughts and Crosses, but it is unclear if von
der Lasa or van der Linde recognised the difference between Three Men's Morris
and Noughts and Crosses.
C. Babbage. Passages from the Life of a
Philosopher. 1864. Chapter XXXIV -- section on Games of Skill,
pp. 465‑471. (= pp. 152‑156
in: Charles Babbage and His Calculating
Engines, Dover, 1961.) Partial
analysis. He calls it tit‑tat‑to.
The Play Room: or, In-door Games
for Boys and Girls. Dick &
Fitzgerald(?), 1866. [Reprinted as: How
to Amuse an Evening Party. Dick &
Fitzgerald, NY, 1869.] ??NX -- the 1869
was seen at Shortz's. P. 22:
Tit-tat-to. Uses O
and +. "This is a game that small boys enjoy, and some big ones who
won't own it."
Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor
games with and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and fifty
illustrations. Routledge, London,
nd. HPL gives c1850, but the text is
clearly derived from Every Boy's Book, whose first edition was 1856. But the main part of the text considered
here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed
editor), but is in the 8th ed of 1868 (published for Christmas 1867), which was
the first seriously revised edition, with Edmund Routledge as editor. So this may be c1868. This is the first published use of the term
Noughts and Crosses found so far -- the OED's 1861 quote is to Oughts and
Crosses..
Pp.
46-47: Slate games: Noughts and crosses.
"This is a capital game, and one which every school-boy truly
enjoys." Though the example shown
is a draw, there is no mention of the fact that the game should always be a tie.
Pp.
85-86: Nine-holes. This has nine holes
in a row and each player has a hole.
The ball is rolled to them and the person in whose hole it lands must
run and pick up the ball and try to hit one of the others who are running
away. So this has nothing to do with
our games or other forms of Nine Holes.
P.
106: Nine-holes or Bridge-board. This
has nine holes in an upright board and the object is get one's marbles through
the holes. (This material is in the
1856 ed. of Every Boy's Book.)
Correspondent to Notes and
Queries (1875) ??NYS -- quoted by
Strutt-Cox 257. Describes a game called
Three Mans' Marriage [sic] in Derbyshire which seems to be Noughts and Crosses
played on a crossed square board.
Pieces are not described as moving, but in the next description of a
Nine Men's Morris, they are specifically described as moving. However, the use of a crossed square board
may indicate that diagonals were not considered.
Cassell's. 1881.
Slate Games: Noughts and Crosses, or Tit‑Tat‑To, p. 84,
with cross reference under Tit-Tat-To, p. 87.
= Manson, 1911, pp. 202-203 & 208.
Albert Norman. Ungdomens Bok [Book for Youth] (in
Swedish). 2nd ed., Stockholm,
1883. Vol. I, p. 162++. ??NYS -- quoted and described in Fiske
134-136. Description of Tripp, Trapp,
Trull, with winning cry: "Tripp,
trapp, trull, min qvarn är full."
(Qvarn = mill.)
Lucas. RM2, 1883. Pp.
73-99. Analysis of Three Men's Morris,
on a board with the main diagonals drawn, with moves of only one square along a
winning line. He shows this is a first
person game. If the first player is not
permitted to play in the centre, then it is a tie game. No mention of Tic-Tac-Toe.
Albert Ellery Berg, ed. The Universal Self‑Instructor. Thomas Kelly, NY, 1883. Tit‑tat‑to, p. 379. Brief description.
Mark Twain. The Adventures of Huckleberry Finn. 1884.
Chap. XXXIV, about half-way through.
"It's as simple as tit-tat-toe, three-in-a-row, ..., Huck
Finn."
"A wrangler and late master
at Harrow school." The science of
naughts and crosses. Boy's Own Paper
10: (No. 498) (28 Jul 1888) 702‑703; (No. 499) (4 Aug 1888) 717; (No. 500) (11 Aug 1888) 735; (No. 501) (18 Aug 1888) 743. Exhaustive analysis, including odds of
second player making a correct response to each opening. For first move in: middle, side, corner, the odds of a correct response are: 1/2,
1/2, 1/8. He implies that the analysis is not widely
known.
"Tom Wilson". Illustred Spelbok (in Swedish). Nd [late 1880s??]. ??NYS -- described by Fiske 136-137. This gives Tripp, Trapp, Trull as a Three Men's Morris game on
the crossed square, with moves "according to one way of playing, to
whatever points they please, but according to another, only to the nearest
point along the lines on which the pieces stand. This last method is always employed when the board has, in
addition to the right lines, or lines joining the middles of the exterior
lines, also diagonals connecting the angles". He then describes a drawn version using the #
board and 0 and
+ (or 1 and 2 in
the North) which seems to be genuinely Noughts and Crosses. Fiske says the book seems to be based on an
early edition of the Encyclopédie des Jeux or a similar book, so it is
uncertain how much the above represents the current Swedish game. Fiske was unable to determine the author's
real name, though he was still living in Stockholm at the time.
Il Libro del Giuochi. Florence, 1894. ??NYS -- described in Fiske, pp. 109-110. Gives doubly crossed square board and
mentions a Three Men's Morris game.
T. de Moulidars. Grande Encyclopédie des Jeux. Montgredien or Librairie Illustree, Paris,
nd. ??NYS -- Fiske 115 (in 1905) says
it appeared 'not very long ago' and that Gelli seems to be based on it. Fiske quotes the clear description of Three
Men's Morris as Marelle Simple, using a doubly crossed square, saying that
pieces move to adjoining cells, following a line, and that the first player
should win if he plays in the centre.
Fiske notes that Noughts and Crosses is not mentioned.
J. Gelli. Come Posso Divertirmi? Milan, 1900. ??NYS -- described in Fiske 107.
Fiske quotes the description of Three Men's Morris as Mulinello
Semplice, essentially a translation from Moulidars.
Dudeney. CP.
1907. Prob. 109: Noughts and
crosses, pp. 156 & 248.
(c= MP, prob. 202: Noughts and crosses, pp. 89 & 175‑176. = 536, prob. 471: Tic tac toe, pp. 185 &
390‑392. Asserts the game is a
tie, but gives only a sketchy analysis.
MP gives a reasonably exhaustive analysis. Looks at Ovid's game.
A. C. White. Tit‑tat‑toe. British Chess Magazine (Jul 1919) 217‑220. Attempt at a complete analysis, but has a
mistake. See Gardner, SA (Mar
1957) = 1st Book, chap. 4.
D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940.
Section V Games and Playthings,
pp. 148-165. On p. 160, he quotes Ovid
and says it is Noughts and Crosses, or in Ireland, Tip-top-castle.
The Home Book of Quizzes, Games
and Jokes. Blue Ribbon Books, NY,
1941. This is excerpted from several
books -- this material is most likely taken from: Clement Wood & Gloria Goddard; Complete Book of Games; same
publisher, nd [late 1930s]. P. 150:
Tit-tat-toe, noughts and crosses. Brief
description of the game on the # board.
"To win requires great ingenuity."
G. E. Felton &
R. H. Macmillan. Noughts and
Crosses. Eureka 11 (Jan 1949) 5-9. Mentions Dudeney's work on the 3 x 3
board and asks for generalizations.
Mentions pegotty = go-bang. Then
studies the 4 x 4 x 4 game -- see 4.B.1.a. Adds some remarks on pegotty, citing
Falkener, Lucas and Tarry.
Stanley Byard. Robots which play games. Penguin Science News 16 (Jun 1950)
65-77. On p. 73, he says D. W.
Davies, of the National Physical Laboratory, has built, and exhibited to the
Royal Society in May 1949, an electro-mechanical noughts and crosses machine. A photo of the machine is plate 16. He also mentions Babbage's interest in such
a machine and an 1874 paper to the US National Academy by a Dr. Rogers --
??NYS.
P. C. Parks. Building a noughts and crosses machine. Eureka 14 (Oct 1951) 15-17. Cites Babbage, Rogers, Davies, Byard. Parks built a simple machine with wire and
tin cans in 1950 at a cost of about £6.
Says G. Eastell of Thetford, Norfolk, built a machine using sixty valves
for the Festival of Britain.
Gardner. Ticktacktoe. SA (Mar 1957) c= 1st
Book, chap. 4. Quotes Wordsworth,
discusses Three Men's Morris (citing Ovid) and its variants (including versions
on 4 x 4 and 5 x 5 boards), the misère version (the person who
makes three in a row loses), three and
n dimensional forms (giving L.
Moser's result on the number of winning lines on a kn board),
go-moku, Babbage's proposed machine, A. C. White's article. Addendum mentions the Opies' assertion that
the name comes from the rhyme starting
"Tit, tat, toe, My first
go,".
C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. A ticktacktoe machine, pp. 384-385. Noel Elliott gives a brief description of a relay logic machine
to play the game.
Donald Michie. Trial and error. Penguin Science Survey 2 (1961) 129-145. ??NYS.
Describes his matchbox and bead learning machine, MENACE (Matchbox
educable noughts and crosses engine), for the game.
Gardner. A matchbox game-learning machine. SA (Mar 1962) c= Unexpected, chap. 8.
Describes Michie's MENACE. Says it
used 300 matchboxes. Gardner adapts it
to Hexapawn, which is much simpler, requiring only 24 matchboxes. Discusses other games playable by
'computers'. Addendum discusses results
sent in by readers including other games.
Barnard. 50 Observer Brain-Twisters. 1962.
Prob. 34: Noughts and crosses, pp. 39‑40, 64 & 93‑94. Asserts there are 1884 final winning
positions. He doesn't consider
equivalence by symmetry and he allows either player to start.
Donald Michie &
R. A. Chambers. Boxes: an experiment
in adaptive control. Machine
Intelligence 2 (1968) 136-152.
Discusses MENACE, with photo of the pile of boxes. Says there are 288 boxes, but doesn't
explain exactly how he found them.
Chambers has implemented MENACE as a general game-learning computer
program using adaptive control techniques designed by Michie. Results for various games are given.
S. Sivasankaranarayana
Pillai. A pastime common among South
Indian school children. In: P. K. Srinivasan, ed.; Ramanujan Memorial
Volumes: 1: Ramanujan -- Letters and
Reminiscences; 2: Ramanujan -- An
Inspiration; Muthialpet High School,
Number Friends Society, Old Boys' Committee, Madras, 1968. Vol. 2, pp. 81-85. [Taken from Mathematics Student, but no date or details given --
??] Shows ordinary tic-tac-toe is a
draw and considers trying to get t in a row on an n x n board. Shows that
n = t ³ 3 is a draw and
that if t ³ n
+ 1 - Ö(n/6), then the game
is a draw.
L. A. Graham. The Surprise Attack in Mathematical
Problems. Dover, 1968. Tic-tac-toe for gamblers, prob. 8, pp.
19-22. Proposed by F. E. Clark,
solutions by Robert A. Harrington & Robert E. Corby. Find the probability of the first player
winning if the game is played at random.
Two detailed analyses shows that the probabilities for first player, second player, tie are
(737, 363, 160)/1260.
[Henry] Joseph & Lenore
Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books (Charter Communications), NY,
1975. P. 143: Ha-ho-ha. Misère version of noughts and crosses proposed. No discussion.
Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Noughts and Crosses, pp. 11-12. Asserts "It's been played all around the world for hundreds,
if not thousands, of years ...."
I've included it as a typical example of popular belief about the
game. = Pencil & Paper Puzzle
Games; Watermill Press, Mahwah, New Jersey, 1989, Tic-Tac-Toe, pp. 11‑12.
Winning Ways. 1982.
Pp. 667-680. Complete and
careful analysis, including various uncommon traps. Several equivalent games.
Discusses extensions of board size and dimension.
Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Tic-tac-toe squared, pp. 88-89. Get
3 in a row on the 4 x 4
board. Also considers
Tic-tac-toe-toe -- get 4 in a row on
5 x 5 board.
George Markovsky. Numerical tic-tac-toe -- I. JRM 22:2 (1990) 114-123. Describes and studies two versions where the
moves are numbered 1, 2, .... One
is due to Ron Graham, the other to P. H. Nygaard and Markowsky sketches the histories.
Ira Rosenholtz. Solving some variations on a variant of
tic-tac-toe using invariant subsets.
JRM 25:2 (1993) 128-135. The
basic variant is to avoid making three in a row on a 4 x 4
board. By playing symmetrically,
the second player avoids losing and 252 of the 256 centrally symmetric
positions give a win for the second player.
Extends analysis to 2n x
2n, 5 x 5, 4 x 4 x 4, etc.
Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24-25
& 33 (Feb 1994) 32. This describes a sliding piece puzzle on the
doubly crossed square board -- see under 5.A.
See: Yuri I. Averbakh; Board
games and real events; 1995; in 5.R.5, for a possible connection.
C. Planck. Four‑fold magics. Part 2 of chap. XIV, pp. 363‑375, of
W. S. Andrews, et al.; Magic Squares and Cubes; 2nd ed., Open Court, 1917; Dover, 1960. On p. 370, he notes that the number of m‑dimensional directions through a cell of the n‑dimensional board is the m‑th term of the binomial expansion
of ½(1+2)n.
Maurice Wilkes says he
played 3-D noughts and crosses at
Cambridge in the late 1930s, but the game was to get the most lines on a 3 x 3 x 3
board. I recall seeing a
commercial version, called Plato?, of this in 1970.
Cedric Smith says he played 3-D
and 4-D versions at Cambridge in the early 1940s.
Arthur Stone (letter to me of 9
Aug 1985) says '3 and 4 dimensional forms of tic-tac-toe produced by Brooks,
Smith, Tutte and myself', but it's not quite clear if they invented these. Tutte became expert on the 43
board and thought it was a first person game.
They only played the 54 game once - it took a long time.
Funkenbusch & Eagle, National
Mathematics Mag. (1944) ??NYR.
G. E. Felton &
R. H. Macmillan. Noughts and
crosses. Eureka 11 (1949) 5‑9. They say they first met the 4 x 4 x 4
game at Cambridge in 1940 and they give some analysis of it, with
tactics and problems.
William Funkenbusch & Edwin
Eagle. Hyper‑spacial tit‑tat‑toe
or tit‑tat‑toe in four dimensions.
National Mathematics Magazine 19:3 (Dec 1944) 119‑122. ??NYR
A. L. Rubinoff, proposer; L. Moser, solver. Problem E773 -- Noughts and crosses. AMM 54 (1947) 281
& 55 (1948) 99. Number of winning lines on a kn board is {(k+2)n ‑ kn}/2. Putting
k = 1 gives Planck's result.
L. Buxton. Four dimensions for the fourth form. MG 26 (1964) 38‑39. 3 x 3 x 3
and
3 x 3 x 3 x 3
games are obviously first person, but he proposes playing for most lines
and with the centre blocked on the 3 x
3 x 3 x 3 board. Suggests
3n and 4 x 4 x 4
games.
Anon. Puzzle page: Noughts and crosses. MTg 33 (1965) 35. Says
practice shows that the 4 x 4 x
4 game is a draw. [I only ever had one drawn game!] Conjectures
nn is first player
and (n+1)n is a draw.
Roland Silver. The group of automorphisms of the game of 3‑dimensional
ticktacktoe. AMM 74 (1967) 247‑254. Finds the group of permutations of cells
that preserve winning lines is generated by the rigid motions of the cube and
certain 'eviscerations'. [It is
believed that this is true for the kn board, but I don't know of a simple proof.]
Ross Honsberger. Mathematical Morsels. MAA, 1978.
Prob. 13: X's and O's, p. 26.
Obtains L. Moser's result.
Kathleen Ollerenshaw. Presidential Address: The magic of
mathematics. Bull. Inst. Math. Appl.
15:1 (Jan 1979) 2-12. P. 6 discusses my
rediscovery of L. Moser's 1948 result.
Paul Taylor. Counting lines and planes in generalised
noughts and crosses. MG 63 (No. 424)
(Jun 1979) 77-82. Determines the number pr(k) of r-sections of a kn board by means of a recurrence
pr(k) = [pr-1(k+2) - pr-1(k)]/2r which generalises L. Moser's 1948
result. He then gets an explicit sum
for it. Studies some other
relationships. This work was done while
he was a sixth form student.
Oren Patashnik. Qubic:
4 x 4 x 4 tic‑tac‑toe. MM 53 (1980) 202‑216. Computer assisted proof that 4 x 4 x 4
game is a first player win.
Winning Ways. 1982.
Pp. 673-679, esp. 678-679.
Discusses getting k in a row on a n x n board. Discusses
43 game
(Tic-Toc-Tac-Toe) and kn game.
Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A Second Mensa Puzzle
Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London,
1991.) Chapter 7: Conceptual conflict
in multi-dimensional space, pp. 80-94 (1991: 98-112) & answers on pp. 99,
100, 106 & 131 (1991: 115, 116, 122 & 147). He considers various higher dimensional noughts and crosses on
the 33, 34 and 35 boards.
He finds that there are 49 winning lines on the 33 and he finds how to determine the number of d-facets on an n-cube as the
coefficients in the expansion of (2x +
1)n. He also considers games
where one has to complete a 3 x 3 plane to win and gives a problem: OXO three hypercube planes, p. 91 (1991:
109) & Answer 29, p. 106 (1991: 122)
which asks for the number of planes in the hypercube 34. The answer says there are
123 of them, but in 1985 I
found 154 and the general formula for the number of d-sections of a kn board. When I wrote to Serebriakoff, he responded
that he could not follow the mathematics and that "I arrived at the
figures ... from a simple formula published in one of Art [sic] Gardner's books
which checked out as far as I could take it.
Several other mathematicians have looked through it and not
disagreed." I wrote for a
reference to Gardner but never had a response.
I presented my work to the British Mathematical Colloquium at Cambridge
on 2 Apr 1985 and discovered that the results were known -- I had found the
explicit sum given by Taylor above, but not the recurrence.
David Fielker sent some pages
from a Danish book on games, but the TP is not present in his copies, so we
don't have details. This says that Hein
introduced the game in a lecture to students at the Institute for Theoretical
Physics (now the Niels Bohr Institute) in Copenhagen in 1942. After its appearance in Politiken, specially
printed pads for playing the game were sold, and a game board was marketed in
the US as Hex in 1952.
Piet Hein. Article or column in Politiken (Copenhagen)
(26 Dec 1942). ??NYR, but the diagrams
show a board of hexagons.
Gardner (1957) and others have
related that the game was independently invented by John Nash at Princeton in
1948-1949. Gardner had considerable
correspondence after his article which I have examined. The key point is that one of Niels Bohr's
sons, who had known the game in Copenhagen, was a visitor at the Institute for
Advanced Study at the time and showed it to friends. I concluded that it was likely that some idea of the game had
permeated to Nash who had forgotten this and later recalled and extensively
developed the idea, thinking it was new to him. I met Harold Kuhn in 1998, who was a student with Nash at the
time and he has no doubt that Nash invented the idea. In particular, Nash started with the triangular lattice, i.e. the
dual of Hein's board, for some time before realising the convenience of the
hexagonal lattice. Nash came to
Princeton as a graduate student in autumn 1948 and had invented the game by the
spring of 1949. Kuhn says he observed
Nash developing the ideas and recognising the connections with the Jordan Curve
Theorem, etc. Kuhn also says that there
was not much connection between students at Princeton and at the Institute and
relates that von Neumann saw the game at Princeton and asked what it was,
indicating that it was not well known at the Institute. In view of this, it seems most likely that
Nash's invention was independent, but I know from my own experience that it can
be difficult to remember the sources of one's ideas -- a casual remark about a
hexagonal game could have re-emerged weeks or months later when Nash was
studying games, as the idea of looking at hexagonal boards in some form, from
which the game would be re-invented.
Sylvester was notorious for publishing ideas which he had actually
refereed or edited some years earlier, but had completely forgotten the earlier
sources. In situations like Hex, we
will never know exactly what happened -- even if we were present at the time,
it is difficult to know what is going on in the mind of the protagonist and the
protagonist himself may not know what subconscious connections his mind is
making. Even if we could discover that
Nash had been told something about a hexagonal game, we cannot tell how his
mind dealt with this information and we cannot assume this was what inspired
his work. In other words, even a time
machine will not settle such historical questions -- we need something that
displays the conscious and the unconscious workings of a person's mind.
Parker Brothers. Literature on Hex, 1952. ??NYS or NYR.
Claude E. Shannon. Computers and automata. Proc. Institute of Radio Engineers 41
(Oct 1953) 1234‑1241. Describes
his Hex machine on p. 1237.
M. Gardner. The game of Hex. SA (Jul 1957) = 1st Book, chap. 8. Description of Shannon's 8 by 7 'Hoax' machine, pp. 81‑82,
and its second person strategy, p. 79.
Anatole Beck, Michael N.
Bleicher & Donald W. Crowe. Excursions
into Mathematics. Worth Publishers, NY,
1969. Chap. 5: Games (by Beck), Section
3: The game of Hex, pp. 327-339 (with photo of Hein on p. 328). Says it has been attributed to Hein and
Nash. At Yale in 1952, they played on
a 14 x 14 board. Shows it is a
first player win, invoking the Jordan Curve Theorem
David Gale. The game of Hex and the Brouwer fixed-point
theorem. AMM 86:10 (Dec 1979)
818-827. Shows that the non-existence
of ties (Hex Theorem) is equivalent to the Brouwer Fixed-Point Theorem in two
and in n dimensions. Says the use
of the Jordan Curve Theorem is unnecessary.
Winning Ways. 1982.
Pp. 679-680 sketches the game and the strategy stealing argument which
is attributed to Nash.
C. E. Shannon. Photo of his Hoax machine sent to me in
1983.
Cameron Browne. Hex Strategy: Making the Right
Connections. A. K. Peters, Natick,
Massachusetts, 2000.
Lucas. Le jeu de l'École Polytechnique.
RM2, 1883, pp. 90‑91. He
gives a brief description, starting: "Depuis quelques années, les élèves
de l'École Polytechnique ont imaginé un nouveaux jeu de combinaison assez
original." He clearly describes
drawing the edges of the game board and that the completer of a box gets to go
again. He concludes: "Ce jeu nous
a paru assez curieux pour en donner ici la description; mais, jusqu'a présent,
nous ne connaissons pas encore d'observations ni de remarques assez importantes
pour en dire davantage."
Lucas. Nouveaux jeux scientifiques de M. Édouard Lucas. La Nature 17 (1889) 301‑303. Clearly describes a game version of La
Pipopipette on p. 302, picture on p. 301, "... un nouveau jeu ... dédié
aux élèves de l'école Polytechnique."
This is dots and boxes with the outer edges already drawn in.
Lucas. L'Arithmétique Amusante.
1895. Note III: Les jeux scientifiques de Lucas,
pp. 203‑209 -- includes his booklet: La Pipopipette, Nouveau jeu de combinaisons, Dédié aux élèves de
l'École Polytechnique, Par un Antique de la promotion de 1861, (1889), on pp.
204‑208. On p. 207, he says
the game was devised by several of his former pupils at the École
Polytechnique. On p. 37, he remarks
that "Pipo est la désignation abrégée de Polytechnique, par les
élèves de l'X, ...."
Robert Marquard & Georg
Frieckert. German Patent 108,830 -- Gesellschaftsspiel. Patented: 15 Jun 1899. 1p + 1p diagrams. 8 x 8 array of boxes on a
board with slots for inserting edges.
No indication that the player who completes a box gets to play
again. They have some squares with
values but also allow all squares to have equal value.
C. Ganse. The dot game. Ladies' Home Journal (Jun 1903) 41. Describes the game and states that one who makes a box gets to go
again.
Loyd. The boxer's puzzle.
Cyclopedia, 1914, pp. 104 & 352.
= MPSL1, prob. 91, pp. 88‑89 & 152‑153. c= SLAHP: Oriental tit‑tat‑toe,
pp. 28 & 92‑93. Loyd
doesn't start with the boundaries drawn.
He asserts it is 'from the East'.
Ahrens. A&N.
1918. Chap. XIV: Pipopipette,
pp. 147‑155, describes it in more detail than Lucas does. He says the game appeared recently.
Blyth. Match-Stick Magic.
1921. Boxes, pp. 84-85. "The above game is familiar to most boys
and girls ...." No indication that
the completer of a box gets to play again.
Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930. Pp. 84-85: Die Käsekiste. Describes a version for two or more players. The first player must start at a corner and
players must always connect to previously drawn lines. A player who completes a box gets to play
again.
Meyer. Big Fun Book. 1940. Boxes, p. 661. Brief description, somewhat vaguely stating that a player who
completes a box can play again.
The Home Book of Quizzes, Games
and Jokes. Op. cit. in 4.B.1,
1941. P. 151: Dots and squares. Clearly says the completer gets to play
again. "The game calls for great
ingenuity."
"Zodiastar". Fun with Matches and Match Boxes. (Cover says: Match Tricks From the 1880s
to the 1940s.) Universal Publications,
London, nd [late 1940s?]. The game of
boxes, pp. 48-49. Starts by laying out
four matches in a square and players put down matches which must touch the
previous matches. Completing a box
gives another play. No indication that
matches must be on lattice lines, but perhaps this is intended.
Readers' Research
Department. RMM 2 (Apr 1961) 38‑41,
3 (Jun 1961) 51‑52, 4 (Aug 1961) 52‑55. On pp. 40‑41 of No. 2, it says that Martin Gardner suggests
seeking the best strategy. Editor notes
there are two versions of the rules -- where the one who makes a box gets an
extra turn, and where he doesn't -- and that the game can be played on other
arrays. On p. 51 of No. 3, there is a
symmetry analysis of the no‑extra‑turn game on a board with an odd
number of squares. On pp. 52‑54
of No. 4, there is some analysis of the extra‑turn case on a board with
an odd number of boxes.
Everett V. Jackson. Dots and cubes. JRM 6:4 (Fall 1973) 273‑279. Studies 3‑dimensional game where a play is a square in the
cubical lattice.
Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Worm, pp. 18-19. This is
a sort of 'anti-boxes' -- one draws segments on the lattice forming a path without
any cycles -- last player wins. =
Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, pp.
18-19.
Winning Ways. 1982.
Chap. 16: Dots-and-Boxes, pp. 507-550
David B. Lewis. Eureka!
Perigee (Putnam), NY, 1983. Pp.
44‑45 suggests playing on the triangular lattice.
Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987.
Eternal
triangles, pp. 80-81. Gives the game on
the triangular lattice.
Snakes,
pp. 81-82. Same as Brandreth's
Worm. I think 'snake' would be a better
title as only one path is drawn.
M. Gardner. SA (Jul
1967) = Carnival, chap. 1. Describes Michael
Stewart Paterson and John Horton Conway's invention of the game on 21 Feb 1967
at tea time in the Department common room at Cambridge. The idea of adding a spot was due to
Paterson and they agreed the credit for the game should be 60% Paterson to 40%
Conway.
Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Sprouts, p. 13. "...
actually born in Cambridge about ten years ago." c= Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New
Jersey, 1989, p. 13: "... was invented about ten years ago."
Winning Ways. 1982.
Sprouts, pp. 564-570 & 573.
Says the game was "introduced by M. S. Paterson and J. H. Conway
some time ago". Also describes
Brussels Sprouts and Stars-and-Stripes.
An answer for Brussels Sprouts and some references are on p. 573.
Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Sprouts, pp. 95-97.
Karl-Heinz Koch. Pencil & Paper Games. (As: Spiele mit Papier und Bleistift, no
details); translated by Elisabeth E. Reinersmann. Sterling, NY, 1992.
Sprouts, pp. 36-37, says it was invented by J. H. Conway & M. S.
Paterson on 21 Feb 1976 [sic -- misprint of 1967] during their five o'clock tea
hour.
4.B.5. OVID'S GAME AND NINE MEN'S MORRIS
See
also 4.B.1 for historical material.
The
classic Nine Men's Morris board consists of three concentric squares with their
midpoints joined by four lines. The
corners are sometimes also joined by another four diagonal lines, but this
seems to be used with twelve men per side and is sometimes called Twelve Men's
Morris -- see 1891 below. Fiske 108
says this is common in America but infrequent in Europe, though on 127 he says
both forms were known in England before 1600, and both were carried to the US,
though the Nine form is probably older.
Murray 615 discusses Nine Men's
Morris. He cites Kurna, Egypt (‑14C),
medieval Spain (Alquerque de Nueve), the Gokstad ship and the steps of the
Acropolis of Athens. He says the board
sometimes has diagonals added and then is played with 9, 11 or 12 pieces.
Dudeney. AM. 1917.
Introduction to Moving Counter Problems, pp. 58-59. This gives a brief survey, mentioning a
number of details that I have not seen elsewhere, e.g. its occurrence in Poland
and on the Amazon. Says the board was
found on a Roman tile at Silchester and on the steps of the Acropolis in Athens
among other sites.
J. A. Cuddon. The Macmillan Dictionary of Sports and
Games. Macmillan, London, 1980. Pp. 563‑564. Discusses the history.
Says there is a c‑1400 board cut in stone at Kurna, Egypt and
similar boards were made in years 9 to 21 at Mihintale, Ceylon. Says Ars Amatoria may be describing Three
Men's Morris and Tristia may be describing a kind of Tic‑tac‑toe. Cites numerous medieval descriptions and
variants.
Claudia Zaslavsky. Tic Tac Toe and Other Three‑in‑a‑Row
Games from Ancient Egypt to the Modern Computer. Crowell, NY, 1982. This
is really a book for children and there are no references for the historical
statements. I have found most of them
elsewhere, and the author has kindly
send me a list of source books, but I have not yet tracked down the following
items -- ??.
There
is an English court record of 1699 of punishment for playing Nine Holes in
church.
There
is a Nine Men's Morris board on a stone on the temple of Seti I (presumably
this is at Kurna). There is a picture
in the 13C Spanish 'Book of Games' (presumably the Alfonso MS -- see below) of
children playing Alquerque de Tres (c= Three Men's Morris). A 14C inventory of the Duc de Berry lists
tables for Mérelles (=? Nine Men's Morris) (see Fiske 113-115 below) and a book
by Petrarch shows two apes playing the game.
H. Parker. Ancient Ceylon. Loc. cit. in 4.B.1. Nine
Men's Morris board in the Temple of Kurna, Egypt, ‑14C. [Rohrbough, below, says this temple was
started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal
Asiatic Soc. (1783) 17.] Two diagrams
for Nine Men's Morris are cut into the great flight of steps at Mihintale,
Ceylon and these are dated c1C. He
cites Bell; Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for
another diagram of similar age.
Jack Botermans, Tony Burrett,
Pieter van Delft & Carla van Spluntern.
The World of Games. (In Dutch,
1987); Facts on File, NY, 1989.
P.
35 describes Yih, a form of Three Men's Morris, played on a doubly crossed
square with a man moving "one step along any line". A note adds that only the French have a rule
forbidding the first player to play in the centre, which makes the game more
challenging and is recommended.
Pp.
103-107 is the beginning of a section:
Games of alignment and configuration and discusses various games, but
rather vaguely and without references.
They mention Al-Qurna, Mihintale, Gokstad and some other early
sites. They say Yih was described by
Confucius, was played c-500 and is "the game, that we now know as
tic-tac-toe, or three men's morris."
They describe Noughts and Crosses in the usual way. They then distinguish Tic-Tac-Toe, saying
"In Britain it is generally known as three men's morris ...." and say
it is the same as Yih, "which was known in ancient Egypt". They say "Ovid mentions
tic-tac-toe" in Ars Amatoria, that several Roman boards have survived and
that it was very popular in 14C England with several boards for this and Nine
Men's Morris cut into cloister seats.
They then describe Three-in-a-Row, which allows pieces to move one step
in any direction, as a game played in Egypt.
They then describe Five or Six Men's Morris, Nine Men's Morris, Twelve
Men's Morris and Nine Men's Morris with Dice, with nice 13C & 15C
illustration of Nine Men's Morris.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pp. 6-8. They discuss the crossed square board -- see
4.B.1 -- and describe Three Men's Morris with moves only along the lines to an
adjacent vacant point. They then
describe Achi, from Ghana, on the doubly crossed square with the same
rules. They then describe Six Men's
Morris which was apparently popular in medieval Europe but became obsolete by
c1600.
Ovid. Ars Amatoria. -1. II, 203-208
& III, 353-366. Translated by J. H. Mozley; Loeb Classical
Library, 1929, pp. 80-81 & 142-145.
Translated by B. P. Moore, 1935, used in A. D. Melville; Ovid The Love Poems; OUP, 1990, pp. 113, 137, 229
& 241.
II,
203-208 are three couplets apparently referring to three games: two dice games
and Ludus Latrunculorum. Mozley's prose
translation is:
"If she be gaming, and throwing
with her hand the ivory dice, do you throw amiss and move your throws amiss; or
if is the large dice you are throwing, let no forfeit follow if she lose; see
that the ruinous dogs often fall to you; or if the piece be marching under the
semblance of a robbers' band, let your warrior fall before his glassy
foe."
'Dogs'
is the worst throw in Roman dice games.
Moore's
verse translation of 207-208 is:
"And when the raiding chessmen
take the field, Your champion to his
crystal foe must yield."
Melville's
note says the original has 'bandits' and says the game is Ludus Latrunculorum.
III,
357-360 is probably a reference to the same game since 'robbers' occurs again,
though translated as brigands by Mozley, and again it immediately follows a
reference to throwing dice. Mozley's
translation of 353-366 is:
"I am ashamed to advise in
little things, that she should know the throws of the dice, and thy powers, O
flung counter. Now let her throw three
dice, and now reflect which side she may fitly join in her cunning, and which challenge, Let her cautiously and not foolishly play
the battle of the brigands, when one piece falls before his double foe and the
warrior caught without his mate fights on, and the enemy retraces many a time
the path he has begun. And let smooth
balls be flung into the open net, nor must any ball be moved save that which
you will take out. There is a sort of
game confined by subtle method into as many lines as the slippery year has
months: a small board has three counters on either side, whereon to join your
pieces together is to conquer."
Moore's
translation of 357-360 is:
"To guide with wary skill the
chessmen's fight, When foemen twain
o'erpower the single knight, And caught
without his queen the king must face
The foe and oft his eager steps retrace".
This
is clearly not a morris game -- Mozley's note above and the next entry make it
clear it is Ludus Latrunculorum, which had a number of forms. Mozley's note on pp. 142-143 refers to
Tristia II, 478 and cites a number of other references for Ludus Latrunculorum.
Moore's
translation of 363-366 is:
"A game there is marked out in
slender zones As many as the fleeting
year has moons; A smaller board with
three a side is manned, And victory's
his who first aligns his band."
Mozley's
notes and Melville's notes say the first two lines refer to the Roman game of
Ludus Duodecim Scriptorum -- the Twelve Line Game -- which is the ancestor of
Backgammon. Mozley says the game in the
latter two lines is mentioned in Tristia, "but we have no information
about it." Melville says it is "a
'position' game, something like Nine Men's Morris" and cites R. C. Bell's
article on 'Board and tile games' in the Encyclopaedia Britannica, 15th ed.,
Macropaedia ii.1152‑1153, ??NYS.
Ovid. Tristia. c10. II, 471‑484. Translated by A. L. Wheeler.
Loeb Classical Library, 1945, pp. 88‑91. This mentions several games and the text parallels that of Ars
Amatoria III.
"Others have written of the arts
of playing at dice -- this was no light sin in the eyes of our ancestors --
what is the value of the tali, with what throw one can make the highest
point, avoiding the ruinous dogs; how the tessera is counted, and when
the opponent is challenged, how it is fitting to throw, how to move according
to the throws; how the variegated soldier steals to the attack along the
straight path when the piece between two enemies is lost, and how he
understands warfare by pursuit and how to recall the man before him and to
retreat in safety not without escort; how a small board is provided with three
men on a side and victory lies in keeping one's men abreast; and the other
games -- I will not describe them all -- which are wont to waste that precious
thing, our time."
A
note says some see a reference to Ludus Duodecim Scriptorum at the beginning of
this. The next note says the next text
refers to Ludus Latrunculorum, a game on a squared board with 30 men on a side,
with at least two kinds of men. The
note for the last game says "This game seems to have resembled a game of draughts
played with few men." and refers to Ars Amatoria and the German
Mühlespiel, which he describes as 'a sort of draughts', but which is Nine Men's
Morris.
R. G. Austin. Roman board games -- I & II. Greece and Rome 4 (No. 10) (Oct 1934) 24‑34 & 4 (No. 11) (Feb
1935) 76-82. Claims the Ovid references
are to Ludus Latrunculorum (a kind of Draughts?), Ludus Duodecim Scriptorum
(later Tabula, an ancestor of Backgammon) and (Ars Amatoria.iii.365-366) a kind
of Three Men's Morris. In the last, he
shows a doubly crossed 3 x 3 board, but it is not clear which rule he
adopts for the later movement of pieces, but he says: "the first player is
always able to force a win if he places his first man on the centre point, and
this suggests that the dice may have been used to determine priority of play,
although there is no evidence of this."
He says no Roman name for this game has survived. He discusses various known artifacts for all
the game, citing several Roman 8 x
8 boards found in Britain. He gives an informal bibliography with
comments as to the value of the works.
D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940.
Section V Games and Playthings,
pp. 148-165. On p. 160, he quotes Ovid,
Ars Amatoria.iii.365-366 and says it is Noughts and Crosses, or in Ireland,
Tip-top-castle.
The British Museum has a Nine
Men's Morris board from the Temple of Artemis, Ephesus, 2C-4C. Item BM GR 1872,8-3,44. This was in a small exhibition of board
games in 1990. I didn't see it on
display in late 1996.
Murray, p. 189. There was an Arabic game called Qirq, which
Murray identifies with Morris.
"Fourteen was a game played with small stones on a wooden board
which had three rows of holes (al‑Qâbûnî)." Abû‑Hanîfa [the H
should have a dot under it], c750, held that Fourteen was illegal and
Qirq was held illegal by writers soon afterward. On p. 194, Murray gives a 10C passage mentioning Qirq being
played at Mecca.
Fiske 255 cites the Kitāb
al Aghāni, c960, for a reference to qirkat, i.e. morris boards.
Paul B. Du Chaillu. The Viking Age. Two vols., John Murray, London, 1889. Vol. II, p.168, fig. 992 -- Fragments of wood from Gokstad ship. Shows a partial board for Nine Men's Morris
found in the Gokstad ship burial. There
is no description of this illustration and there is only a vague indication
that this is 10C, but other sources say it is c900.
Gutorm Gjessing. The Viking Ship Finds. Revised ed., Universitets Oldsaksamling,
Oslo, 1957. P. 8: "... there are two boards which were
used for two kinds of games; on one side figures appear for use in a game which
is frequently played even now (known as "Mølle")."
Thorlief Sjøvold. The Viking Ships in Oslo. Universitets Oldsaksamling, Oslo, 1979. P. 54:
"... a gaming board with one antler gaming piece, ...."
In medieval Europe, the game is
called Ludus Marellorum or Merellorum or just Marelli or Merelli or Merels,
meaning the game of counters. Murray
399 says the connection with Qirq is unclear.
However, medieval Spain played various games called Alquerque, which is
obviously derived from Qirq. Alquerque
de Nueve seems to be Nine Men's Morris.
However, in Italy and in medieval France, Marelle or Merels could mean
Alquerque (de Doze), a draughts‑like game with 12 men on a side played on
a 5 x 5 board (Murray 615). Also
Marro, Marella can refer to Draughts which seems to originate in Europe
somewhat before 1400.
Stewart Culin. Korean Games, with Notes on the
Corresponding Games of China and Japan.
University of Pennsylvania, Philadelphia, 1895. Reprinted as: Games of the Orient; Tuttle, Rutland, Vermont, 1958. Reprinted under the original title, Dover
and The Brooklyn Museum, 1991. P. 102,
section 80: Kon-tjil -- merrells. This
is the usual Nine Men's Morris. The
Chinese name is Sám-k'i (Three Chess).
"I am told by a Chinese merchant that this game was invented by
Chao Kw'ang-yin (917-975), founder of the Sung dynasty." This is the only indication of an oriental
source that I have seen.
Gerhard Leopold. Skulptierte Werkstücke in der Krypta der
Wipertikirche zu Quedlinburg. IN:
Friedrich Möbius & Ernst Schubert, eds.; Skulptur des Mittelalters; Hermann
Böhlaus Nachfolger, Weimar, 1987, pp. 27-43; esp. pp. 37 & 43. Describes and gives photos of several
Nine-Men's-Morris boards carved on a pillar of the crypt of the Wipertikirche,
Quedlinburg, Sachsen-Anhalt, probably from the 10/11 C.
Richard de Fournivall. De Vetula.
13C. This describes various
games, including Merels. Indeed the
French title is: Ci parle du gieu des
Merelles .... ??NYS -- cited by Murray,
pp. 439, 507, 520, 628. Murray 620
cites several MSS and publications of the text.
"Bonus Socius"
[Nicolas de Nicolaï?]. This is a
collection of chess problems, compiled c1275, which exists in many manuscript
forms and languages. See 5.F.1 for more
details of these MSS. See Murray 618‑642. On pp. 619‑624 & 627, he mentions
several MSS which include 23, 24, 25 or 28 Merels problems. On p. 621, he cites "Merelles a
Neuf" from 14C. Fiske 104 &
110-111 discusses some MSS of this collection.
The Spanish Treatise on
Chess-Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of
Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototropic
Plates. 2 vols., Karl W. Horseman, Leipzig, 1913. (See in 4.A.1 for another ed.) This is a collection of chess problems
produced for Alfonso X, the Wise, King of Castile (Castilla). Vol. 2, ff. 92v‑93r,
pp. CLXXXIV‑CLXXXV, shows Nine Men's Morris boards. ??NX -- need to study text. See:
Murray 568‑573; van der
Linde I 137 & 279 ??NYS & Quellenstudien 73 & 277‑278, ??NYS
(both cited by Fiske 98); van der Lasa
116, ??NYS (cited by Fiske 99).
Fiske
98-99 says that the MS also mentions Alquerque, Cercar de Liebre and Alquerque
de Neuve (with 12 men against one).
Fiske 253-255 gives a more detailed study of the MS based on a
transcript. He also quotes a
communication citing al Querque or al Kirk in Kazirmirski's Arabic dictionary
and in the Kitāb al Aghāni, c960.
José
Brunet y Bellet. El Ajedrez. Barcelona, 1890. ??NYS -- described by Fiske 98.
This has a chapter on the Alfonso MS and refers to Alquerque de Doce,
saying that it is known as Tres en Raya in Castilian and Marro in Catalan
(Fiske 102 says this word is no longer used in Spanish). Brunet notes that there are five miniatures
pertaining to alquerque. Fiske says
that all this information leaves us uncertain as to what the games were. Fiske says Brunet's chapter has an appendix
dealing with Carrera's 1617 discussion of 'line games' and describing Riga di
Tre as the same as Marro or Tres en Raya as a form of Three Men's Morris
Murray gives many brief
references to the game, which I will note here simply by his page number and
the date of the item.
438‑439
(12C); 446 (14C);
449
(c1400 -- 'un marrelier', i.e. a Merels board);
431
(c1430); 447 (1491); 446 (1538).
Anon. Romance of Alexander.
1338. (Bodleian Library, Mss
Bodl. 264). ??NYS. Nice illustration clearly showing Nine Men's
Morris board. I. Disraeli (Amenities of
Literature, vol. I, p. 86) also cites British Museum, Bib. Reg. 15, E.6 as a
prose MS version with illustrations.
Prof. D. J. A. Ross tells me there is nothing in the text corresponding
to the illustrations and that the Bodleian text was edited by M. R. James,
c1920, ??NYS. Illustration reproduced
in: A. C. Horth; 101 Games to Make and
Play; Batsford, London, (1943; 2nd ed.,
1944); 3rd ed., 1946; plate VI facing
p. 44, in B&W. Also in: Pia Hsiao et al.; Games You Make and Play;
Macdonald and Jane's, London, 1975, p. 7, in colour.
Fiske 113-115 gives a number of
quotations from medieval French sources as far back as mid 14C, including an
inventory of the Duc de Berry in 1416 listing two boards. Fiske notes that the game has given rise to
several French phrases. He quotes a
1412 source calling it Ludus Sanct Mederici or Jeu Saint Marry and also
mentions references in city statutes of 1404 and 1414.
MS, Montpellier, Faculty of
Medicine, H279 (Fonts de Boulier, E.93).
14C. This is a version of the
Bonus Socius collection. Described in
Murray 623-624, denoted M, and in van der Linde I 301, denoted K. Lucas, RM2, 1883, pp. 98-99 mentions it and
RM4, 1894, Quatrième Récréation: Le jeu des mérelles au XIIIe
siècle, pp. 67-85 discusses it extensively.
This includes 28 Merels problems which are given and analysed by
Lucas. Lucas dates the MS to the 13C.
Household accounts of Edward IV,
c1470. ??NYS -- see Murray 617. Record of purchase of "two foxis and 46
hounds" to form two sets of "marelles".
Civis Bononiae [Citizen of
Bologna]. This is a collection of chess
problems compiled c1475, which exists in several MSS. See Murray 643‑703.
It has 48 or 53 merels problems.
On p. 644, 'merelleorum' is quoted.
A Hundred Sons. Chinese scroll of Ming period
(1368-1644). 18C copy in BM. ??NYS -- extensively reproduced and
described in: Marguerite Fawdry;
Chinese Childhood; Pollock's Toy Theatres, London, 1977. On p. 12 of Fawdry is a scene, apparently
from the scroll, in which some children appear to be playing on a Twelve Men's
Morris board.
Elaborate boards from Germany
(c1530) and Venice (16C) survive in the National Museum, Munich and in South
Kensington (Murray 757‑758).
Murray shows the first in B&W facing p. 757.
William Shakespeare. A Midsummer Night's Dream. c1610.
Act II, scene I, lines 98-100:
"The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton
green For lack of tread are
indistinguishable." Fiske 126
opines that the latter two lines may indicate that the board was made in the
turf, though he admits that they may refer just to dancers' tracks, but to me
it clearly refers to turf mazes.
J. C. Bulenger. De Ludis Privatis ac Domesticus
Veterum. Lyons, 1627. ??NYS
Fiske 115 & 119 quote his description of and philological note on
Madrellas (Three Men's Morris).
Paul Fleming (1609-1640). In one of his lyrics, he has Mühlen. ??NYS -- quoted by Fiske 132, who says this is
the first German mention of Morris.
Fiske 133 gives the earliest
Russian reference to Morris as 1675.
Thomas Hyde. Historia Nerdiludii, hoc est dicere,
Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From
the Sheldonian Theatre (i.e. OUP), Oxford, 1694. Historia Triodii, pp. 202-214, is on morris games. (Described in Fiske 118-124, who says there
is further material in the Elenchus at the end of the volume -- ??NYS) Hyde asserts that the game was well known to
the Romans, though he cannot find a Roman name for it! He cites and discusses Bulenger, but
disagrees with his philology. Gives
lots of names for the game, ranging as far as Russian and Armenian. He gives both the Nine and Twelve Men's
Morris boards on p. 210, but he has not found the Twelve board in Eastern
works. On p. 211, he gives the doubly
crossed square board with a title in Chinese characters, pronounced 'Che-lo',
meaning 'six places', and having three white and three black men already placed
along two sides. He says the Irish name
is Cashlan Gherra (Short Castle) and that the name Copped Crown is common in
Cumberland and Westmorland. He then
describes playing the Twelve Man and Nine Man games, and then he considers the
game on the doubly crossed square board.
He seems to say there are different rules as to how one can move. ??need to study the Latin in detail. This is said to throw light on the Ovid
passages. Hyde believes the game was
well known to the Romans and hence must be much older. Fiske remarks that this is history by
guesswork.
Murray 383 describes Russian
chess. He says Amelung identifies the
Russian game "saki with Hölzchenspiel (?merels)". Saki is mentioned on this page as being
played at the Tsar's court, c1675.
Archiv der Spiele. 3 volumes, Berlin, 1819-1821. Vol. 2 (1820) 21-27. ??NYS
Described and quoted by Fiske 129-132.
This only describes the crossed square and the Nine Men's Morris
boards. It says that the Three Men's
Morris on the crossed square board is a tie, i.e. continues without end, but it
is not clear how the pieces are allowed to move. Fiske says this gives the most complete explanation he knows of
the rules for Nine Men's Morris.
Charles Babbage. Notebooks -- unpublished collection of MSS
in the BM as Add. MS 37205. ??NX. For more details, see 4.B.1. On ff. 347.r-347.v, 8 Sep 1848, he suggests
Nine Men's Morris boards in triangular and pentagonal shapes and does various
counting on the different shapes.
The Family Friend (1856)
57. Puzzle 17. -- Two and a Bushel. Shows the standard # board. "This very simple and amusing games, --
which we do not remember to have seen described in any book of games, -- is
played, like draughts, by two persons with counters. Each player must have three, ...
and the game is won when one of the players succeeds in placing his
three men in a row; ...." There is
no specification of how the men move.
The word 'bushel' occurs in some old descriptions of Three Men's Morris
and Nine Men's Morris as the name of the central area.
The Sociable. 1858.
Merelles: or, nine men's morris, pp. 279-280. Brief description, notable for the use of Merelles in an English
book.
Von der Lasa. Ueber die griechischen und römischen Spiele,
welche einige ähnlichkeit mit dem Schach hatten. Deutsche Schachzeitung (1863) 162-172, 198-199, 225-234, 257‑264. ??NYS -- described on Fiske 121-122 &
137, who says van der Linde I 40-47 copies much of it. He asserts that the Parva Tabella of Ovid is
Kleine Mühle (Three Men's Morris). Von
der Lasa says the game is called Tripp, Trapp, Trull in the Swedish book
Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57??). Van der Linde says that the Dutch name is
Tik, Tak, Tol. Fiske notes that both of
these refer to Noughts and Crosses, but it is unclear if von der Lasa or van
der Linde recognised the difference between Three Men's Morris and Noughts and
Crosses.
Albert Norman. Ungdomens Bok [Book for Youth] (in
Swedish). 2nd ed., Stockholm,
1883. Vol. I, p. 162++. ??NYS -- quoted and described in Fiske
134-136. Plays Nine Men's Morris on a
Twelve Men's Morris board.
Webster's Dictionary. 1891.
??NYS -- Fiske 118 quotes a definition (not clear which) which includes
"twelve men's morris". Fiske
says: "Here we have almost the
only, and certainly the first mention of the game by its most common New
England name, "twelve men's morris," and also the only hint we have
found in print that the more complicated of the morris boards -- with the
diagonal lines ... -- is used with twelve men, instead of nine, on each
side." Fiske 127 says the name
only appears in American dictionaries.
Dudeney. CP.
1907. Prob. 110: Ovid's game,
pp. 156‑157 & 248. Says the
game "is distinctly mentioned in the works of Ovid." He gives Three Men's Morris, with moves to
adjacent cells horizontally or diagonally, and says it is a first player win.
Blyth. Match-Stick Magic.
1921. Black versus white, pp.
79-80. 4 x 4 board with four men each.
But the men must be initially placed
WBWB in the first row and BWBW
in the last row. They can move
one square "in any direction" and the object is to get four in a row
of your colour.
Games and Tricks -- to make the
Party "Go". Supplement to
"Pearson's Weekly", Nov. 7th, no year indicated [1930s??]. A matchstick game, p. 11. On a
4 x 4 board, place eight
men, WBWB on the top row and
BWBW on the bottom row. Players alternately move one of their men by
one square in any direction -- the object is to make four in a line.
Lynn Rohrbough, ed. Ancient Games. Handy Series, Kit N, Cooperative Recreation Service, Delaware,
Ohio, (1938), 1939.
Morris
was Player [sic] 3,300 Years Ago, p. 27.
Says the temple of Kurna was started by Ramses I and completed by Seti
in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.
Three
Men's Morris, p. 27. After placing
their three men, players 'then move trying to get three men in a row.' Contributor says he played it in Cardiff
more than 50 years ago.
Winning Ways. 1982.
Pp. 672-673. Says Ovid's Game is
conjectured to be Three Men's Morris.
The current version allows moves by one square orthogonally and is a
first person win if the first person plays in the centre. If the first player cannot play in the
centre, it is a draw. They use Three
Men's Morris for the case with one step moves along winning lines, i.e.
orthogonally or along main diagonals.
An American Indian game, Hopscotch, permits one step moves orthogonally
or diagonally (along any diagonal). A
French game, Les Pendus, allows any move to a vacant cell. All of these are draws, even allowing the
first player to play in the centre.
They briefly describe Six and Nine Men's Morris.
Ralph Gasser &
J. Nievergelt. Es ist
entscheiden: Das Muehle-Spiel ist unentscheiden. Informatik Spektrum 17 (1994) 314-317. ??NYS -- cited by Jörg Bewersdorff [email of 6 Jun 1999].
L. V. Allis. Beating the World Champion -- The state of
the art in computer game playing. IN:
Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian
Origins and Future Perspectives; International Institute for Asian Studies,
Leiden, 1995; pp. 155-175. On p. 163,
he states that Ralph Gasser showed that Nine Men's Morris is a draw in Oct
1993, but the only reference is to a letter from Gasser.
Ralph Gasser. Solving Nine Men's Morris. IN: Games of No Chance; ed. by Richard
Nowakowski; CUP, 1996, pp. 101-113.
??NYS -- cited by Bewersdorff [loc. cit.] and described in William Hartston; What mathematicians get up
to; The Independent Long Weekend (29 Mar 1997) 2. Demonstrates that Nine Men's Morris is a draw. Gasser's abstract: "We describe the
combination of two search methods used to solve Nine Men's Morris. An improved analysis algorithm computes
endgame databases comprising about 1010 states. An 18-ply alpha-beta search the used these databases
to prove that the value of the initial position is a draw. Nine Men's Morris is the first non-trivial
game to be solved that does not seem to benefit from knowledge-based
methods." I'm not sure about the
last statement -- 4 x 4 x 4 noughts and crosses (see 4.B.1.a) and
Connect-4 were solved in 1980 and 1988, though the first was a computer aided
proof and the original brute force solution of Connect-4 by James Allen in Sep
1988 was improved to a knowledge-based approach by L. V. Allis by Aug 1989. The five-in-a-row version of Connect-4 was
shown to be a first person win in 1993.
Bewersdorff [email of 11 Jun 1999] clarifies this by observing that draw
here means a game that continues forever -- one cannot come to a stalemate
where neither side can move.
Winning Ways. 1982.
Philosopher's football, pp. 688‑691. In 1985, Guy said this was the only published description of the
game.
This is best viewed as played on
a n x n array of squares.
The n(n+1) vertical edges belong to one player, say
red, while the n(n+1) horizontal edges belong to black. Players alternate marking a square with a
line of their colour between edges of their colour. A square cannot be marked twice.
The object is to complete a path across the board. In practice, the edges are replaced by
coloured dots which are joined by lines.
As with Hex, there can be no ties and there must be a first person strategy.
M. Gardner. SA (Oct 1958) c= 2nd Book, Chap. 7. Introduces David Gales's game, later called
Bridg‑it. Addendum in the book
notes that it is identical to Shannon's 'Bird Cage' game of 1951 and that it
was marketed as Bridg‑it in 1960.
M. Gardner. SA (Jul 1961) c= New MD, Chap. 18. Describes Oliver Gross's simple strategy for
the first player to win. Addendum in
the book gives references to other solutions and mentions.
M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Article says Bridg‑it was still on the
market.
Winning Ways. 1982.
Pp. 680-682. Covers Bridg-it and
Shannon Switching Game.
In Oct 2000, I bought a
second-hand copy of a 5 x 5 version called Connections, attributed to
Tom McNamara, but with no date.
Fred Schuh. Spel van delers (Game of divisors). Nieuw Tijdschrift vor Wiskunde 39 (1951‑52)
299‑304. ??NYS -- cited by Gardner, below.
M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Gives David Gale's description of his game
and results on it. Addendum in Knotted
points out that it is equivalent to Schuh's game and gives other references.
David Gale. A curious Nim-type game. AMM 81 (1974) 876-879. Describes the game and the basic
results. Wonders if the winning move is
unique. Considers three dimensional and
infinite forms. A note added in proof
refers to Gardner's article, says two programmers have consequently found that
the 8 x 10 game has two winning first moves and mentions Schuh's game.
Winning Ways. 1982.
Pp. 598-600. Brief description
with extensive table of good moves.
Cites an earlier paper of Gale and Stewart which does not deal with this
game.
I
have included this because it has an interesting history and because I found a
nice way to express it as a kind of Markov process or random walk, and this
gives an expression for the average time the game lasts. I then found that the paper by Daykin et al.
gives most of these ideas.
The
game has two or three rules for finishing.
A. One finishes by going exactly to the last
square, or beyond it.
B. One finishes by going exactly to the last
square. If one throws too much, then
one stands still.
C. One finishes by going exactly to the last
square. If one throws too much, one
must count back from the last square.
E.g., if there are 100 squares and one is at 98 and one throws 6, then
one counts: 99, 100, 99, 98, 97,
96 and winds up on 96. (I learned this from a neighbour's child but
have only seen it in one place -- in the first Culin item below.)
Games
of this generic form are often called promotion games. If one considers the game with no snakes or
ladders, then it is a straightforward race game, and these date back to
Egyptian and Babylonian times, if not earlier.
In fact, the same theory applies to
random walks of various sorts, e.g. random walks of pieces on a chessboard,
where the ending is arrival exactly at the desired square.
In the British Museum, Room 52,
Case 24 has a Babylonian ceramic board (WA 1991-7.20,I) for a kind of snakes
and ladders from c-1000. The label says
this game was popular during the second and first millennia BC.
Sheng-kuan t'u [The game of
promotion]. 7C. Chinese game. This is described in:
Nagao Tatsuzo; Shina Minzoku-shi [Manners and Customs of the Chinese];
Tokyo, 1940-1942, perhaps vol. 2, p. 707, ??NYS This is cited in:
Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London,
1977, p. 183, where the game is described as "played on a board or plan
representing an official career from the lowest to the highest grade, according
to the imperial examination system. It
is a kind of Snakes and Ladders, played with four dice; the object of each
player being to secure promotion over the others."
Thomas Hyde. Historia Nerdiludii, hoc est dicere,
Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From
the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De ludo promotionis mandarinorum, pp. 70-101 -- ??NX. This
is a long description of Shing quon
tu, a game on a board of 98 spaces,
each of which has a specific description which Hyde gives. There is a folding plate showing the Chinese
board, but the copy in the Graves collection is too fragile to photocopy. I did not see any date given for the game.
Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S. National Museum
for 1893, pp. 489‑537. Pp.
502-507 describes several versions of the Japanese Sugoroku (Double Sixes)
which is a generic name for games using dice to determine moves, including
backgammon and simple race games, as well as Snakes and Ladders games. One version has ending in the form C. Then says
Shing Kún T’ò (The Game of the
Promotion of Officials) is described by
Hyde as The Game of the Promotion of the Mandarins and gives an extended
description of it. There is a legend
that the game was invented when the Emperor Kienlung (1736-1796) heard a
candidate playing dice and the candidate was summoned to explain. He made up a story about the game, saying
that it was a way for him and his friends to learn the different ranks of the
civil service. He was sent off to bring
back the game and then made up a board overnight. However Hyde had described the game a century before this
date. It seems that this is not really
a Snakes and Ladders game as the moves are determined by the throw of the dice
and the position -- there are no interconnections between cells. But Culin notes that the game is complicated
by being played for money or counters which permit bribery and rewards, etc.
Culin. Chess and Playing Cards.
Op. cit. in 4.A.4. 1898.
Pp.
820-822 & plates 24 & 25 between 821 & 822. Says
Shing Kún T’ò (The Game of the
Promotion of Officials) is described by
Hyde as The Game of the Promotion of
the Mandarins and refers to the above
for an extended description. Describes
the Korean version:
Tjyong-Kyeng-To (The Game of Dignitaries) and several others from Korea and Tibet,
with 108, 144, 169 and 64 squares.
Pp.
840-842 & plate 28, opp. p. 841 describes
Chong ün Ch’au (Game of the
Chief of the Literati) as 'in many
respects analogous' to Shing Kún
T’ò and the Japanese game Sugoroku
(Double Sixes) -- in several versions.
Then mentions modern western versions -- Jeu de L'Oie, Giuoco
dell'Oca, Juego de la Oca, Snake Game.
Pp. 843-848 is a table listing 122 versions of the game in the
University of Pennsylvania Museum of Archaeology and Paleontology. These are in 11 languages, varying from 22
to 409 squares.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Snakes and Ladders
and the Chinese Promotion Game, pp. 65‑67. They describe the Hindu version of Snakes and Ladders,
called Moksha-patamu. Then they discuss Shing Kun t'o (Promotion
of the Mandarins), which was played in
the Ming (1368-1616) with four or more players racing on a board with 98 spaces
and throwing 6 dice to see how many equal faces appeared. They describe numerous modern variants.
Deepak Shimkhada. A preliminary study of the game of Karma in
India, Nepal, and Tibet. Artibus Asiae
44 (1983) 4. ??NYS - cited in Belloli
et al, p. 68.
Andrew Topsfield. The Indian game of snakes and ladders. Artibus Asiae 46:3 (1985) 203‑214 + 14
figures. Basically a catalogue of
extant Indian boards. He says the game
is called Gyān caupad [the d
should have an underdot] or Gyān chaupar in Hindi. He states
that Moksha-patamu sounds like it is Telugu and that this name
appeared in Grunfield's Games of the World (1975) with no reference to a source
and that Bell has repeated this. Game
boards were drawn or painted on paper or cloth and hence were perishable. The oldest extant version is believed to be
an 84 square board of 1735, in the Museum of Indology, Jaipur. There were Hindu, Jain, Muslim and Tibetan
versions representing a kind of Pilgrim's Progress, finally arriving at God or
Heaven or Nirvana. The number of squares
varies from 72 to 360.
He
gives many references and further details.
An Indian version of the game was described by F. E. Pargiter; An Indian
game: Heaven or Hell; J. Royal Asiatic Soc. (1916) 539-542, ??NYS. He cites the version by Ayres (and Love's
reproduction of it -- see below) as the first English version. He cites several other late 19C versions.
F. H. Ayres. [Snakes and ladders game.] No. 200682 Regd. Example in the Bethnal Green Museum, Misc. 8 - 1974. Reproduced in: Brian Love; Play The Game; Michael Joseph, London, 1978; Snakes
& Ladders 1, pp. 22-23. This is the
earliest known English version of the game, with 100 cells in a spiral
and 5
snakes and 5 ladders.
N. W. Bazely &
P. J. Davis. Accuracy of Monte
Carlo methods in computing finite Markov chains. J. of Res. of the Nat. Bureau of Standards -- Mathematics and
Mathematical Physics 64B:4 (Oct-Dec 1960) 211-215. ??NYS -- cited by Davis & Chinn and Bewersdorff. Bewersdorff [email of 6 Jun 1999] brought
these items to my attention and says it is an analysis based on absorbing
Markov chains.
D. E. Daykin, J. E. Jeacocke
& D. G. Neal. Markov chains and
snakes and ladders. MG 51 (No. 378)
(Dec 1967) 313-317. Shows that the game
can be modelled as a Markov process and works out the expected length of play
for one player (47.98 moves) or two players (27.44
moves) on a particular board with finishing rule A.
Philip J. Davis &
William G. Chinn. 3.1416 and All
That. S&S, 1969, ??NYS; 2nd ed, Birkhäuser, 1985, chap. 23 (by
Davis): "Mr. Milton, Mr. Bradley -- meet Andrey Andreyevich Markov",
pp. 164-171. Simply describes how to
set up the Markov chain transition matrix for a game with 100
cells and ending B. Doesn't give
any results.
Lewis Carroll. Board game for one. In: Lewis Carroll's Bedside Book; ed. by
Gyles Brandreth (under the pseud. Edgar Cuthwellis); Methuen, 1979, pp.
19-21. ??look for source; not in
Carroll-Wakeling, Carroll-Wakeling II or Carroll-Gardner. Board of 27 cells with pictures in the odd cells. If you land on any odd cell, except the last
one, you have to return to square 1.
"Sleep is almost certain to have overwhelmed the player before he
reaches the final square." Ending
A is probably intended. (The average
duration of this game should be computable.)
S. C. Althoen, L. King & K.
Schilling. How long is a game of snakes
and ladders? MG 77 (No. 478) (Mar 1993)
71-76. Similar analysis to Daykin,
Jeacocke & Neal, using finishing rule B, getting 39.2 moves. They also use a simulation to find the
number of moves is about 39.1.
David Singmaster. Letter [on Snakes and ladders]. MG 79 (No. 485) (Jul 1995) 396-397. In response to Althoen et al. Discusses history, other ending rules and
wonders how the length depends on the number of snakes and ladders.
Irving L. Finkel. Notes on two Tibetan dice games. IN: Alexander J. de Voogt, ed.; New
Approaches to Board Games Research: Asian Origins and Future Perspectives;
International Institute for Asian Studies, Leiden, 1995; pp. 24-47. Part II: The Tibetan 'Game of Liberation',
pp. 34-47, discusses promotion games with many references to the literature and
describes a particular game.
Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse
und Grenzen. Vieweg, 1998. Das Leiterspiel, pp. 67-68 &
Das Leiterspiel als Markow‑Kette.
Discusses setting up the Markov chain, citing Bazley & Davis, with
the same board as in Davis & Chinn, then states that the average duration
is 39.224 moves.
Jay Belloli, ed. The Universe A Convergence of Art, Music, and Science. [Catalogue for a group of exhibitions and
concerts in Pasadena and San Marino, Sep 2000 - Jun 2001.] Armory Center for the Arts, Pasadena,
2001. P. 68 has a discussion of the Jain
versions of the game, called 'gyanbazi', with a colour plate of a 19C example
with a 9 x 9 board with three extra cells.
This
is a Maori game which can be found in several books on board games. I have included it because it has been
completely analysed. There are eight
(or 2n) points around a central area.
Each player has four (or n) markers, originally placed on consecutive
points. One can move from a point to an
adjacent point or to the centre, or one can move from the centre to a point,
provided the position moved to is empty.
The first player who cannot move is the loser. To prevent the game becoming trivial, it is necessary to require
that the first two (or one) moves of each player involve his end pieces, though
other restrictions are sometimes given.
Marcia Ascher. Mu Torere: An analysis of a Maori game. MM 60 (1987) 90-100. Analyses the game with 2n
points. For n = 1,
there are 6 inequivalent positions (where equivalence is by rotation or
reflection of the board) and play is trivially cyclic. For
n = 2, there are 12
inequivalent positions, but there are no winning positions. For
n = 3, there are 30
inequivalent positions, some of which are wins, but the game is a
tie. Obtains the number of positions
for general n. For the traditional version with n = 4, there are 92 inequivalent positions, some of which are
wins, but the game is a tie, though this is not at all obvious to an
inexperienced player. In 1856, it was
reported that no foreigner could win against a Maori. For n = 5, there are
272 inequivalent positions, but
the game is a easy win for the first player -- the constraints on first moves
need to be revised. Ascher gives
references to the ethnographic literature for descriptions of the game.
Marcia Ascher. Ethnomathematics. Brooks/Cole Publishing, Pacific Grove, California, 1991. Sections 4.4-4.7, pp. 95-109 &
Notes 4-7, pp. 118-119.
Amplified version of her MM article.
There
were a number of earlier guessing games of the Mastermind type before the
popular version devised by Marco Meirovitz in 1973 -- see: Reddi.
One of these was the English Bulls and Cows, but I haven't seen anything
written on this and it doesn't appear in Bell, Falkener or Gomme. Since 1975 there have been several books on
the game and a number of papers on optimal strategies. I include a few of the latter.
NOTATION. If there are h holes and c
choices at each hole, then I abbreviate this as ch.
A. K. Austin. How do You play 'Master Mind'. MTg 71 (Jun 1975) 46-47. How to state the rules correctly.
S. S. Reddi. A game of permutations. JRM 8:1 (1975) 8-11. Mastermind type guessing of a permutation of 1,2,3,4
can win in 5 guesses.
Donald E. Knuth. The computer as Master Mind. JRM 9:1 (1976-77) 1-6. 64 can be won in 5
guesses.
Robert W. Irving. Towards an optimum Mastermind strategy, JRM 11:2 (1978-79) 81-87. Knuth's algorithm takes an average of 5804/1296 = 4.478 guesses. The author
presents a better strategy that takes an average of 5662/1296 = 4.369
guesses, but requires six guesses in one case. A simple adaptation eliminates this, but increases the average
number of guesses to 5664/1296 =
4.370. An intelligent setter will
choose a pattern with a single repetition, for which the average number of
guesses is 3151/720 = 4.376.
A. K. Austin. Strategies for Mastermind. G&P 71 (Winter 1978) 14-16. Presents Knuth's results and some other
work.
Merrill M. Flood. Mastermind strategy. JRM 18:3 (1985-86) 194-202. Cites five earlier papers on strategy,
including Knuth and Irving. He
considers it as a two-person game and considers the setter's strategy. He has several further papers in JRM
developing his ideas.
Antonio M. Lopez, Jr. A PROLOG Mastermind program. JRM 23:2 (1991) 81-93. Cites Knuth, Irving, Flood and two other
papers on strategy.
Kenji Koyama and Tony W.
Lai. An optimal Mastermind
strategy. JRM 25:4 (1994) 251‑256. Using exhaustive search, they find the
strategy that minimizes the expected number of guesses, getting expected
number 5625/1296 = 4.340. However, the worst case in this problem
requires 6 guesses. By a slight
adjustment, they find the optimal strategy with worst case requiring 5 guesses
and its expected number of guesses is
5626/1296 = 4.341. 10 references
to previous work, not including all of the above.
Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse
und Grenzen. Vieweg, 1998. Section 2.15 Mastermind: Auf Nummer sicher, pp. 227-234 &
Section 3.13 Mastermind:
Farbcodes und Minimax, pp. 316-319. Surveys
the work on finding optimal strategies.
Then studies Mastermind as a two-person game. Finds the minimax strategy for the 32 game and
describes Flood's approach.
4.B.12. RITHMOMACHIA = THE PHILOSOPHERS' GAME
I have generally not tried to include board games in any
comprehensive manner, but I have recently seen some general material on this
which will be useful to anyone interested in the game. The game is one of the older and more
mathematical of board games, dating from c1000, but generally abandoned about
the end of the 16C along with the Neo-Pythagorean number theory of Boethius on
which the game was based.
Arno Borst. Das mittelalterliche Zahlenkampfspiel. Sitzungsberichten der Heidelberger Akademie
der Wissenschaften, Philosophisch-historische Klasse 5 (1986) Supplemente. Available separately: Carl Winter, Heidelberg, 1986. Edits the surviving manuscripts on the
game. ??NYS -- cited by Stigter &
Folkerts.
Detlef Illmer, Nora Gädeke,
Elisabeth Henge, Helen Pfeiffer & Monika Spicker-Beck. Rhythmomachia. Hugendubel, Munich, 1987.
Jurgen Stigter. Emanuel Lasker: A Bibliography AND
Rithmomachia, the Philosophers' Game: a reference list. Corrected, 1988 with annotations to 1989, 1
+ 15 + 16pp preprint available from the author, Molslaan 168, NL‑2611 CZ
Delft, Netherlands. Bibliography of the
game.
Jurgen Stigter. The history and rules of Rithmomachia, the
Philosophers' Game. 14pp preprint available
from the author, as above.
Menso Folkerts. 'Rithmimachia'. In: Die deutsche
Litteratur des Mittelalters: Verfasserlexikon; 2nd ed., De Gruyter, Berlin,
1990; vol. 8, pp. 86-94. Sketches
history and describes the 10 oldest texts.
Menso Folkerts. Die Rithmachia des Werinher von
Tegernsee. In: Vestigia Mathematica, ed. by M. Folkerts
& J. P. Hogendijk, Rodopi, Amsterdam, 1993, pp. 107-142. Discusses Werinher's work (12C), preserved
in one MS of c1200, and gives an edition of it.
This
is a very broad field and I will only mention a few early items. Four row mancala games are played in south
and east Africa. Three row games are
played in Ethiopia and adjacent parts of Somaliland. Two row games are played everywhere else in Africa, the Middle
East and south and south-east Asia. See
the standard books by R. C. Bell and Falkener for many examples. Many general books mention the game, but I
only know a few specific books on the game -- these are listed first below.
One
article says that game boards have been found in the pyramids of Khamit (-1580)
and there are numerous old boards carved in rocks in several parts of Africa.
An anonymous article, by a member of the Oware Society in
London, [Wanted: skill, speed, strategy; West Africa (16-22 Sep 1996)
1486-1487] lists the following names for variants of the game: Aditoe (Volta
region of Ghana), Awaoley (Côte d'Ivoire), Ayo (Nigeria), Chongkak (Johore),
Choro (Sudan), Congclak (Indonesia), Dakon (Philippines), Guitihi (Kenya),
Kiarabu (Zanzibar), Madji (Benin), Mancala (Egypt), Mankaleh (Syria), Mbau
(Angola), Mongola (Congo), Naranji (Sri Lanka), Qai (Haiti), Ware (Burkina
Faso), Wari (Timbuktu), Warri (Antigua),
Stewart Culin. Mancala, The National Game of Africa. IN: US National Museum Annual Report 1894,
Washington, 1896, pp. 595-607.
Chief A. O. Odeleye. Ayo
A Popular Yoruba Game.
University Press Ltd., Ibadan, Nigeria, 1979. No history.
Laurence Russ. Mancala Games. Reference Publications, Algonac, Michigan, 1984. Photocopy from Russ, 1995.
Kofi Tall. Oware
The Abapa Version. Kofi Tall
Enterprise, Kumasi, Ghana, 1991.
Salimata Doumbia &
J. C. Pil. Les Jeux de
Cauris. Institut de Recherches
Mathématiques, 08 BP 2030, Abidjan 08, Côte d'Ivoire, 1992.
Pascal Reysset &
François Pingaud. L'Awélé. Le jeu des semailles africaines. 2nd ed., Chiron, Paris, 1995 (bought in Dec
1994). Not much history.
François Pingaud. L'awélé
jeu de strategie africain.
Bornemann, 1996.
Alexander J. de Voogt. Mancala
Board Games. British Museum
Press, 1997. ??NYR.
Larry (= Laurence) Russ. The Complete Mancala Games Book How to Play the World's Oldest Board
Games. Foreword by Alex de Voogt. Marlowe & Co., NY, 2000. His 1984
book is described as an earlier edition of this.
William Flinders Petrie. Objects of Daily Use. (1929);
Aris & Phillips, London??, 1974.
P. 55 & plate XLVII.
??NYS -- described with plate reproduced in Bell, below. Shows and describes a 3 x 14
board from Memphis, ancient Egypt, but with no date given, but Bell
indicates that the context implies it is probably earlier than ‑1500. Petrie calls it 'The game of forty-two and
pool' because of the 42 holes and a large hole on the side, apparently for
storing pieces either during play or between games.
R. C. Bell. Games to Play. Michael Joseph (Penguin), 1988.
Chap. 4, pp. 54-61, Mancala games.
On pp. 54-55, he shows the ancient Egyptian board from Petrie and his
own photo of a 3 x 6 board cut into the roof of a temple at
Deir-el-Medina, probably about ‑87.
Thomas Hyde. Historia Nerdiludii, hoc est dicere,
Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From
the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo Mancala, pp. 226-232.
Have X of part of this.
R. H. Macmillan. Wari.
Eureka 13 (Oct 1950) 12. 2 x
6 board with each cup having four to
start. Says it is played on the Gold
Coast.
Vernon A. Eagle. On some newly described mancala games from
Yunnan province, China, and the definition of a genus in the family of mancala
games. IN: Alexander J. de Voogt, ed.;
New Approaches to Board Games Research: Asian Origins and Future Perspectives;
International Institute for Asian Studies, Leiden, 1995; pp. 48-62. Discusses the game in general, with many references. Attempts a classification in general. Describes six forms found in Yunnan.
Ulrich Schädler. Mancala in Roman Asia Minor? Board Games Studies International Journal for the Study of Board
Games 1 (1998) 10-25. Notes that mancala could have been played on
a flat board of two parallel rows of squares, i.e. something like a 2 x n
chessboard, but that archaeologists have tended to view such patterns as
boards for race games, etc. Describes
52 examples from Asia Minor. Some
general discussion of Greek and Roman games.
John Romein &
Henri E. Bal (Vrije Universiteit, Amsterdam). New computer cluster solves 3500-year old game. Posted on
www.alphagalileo.org on 29 Aug
2002. They show that Awari is a tie
game. They determined all 889,063,398,406 possible positions and stored them in a 778 GByte
database. They then used a 144
processor cluster to analyse the graph, which 'only' took 51 hours.
R. C. Bell. Games to Play. 1988. Op. cit. in
4.B.13. P. 136 gives some history. The Académie Français adopted the word for
both the pieces and the game in 1790 and it was generally thought that they
were an 18C invention. However, a
domino was found on the Mary Rose, which sank in 1545, and a record of
Henry VIII (reigned 1509-1547) losing £450 at dominoes has been found.
Bell, p. 131, describes the
modern variant Tri-Ominos which are triangular pieces with values at the
corners. They were marketed c1970 and
marked © Pressman Toy Corporation, NY.
Hexadoms are hexagonal pieces
with numbers on the edges -- opposite edges have the same numbers. These were also marketed in the early 1970s
-- I have a set made by Louis Marx, Swansea, but there is no date on it.
Anonymous [R. S. &
J. M. B[rew ?]]. Svoyi kosiri is
an easy game. Eureka 16 (Oct 1953) 8‑12. This is an intriguing game of pure strategy
commonly played in Russia and introduced to Cambridge by Besicovitch. It translates roughly as 'One's own trumps'. There are two players and the hands are
exposed, with one's spades and clubs being the same as the other's hearts and
diamonds. At Cambridge, the cards below
6 are removed, leaving 36 cards in the deck.
The article doesn't explain how trumps are chosen, but if one has spades
as trumps, then the other has hearts as trumps! Players alternate playing to a central discard pile. A player can take the pile and start a new
pile with any card, or he can 'cover' the top card and then play any card on
that. 'Covering' is done by playing a
higher card of the same suit or one of the player's own trumps -- if this
cannot be done, e.g. if the ace of the player's own trumps has been played, the
player has to take the pile. The object
is to get rid of all one's cards.
7.AZ
is actually combinatorial rather than arithmetical and I may shift it.
Pictorial versions: The Premier (1880), Lemon (1890), Stein (1898), King
(1927).
Double-sided versions: The Premier (1880), Brown (1891).
Relation to Magic Squares: Loyd (1896), Cremer (1880), Tissandier
(1880 & 1880?), Cassell's (1881), Hutchison (1891).
Making a magic square with the
Fifteen Puzzle: Dudeney (1898),
Anon & Dudeney (1899), Loyd (1914), Dudeney
(1917), Gordon (1988). See also: Ollerenshaw & Bondi
in 7.N.
Peter Hajek. 1995 report of his 1992 visit to the Museum
of Money, Montevideo, Uruguay, with later pictures by Jaime Poniachik. In this Museum is a metal chest made in England
in 1870 for the National State Bank of Uruguay. The front has a 7 x
7 array of metal squares with bolt
heads. These have to be slid in a 12
move sequence to reveal the three keyholes for opening the chest. This opens up a whole new possible background
for the 15 Puzzle -- can anyone provide details of other such sliding devices?
S&B, pp. 126‑129,
shows several versions of the puzzle.
L. Edward Hordern. Sliding Piece Puzzles. OUP, 1986.
Chap. 2: History of the sliding block puzzle, pp. 18‑30. This is the most extensive survey of the
history. He concludes that Loyd did not
invent the general puzzle where the 15 pieces are placed at random, which
became popular in 1879(?). Loyd may
have invented the 14‑15 version or he may have offered the $1000 prize
for it, but there is no evidence of when (1881??) or where. However, see the entries for Loyd's Tit‑Bits
article and Dudeney's 1904 article which seem to add weight to Loyd's
claims. Most of the puzzles considered
here are described by Hordern and have code numbers beginning with a letter,
e.g. E23, which I will give.
I
contributed a note about computer techniques of solving such puzzles and hoping
that programmers would attack them as computer power increased.
In 1993-1995, I produced four
Sliding Block Puzzle Circulars, totalling 24 pages (since reformatted to 21),
largely devoted to reporting on computer solutions of puzzles in Hordern. Since then, a large number of solution
programs have appeared and many more puzzles have appeared. The best place to look is on Nick Baxter's
Sliding Block Home Page:
http://www.johnrausch.com/slidingblockpuzzles/index.html .
Embossing Co. Puzzle labelled "No. 2 Patent Embossed
puzzle of Fifteen and Magic Sixteen.
Manufactured by the Embossing Co.
Patented Oct 24 1865".
Illustrated in S&B, p. 127.
Examples are in the collections of Slocum and Hordern. Hordern, p. 25, says that searching has not
turned up such a patent.
Edward F. [but drawing gives E.]
Gilbert. US Patent 91,737 --
Alphabetical Instruction Puzzle.
Patented 22 Jun 1869. 1p + 1p
diagrams. Described by Hordern, p. 26. This is not really a puzzle -- it has the
sliding block concept, but along several tracks and with many blank
spaces. I recall a similar toy from
c1950.
Ernest U. Kinsey. US Patent 207,124 -- Puzzle-Blocks. Applied: 22 Nov 1877; patented: 20 Aug 1878. 2pp + 1p diagrams. Described by Hordern, p. 27.
6 x 6 square sliding
block puzzle with one vacant space and tongue & grooving to prevent falling
out. Has letters to spell words. He suggests use of triangular and diamond‑shaped
pieces. This seems to be the most
likely origin of the Fifteen Puzzle craze.
Montgomery Ward & Co. Catalogue.
1889. Reproduced in: Joseph J.
Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois,
1977?, p. 34. Spelling Boards. Like Gilbert's idea, but a more compact
layout.
Loyd prize puzzle: One hundred pounds. Tit-Bits (14 Oct 1893) 25 &
(18 Nov 1893) 111. Loyd is
described as "author of "Fifteen Puzzle," ...."
Loyd. Tit‑Bits 31 (24 Oct 1896) 57. Loyd asserts he developed the 15 puzzle from a 4 x 4
magic square. "[The fifteen
block puzzle] had such a phenomenal run some twenty years ago. ... There was one of the periodical revivals of
the ancient Hindu "magic square" problem, and it occurred to me to
utilize a set of movable blocks, numbered consecutively from 1 to 16, the conditions
being to remove one of them and slide the others around until a magic square
was formed. The "Fifteen Block
Puzzle" was at once developed and became a craze.
I
give it as originally promulgated in 1872 ..." and he shows it with the 15
and 14 interchanged. "The puzzle
was never patented" so someone used round blocks instead of square
ones. He says he would solve such
puzzles by turning over the 6 and the 9.
"Sphinx" [= Dudeney] says he well remembers the sensation and
hopes "Mr. Loyd is duly penitent."
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the
"Fifteen Puzzle" that in 1872 and 1873 was sold by millions,
.... When this puzzle was brought out
by its inventor, Mr. Sam Loyd, ... he thought so little of it that he did not
even take any steps to protect his idea, and never derived a penny profit from
it.... We have recently tried all over
the metropolis to obtain a single example of the puzzle, without success." Dudeney says the puzzle came with 16 pieces
and you removed the 16. He also says he
recently could not find a single example in London.
Loyd. The 14‑15 puzzle in puzzleland. Cyclopedia, 1914, pp. 235 & 371 (= MPSL1, prob. 21, pp.
19‑20 & 128). He says he
introduced it 'in the early seventies'.
One problem asks to move from the wrong position to a magic square with
sum = 30 (i.e. the blank is counted
as 0).
This is c= SLAHP, pp. 17‑18 & 89.
G. G. Bain. Op. cit. in 1, 1907. Story of Loyd being unable to patent it.
Anonymous & Sam Loyd. Loyd's puzzles, op. cit. in 1, 1896. Loyd "owns up to the great sin of
having invented the "15 block puzzle"", but doesn't refer to the
patent story or the date.
W. P. Eaton. Loc. cit. in 1, 1911. Loyd refers to it as the 'Fifteen block'
puzzle, but doesn't say he couldn't patent it.
Loyd Jr. SLAHP.
1928. Pp. 1‑3 &
87. "It was in the early 80's, ...
that the world‑disturbing "14‑15 Puzzle" flashed across
the horizon, and the Loyds were among its earliest victims." He gives many of the stories in the
Cyclopedia and two of the same problems.
He doesn't mention the patent story.
W. W. Johnson. Notes on the 15‑Puzzle -- I. Amer. J. Math. 2 (1879) 397‑399.
W. E. Story. Notes on the 15‑Puzzle -- II. Ibid., 399‑404.
J. J. Sylvester. Editorial comment. Ibid., 404.
(This
issue may have been delayed to early 1880??
Johnson & Story are not terribly readable, but Sylvester is
interesting, asserting that this is the first time that the parity of a
permutation has become a popular concept.)
Anonymous. Untitled editorial. New York Times (23 Feb 1880) 4. "... just now the chief amusement of
the New York mind, ... a mental epidemic ....
In a month from now, the whole population of North America will be at
it, and when the 15 puzzle crosses the seas, it is sure to become an English
mania."
Anonymous. EUREKA!
The Popular but Perplexing Problem Solved at Last. "THIRTEEN -- FOURTEEN -- FIFTEEN" New York Herald (28 Feb 1880) 8. ""Fifteen" is a puzzle of
seeming simplicity, but is constructed with diabolical cunning. At first sight the victim feels little or no
interest; but if he stops for a single moment to try it, or to look at any one
else who is trying it, the mania strikes him.
... As to the last two numbers,
it depends entirely upon the way in which the blocks happen to fall in the
first place .... Two or three
enterprising gamblers took up the puzzle and for a time made an excellent
living.... The subject was brought up
in the Academy of Sciences by the veteran scientist Dr. P. H. Vander
Weyde", who showed it could not be solved. The Herald reporter discovered that the problem is solvable if
one turns the board 90o, i.e. runs the numbers down instead of
across, and Vander Weyde was impressed.
The article implies the puzzle had already been widely known for some
time.
Mary T. Foote. US Patent 227,159 -- Game apparatus. Filed: 4 Mar 1880; patented: 4 May 1880. 1p
+ 1p diagrams. The patent is for a box
with sliding numbered blocks for teaching the multiplication tables. Lines 57-63: "I am aware that it is not novel to produce a game apparatus
in which blocks are to be mixed and then replaced by a series of moves; also,
that it is not novel to number such blocks, as in the "game of 15,"
so called, where the fifteen numbers are first mixed and then moved into
place."
Persifor Frazer Jr. Three methods and forty‑eight
solutions of the Fifteen Problem. Proc.
Amer. Philos. Soc. 18 (1878‑1880) 505‑510. Meeting of 5 Mar 1880. Rather cryptic presentation of some possible
patterns. Asserts his 26 Feb article in
the Bulletin (??NYS -- ??where -- Philadelphia??) was the first "solution
for the 13, 15, 14 case".
J. A. Wales. 15 - 14 - 13 -- The Great Presidential
Puzzle. Puck 7 (No. 158) (17 Mar 1880)
back cover.
Anonymous. Editorial:
"Fifteen". New York
Times (22 Mar 1880) 4. "No
pestilence has ever visited this or any other country which has spread with the
awful celerity of what is popularly called the "Fifteen Puzzle." It is only a few months ago that it made its
appearance in Boston, and it has now spread over the entire country." Asserts that an unregenerate Southern
sympathiser has introduced it into the White House and thereby disrupted a
meeting of President Hayes' cabinet.
Sch. [H. Schubert]. The Boss Puzzle. Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des
Hamburgischen unpartheyeischen Correspondent) No. 82 (6 Apr 1880) 11, with
response on 87 (11 Apr 1880) 12
(Sprechsaal). Gives a fairly careful description
of odd and even permutations and shows the puzzle is solvable if and only if it
is in an even permutation. The response
is signed X and says that when the problem is insoluble, just turn the box by
90o to see another side of the problem!
Gebr. Spiro, Hofliefer (Court
supplier), Jungfernsteig 3(?--hard to read), Hamburg. Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des
Hamburgischen unpartheyeischen Correspondent) No. 88 (13 Apr 1880) 7. Advertises Boss Puzzles: "Kaiser-Spiel 50Pf. Bismarck-Spiel 50 Pf. Spiel der 15 u.
16, 50 Pf. Spiel der 16 separat, 15
Pf. System und Lösung, 20 Pf."
G. W. Warren. Letter:
Clew to the Fifteen Puzzle. The
Nation 30 (No. 774) (29 Apr 1880) 326.
Anon. Shavings. The London
Figaro (1 May 1880) 12. "The
"15 Puzzle," which has for some months past been making a sensation in
New York equal to that aroused by "H. M. S. Pinafore" last
year, has at length reached this country, and bids fair to become the rage here
also." (Complete item!)
George Augustus Sala. Echoes of the Week. Illustrated London News 76 (No. 2138) (22
May 1880) 491.
Mary T. Foote. US Patent 227,159 -- Game Apparatus. Applied: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. Described in Hordern, p. 27.
3 x 12 puzzles based on
multiplication tables. Refers to the
"game of 15" and Kinsey.
Arthur Black. ??
Brighton Herald (22 May 1880).
??NYS -- mentioned by Black in a letter to Knowledge 1 (2 Dec 1881) 100.
Anonymous. Our latest gift to England. From the London Figaro. New York Times (11 Jun 1880) 2(?). ??page
The Premier. First (?) double‑sided version, with
pictures of Gladstone and Beaconsfield, apparently produced for the 1880 UK
election. Described in Hordern, pp. 32‑33
& plate I.
Ahrens. MUS II 227.
1918. Story of Reichstag being
distracted in 1880.
P. G. Tait. Note on the Theory of the "15
Puzzle". Proc. Roy. Soc. Edin. 10
(1880) 664‑665. Brief but valid
analysis. Mentions Johnson &
Story. First mention of the possibility
of a 3D version.
T. P. Kirkman. Question 6489 and Note on the solution
of the 15‑puzzle in question 6489.
Mathematical Questions with their Solutions from the Educational Times
34 (1880) 113‑114 & 35 (1881) 29‑30. The question considers the n x n
problem. The note is rather
cryptic. (No use??)
Messrs. Cremer (210 Regent St.
and 27 New Bond St., London). Brilliant
Melancholia. Albrecht Durer's Game of
the Thirty Four and "Boss" Game of the Fifteen. 1880.
Small booklet, 16pp + covers, apparently instructions to fit in a box
with pieces numbered 1 to 16 to be used for making magic squares as well as for
the 15 puzzle. Explains that only half
the positions of the 15 puzzle are obtainable and describes them by
examples. (Photo in The Hordern
Collection of Hoffmann Puzzles, p. 74, and in Hordern, op. cit. above, plate
IV.) Possibly written by "Cavendish"
(Henry Jones).
H. Schubert. Theoretische Entscheidung über das Boss‑Puzzle
Spiel. 2nd ed., Hamburg, 1880. ??NYS
(MUS, II, p. 227)
Gaston Tissandier. Les carrés magiques -- à propos du
"Taquin," jeu mathématique.
La Nature 8 (No. 371) (10 Jul 1880) 81‑82. Simple description of the puzzle called
'Taquin' which came from America and has had a very great success for several
weeks. Says it had 16 squares and was
usable as a sliding piece puzzle or a magic square puzzle. Cites Frénicle's 880 magic squares of order
4.
Anon. & C. Henry. Gaz. Anecdotique Littéraire, Artistique et
Bibliographique. (Pub. by
G. d'Heylli, Paris) Year 5, t. II,
1880, pp. 58‑59 & 87‑92.
??NYS
Piarron de Mondésir. Le dernier mot du taquin. La Nature 8 (No. 382) (25 Sep 1880) 284‑285. Simple description of parity decision for
the 15 puzzle. Says 'la Presse
illustrée' offered 500 francs for achieving the standard pattern from a random
pattern, but it was impossible, or rather it was possible in only half the
cases.
Jasper W. Snowdon. The "Fifteen" Puzzle. Leisure Hour 29 (1880) 493‑495.
Gwen White. Antique Toys. Batsford, London, 1971;
reprinted by Chancellor Press, London, nd [1982?]. On p. 118, she says: "The French game of Taquin was played
in 1880, in which 15 pieces had to be moved into 16 compartments in as few
moves as possible; the word 'taquin' means 'a teaser'." She gives no references.
Tissandier. Récréations Scientifiques. 1880?
2nd
ed., 1881 -- unlabelled section, pp. 143-153.
As: Le taquin et les carrés magiques;
seen in 1883 ed., ??NX; 1888:
pp. 208-215. Adapted from the 1880 La
Nature articles of Tissandier and de Mondésir.
1881 says it came from America -- 'récemment une nouvelle apparition',
but this is dropped in 1888 -- otherwise the two versions are the same.
Translated
in Popular Scientific Recreations, nd [c1890], pp. 731‑735. Text says "Mathematical games, ...,
have recently obtained a new addition ....
... from America, ...." The
references to contemporary reactions are deleted and the translation is
confused. E.g. the newspaper is now
just "a French paper" and the English says the problem is impossible
in nine cases out of ten!
Lucas. Récréations scientifiques sur l'arithmétique et sur la géométrie
de situation. Sixième récréation: Sur
le jeu du taquin ou du casse‑tête américain. Revue scientifique de France et de l'étranger (3) 27 (1881) 783‑788. c= Le jeu du taquin, RM1, 1882, pp. 189‑211. Revue says that Sylvester told him that it
was invented 18 months ago by an American deaf‑mute. RM1 says "vers la fin de
1878". Cf Schubert, 1895.
Cassell's. 1881.
Pp. 96‑97: American puzzles "15" and
"34". = Manson, pp.
246-248. Says "articles ... have
appeared in many periodicals, but no one has ... publish[ed] a solution."
Then sketches the parity concept and its application.
Richard A. Proctor. The fifteen puzzle. Gentlemen's Magazine 250 (No. 1801) (1881)
30‑45.
"Boss". Letter:
The fifteen puzzle. Knowledge 1
(11 Nov 1881) 37-38, item 13. This magazine
was edited by Proctor. The letter
starts: "I am told that in a
magazine article which appeared some time ago, you have attempted to show that
there are positions in the Fifteen Puzzle from which the won position can never
be obtained." I suspect the letter
was produced by Proctor. The response
is signed Ed. and begins: "I
thought the Fifteen Puzzle was dead, and hoped I had had some share in killing
the time-absorbing monster." Notes
that many people get to the position starting
blank, 1, 2, 3 and view this as
a win. Sketches parity argument and
suggests "Boss" work on the
3 x 3 or 3 x 2 or even the 2 x 2 version.
Editorial comment. The fifteen puzzle. Knowledge 1 (25 Nov 1881) 79. "I supposed every one knew the Fifteen
Puzzle." Proceeds to explain, obviously
in response to readers who didn't know it.
Arthur Black. Letter:
The fifteen puzzle. Knowledge 1
(2 Dec 1881) 100, item 80. Sketches a
proof which he says he published in the Brighton Herald of 22 May 1880.
"Yawnups". Letter:
The fifteen puzzle. Knowledge 1
(30 Dec 1881) 185. Solution from the
15-14 position obtained by turning the box.
Editorial comment says the solution uses 102 moves and the editor gets
an easy solution in 57 moves. Adds that
a 60 move solution has been received.
Arthur Black. Letter:
The fifteen puzzle. Knowledge 1
(13 Jan 1882) 230. Finds a solution
from the 15-14 position in 39 moves by turning the box and asserts no shorter
solution is possible. Says he also gave
this in the Brighton Herald in May 1880.
An addition says J. Watson has provided a similar solution, which takes
38 moves??
A. B. Letter: The fifteen
puzzle. Knowledge 2 (20 Oct 1882) 345,
item 598. Finds a box-turning solution
in 39 moves.
C. J. Malmsten. Göteborg Handl 1882, p. 75. ??NYS -- cited by Ahrens in his Encyklopadie
article, op. cit. in 3.B, 1904.
Anonymous. Enquire Within upon Everything. Houlston and Sons, London. This was a popular book with editions almost
every year -- I don't know when the following material was added. Section 2591: Boss; or the Fifteen Puzzle,
p. 363. Place the pieces
'indifferently' in the box. Half the
positions are unsolvable. Cites
Cavendish for the solution by turning the box 90o but notes this
only works with round pieces. Goes on
to The thirty-four puzzle, citing Dürer.
I found this material in the 66th ed., 862nd thousand, of 1883, but I
didn't find the material in the 86th ed of 1892.
Letters received and short
answers. Knowledge 4 (16 Nov 1883)
310. 'Impossible'.
P. G. Tait. Listing's Topologie. Philosophical Mag. (5) 17 (No. 103) (Jan
1884) 30‑46 & plate opp. p. 80. Section 11, p. 39. Simple
but cryptic solution.
Letters received and short
answers. Letter from W. S. B. asks how
to solve the problem when the last row has 13, 14, 15 [sic!]; Answer by Ed. points out the misprint and
says the easiest solution is to remove the 15 and put it after the 14, or to
invert the 6 and 9. Knowledge
6 (No. 159) (14 Nov 1884) 412
& 6 (No. 160) (21 Nov 1884)
429.
Don Lemon. Everybody's Pocket Cyclopedia .... Saxon & Co., London, (1888), revised 8th
ed., 1890. P. 137: The fifteen
puzzle. Brief description, with pieces
placed randomly in the box -- "to get the last three into order is often a
puzzle indeed".
John D. Champlin & Arthur E. Bostwick. The
Young Folk's Cyclopedia of Games and Sports.
1890. ??NYS Cited in Rohrbough; Brain Resters and
Testers; c1935; Fifteen Puzzle, p. 20.
Describes idea of parity of number of exchanges. [Another reference provided more details of
Champlin & Bostwick.]
Lemon. 1890. A trick puzzle, no.
202, pp. 31 & 105 (= Sphinx, no. 422, pp. 60 & 112). 15 puzzle with lines on the pieces to arrange
as "a representation of a president with only one eye". The solution is a spelling of the word 'president'. Attributed to Golden Days -- ??. After The Premier puzzle of c1880, this is
the second suggestion of using a picture and the first publication of the idea
that I have seen.
G. A. Hutchison, ed. Indoor Games and Recreations. The Boy's Own Bookshelf. (1888);
New ed., Religious Tract Society, London, 1891. (See M. Adams; Indoor Games for a much
revised version, but which doesn't contain this material.) Chap. 19: The American Puzzles., pp. 240‑241. "These puzzles, known as the 'Thirty‑four
Game' and the 'Fifteen Game,' on their introduction amongst us some years ago
...." "The '15' puzzle would
appear to have been, on its coming to England a few years ago, strictly a new
introduction ...." He sketches the
parity concept. [NOTE. I have seen a reference to the editor as
Hutchinson, but the book definitely omits the first n.]
Daniel V. Brown. US Patent 471,941 -- Puzzle. Applied: 23 Apr 1891; patented: 29 Mar 1892, 2pp + 1p diagrams. Double-sided 16 block puzzle to spell George
Washington on one side and Benjamin Harrison on the other. No sliding involved.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
American fifteen puzzle, pp. 105-107.
"The Fifteen Puzzle was introduced by a shrewd American some ten
years ago, ...." Refers to Tait's
1880 paper. Says half the positions are
impossible, but solves them by turning the box 90o or by inverting
the 6 and the 9.
Hoffmann. 1893.
Chap IV, no. 69: The "Fifteen" or "Boss" puzzles,
pp. 161‑162 & 217‑218 = Hoffmann-Hordern, pp. 142-144, with
photo of five early examples, two or three of which also are thirty-four
puzzles. (Hordern Collection, p. 74,
has a photo of a version by Cremer, cf above.)
"This, like a good many of the best puzzles, hails from America,
where, some years ago, it had an extraordinary vogue, which a little later
spread to this country, the British public growing nearly as excited over the
mystic "Fifteen" as they did at a later date over the less innocent
"Missing Word" competitions."
He distinguishes between the ordinary Fifteen where one puts the pieces
in at random, and the Boss or Master puzzle which has the 14 and 15
reversed. "Notwithstanding the
enormous amount of energy that has been expended over the "Fifteen"
Puzzle, no absolute rule for its solution has yet been discovered and it
appears to be now generally agreed by mathematicians that out of the vast
number of haphazard positions ... about half admit [of solution]. To test whether ... the following rule has
been suggested." He then says to
count the parity of the number of transpositions.
Hoffmann. 1893.
Chap. IV, no. 70: The peg‑away puzzle, pp. 163 & 218
= Hoffmann‑Hordern, p. 145.
This is a 3 x 3 version of the Fifteen puzzle, made by Perry
& Co. Start with a random pattern
and get to standard form. "The
possibility of success in solving this puzzle appears to be governed by
precisely the same rule as the "Fifteen" Puzzle." Hoffmann-Hordern has no photo of this -- do
any examples exist??
H. Schubert. Zwölf Geduldspiele. Dümmler, Berlin, 1895. [Taken from his columns in
Naturwissenschaftlichen Wochenschrift, 1891-1894.] Chap. VII: Boss-Puzzle oder Fünfzehner-Spiel, pp. 75-94?? Pp. 75-77 sketches the history, saying it
was called "Jeu du Taquin" (Neck-Spiel) in France and was popular in
1879-1880 in Germany. Cites Johnson
& Story and his own 1880 booklet.
Gives the story of a deaf and dumb American inventing it in Dec 1878,
saying "Sylvester communicated this at the annual meeting of the
Association Française pour l'Avancement des Sciences at Reims". Cf Lucas, 1881. [There is a second edition, Teubner??,
Leipzig, 1899, ??NYS. However this
material is almost identical to the beginning of Chap. 15 in Schubert's
Mathematische Mussestunden, 3rd ed., Göschen, Leipzig, 1909, vol. 2. The later version omits only some of the
Hamburg details of 1879-1880. Hence the
2nd ed. of Zwölf Geduldspiele is probably very close to these versions.]
Dudeney. Problem 49: The Victoria Cross puzzle. Tit‑Bits 32 (4 &
25 Sep 1897) 421 & 475.
= AM, 1917, prob. 218, pp. 60 & 194. B7. 3 x 3 board with letters Victoria going clockwise
around the edges, leaving the middle empty, and starting with V in
a corner. Slide to get Victoria
starting at an edge cell, in the fewest moves.
Does it in 18 moves, by interchanging the i's and says there are 6 such solutions.
Dudeney. Problem 65: The Spanish dungeon. Tit‑Bits 33 (1 Jan &
5 Feb 1898) 257 & 355.
= AM, 1917, prob. 403, pp. 122-123 & 244. B14.
Convert 15 Puzzle, with pieces in correct order, into a magic
square. Does it in 37 moves.
Conrad F. Stein. US Design 29,649 -- Design for a
Game-Board. Applied: 29 Sep 1898; patented: 8 Nov 1898 as No. 692,242. 1p + 1p diagrams. This
appears to be a 3 x 4 puzzle with a picture of a city with a Spanish
flag on a tower. Apparently the object
is to move an American flag to the tower.
Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 &
3:1 (Apr 1900) 89. The eight fat
boys. 3 x 3 square with pieces: 1 2
3; 4 X 5; 6 7 8 to be shifted into
a magic square. Two solutions in 19
moves. Cf Dudeney, 1917.
Addison Coe. US Patent 785,665 -- Puzzle or Game
Apparatus. Applied: 17 Nov 1904; patented: 1 Mar 1905. 4pp + 3pp diagrams. Gives a
3 x 5 flat version and a 3‑dimensional
version -- cf 5.A.2.
Dudeney. AM.
1917.
Prob.
401: Eight jolly gaol birds, pp. 122 & 243. E23. Same as 'The eight
fat boys' (see Anon. & Dudeney, 1899) with the additional condition that
one person refuses to move, which occurs in one of the two previous solutions.
Prob.
403: The Spanish dungeon, pp. 122-123 & 244. = Tit-Bits prob. 65 (1898).
B14.
Prob.
404: The Siberian dungeons, pp. 123 & 244.
B16. 2 x 8 array with prisoners 1, 2, ..., 8 in top row and 9, 10,
..., 16 in bottom row. Two extra rows of 4 above the right hand end
(i.e. above 5, 6, 7, 8) are empty.
Slide the prisoners into a magic square. Gives a solution in 14 moves, due to G. Wotherspoon, which they
feel is minimal. This allows long moves
-- e.g. the first move moves 8 up two and left 3.
"H. E. Licks" [pseud.
of Mansfield Merriman]. Recreations in
Mathematics. Van Nostrand, NY,
1917. Art. 28, pp. 20‑21. 'About the year 1880 ... invented in 1878 by
a deaf and dumb man....'
[From
sometime in the 1980s, I suspected the author's name was a pseudonym. On pp. 132-138, he discusses the Diaphote
Hoax, from a Pennsylvania daily newspaper of 10 Feb 1880, which features the
following people: H. E. Licks, M. E. Kannick, A. D. A. Biatic, L. M.
Niscate. The diaphote was essentially a
television. He says this report was
picked up by the New York Times and the New York World. An email from Col. George L. Sicherman on 5
Jun 2000 agrees that the name is false and suggested that the author was
"the eminent statistician Mansfield Merriman" who wrote the article
on The Cattle Problem of Archimedes in Popular Science Monthly (Nov
1905), which is abridged on pp. 33-39 of the book, but omitting the author's
name. Sichermann added that Merriman was
one of the authors of Pillsbury's List.
William Hartston says this was an extraordinary list of some 30 words
which Pillsbury, who did memory feats, was able to commit to memory quite
rapidly. Sicherman continued to
investigate Merriman and got Prof. Andri Lange interested and Lange
corresponded with a James A. McLennan, author of a history of the physics
department at Lehigh University where Merriman had been. McLennan found Merriman's obituary from the
American Society of Civil Engineers which states that Merriman used H. E. Licks
as a pseudonym. [Email from Sicherman
on 25 Feb 2002.]]
Stephen Leacock. Model Memoirs and Other Sketches from Simple
to Serious. John Lane, The Bodley Head,
1939, p. 300. "But this puzzle
stuff, as I say, is as old as human thought.
As soon as mankind began to have brains they must have loved to exercise
them for exercise' sake. The 'jig-saw'
puzzles come from China where they had them four thousand years ago. So did the famous 'sixteen puzzle' (fifteen
movable squares and one empty space) over which we racked our brains in the
middle eighties."
G. Kowalewski. Boss‑Puzzle und verwandte Spiele. K. F. Kohler Verlag, Leipzig, 1921
(reprinted 1939). Gives solution of
general polygonal versions, i.e. on a graph with a Hamilton circuit and one or
more diagonals.
Hummerston. Fun, Mirth & Mystery. 1924.
1 2
9 Push,
pp. 22 & 25. This is played on the
board
3 10 11
4 shown at the left
with its orthogonal lines, like
12 13 3,
10, 11, 4, and its diagonal lines, like
5 14 15
6 1, 9, 11, 13,
6. 10, 15 and 11, 14 are not
16 connected,
so this is an octagram. Take 16
7 8 numbered counters and place
them at random on
the
board and remove counter 16. Move the
pieces
to
their correct locations. He asserts
that 'unlike the original ["Sixteen" Puzzle],
no
position can be set up in "Push" that cannot be solved'.
The six bulls puzzle, Puzzle no.
34, pp. 90 & 177. This uses
the 2 x 3 + 1 0
board
shown at the right, where the 0 is the blank space. Exchange 1 2 3
3
and 6 and 4 and 5. He does it
in 20
moves. [This is Hordern's 4 5 6
B3,
first known from 1977 under the name Bull Pen, but is a variant of
Hordern's B2, first known from 1973.]
Q. E. D. -- The sergeant's
problem, Puzzle no. 40, pp. 106 & 178.
Take a 2 x 3 board, with the centre of one long side
blank. Interchange the men along one
short side. He does this in 17
moves, but the blank is not in its initial position nor are the other
men. [This is Hordern's B1, first known
from Loyd's Cyclopedia, 1914.]
King. Best 100. 1927. No. 26, p. 15. = Foulsham's, no. 9, pp. 7 & 10. "An entertaining variation ... is to draw, and colour, if
you like, a small picture; then cut it into sixteen squares and discard the
lower right hand square."
G. Kowalewski. Alte und neue mathematische Spiele. Teubner, Leipzig, 1930, pp. 61‑81. Gives solution of general polygonal
versions.
Dudeney. PCP.
1932. The Angelica puzzle, prob.
253, pp. 76 & 167. = 435, prob.
378, pp. 136 & 340. B8. 3 x 3
problem -- convert: A C I L E G
N A X to A N G
E L I C A X. Requires interchanging the As.
Solution in 36 moves.
In the answer in 435, Gardner notes that it can be done in 30
moves.
H. V. Mallison. Note 1454:
An array of squares. MG 24 (No.
259) (May 1940) 119‑121.
Discusses 15 Puzzle and says any legal position can be achieved in at
most about 150 moves. But if one fixes
cells 6, 7, 11, then a simple problem requires about 900
moves.
McKay. At Home Tonight.
1940. Prob. 44: Changing the
square, pp. 73 & 88. In the usual
formation, colour the pieces alternately blue and red, as on a chessboard, with
the blank at the lower right position 16 being a missing red, so there are 7
reds. Move so the colours are still
alternating but the blank is at the lower left, i.e. position 13. Takes 15 moves.
Sherley Ellis Stotts. US Patent 3,208,753 -- Shiftable Block
Puzzle Game. Filed: 7 Oct 1963; patented: 28 Sep 1965. 4pp + 2pp diagrams. Described in Hordern, pp. 152-153, F10‑12. Rectangular pieces of different sizes. One can also turn a piece.
Gardner. SA (Feb 1964) = 6th Book, chap. 7.
Surveys sliding-block puzzles with non-square pieces and notes there is
no theory for them. Describes a number
of early versions and the minimum number of moves for solution, generally done
by hand and then confirmed by computer.
Pennant Puzzle, C19; L'Âne
Rouge, C27d; Line Up the Quinties,
C4; Ma's Puzzle, D1; a form of Stotts' Baby Tiger Puzzle, F10.
Gardner. SA (Mar & Jun 1965) c= 6th Book, chap. 20. Prob. 9: The eight-block puzzle. B5.
3 x 3 problem -- convert: 8 7 6
5 4 3 2 1 X to
1 2 3 4 5 6 7 8 X.
Compares it with Dudeney's Angelica puzzle (1932, B8) but says it can be
done if fewer than 36 moves.
Many readers found solutions in
30 moves; two even found all 10 minimal solutions by
hand! Says Schofield (see next entry)
has been working on this and gives the results below, but this did not quite
resolve Gardner's problem. William F.
Dempster, at Lawrence Radiation Laboratory, programmed a IBM 7094 to find all
solutions, getting 10 solutions in 30 moves; 112
in 32 moves and 512
in 34 moves. Notes it is
unknown if any problem with the blank in a side or corner requires more
than 30 moves. (The description
of Schofield's work seems a bit incorrect in the SA solution, and is changed in
the book.)
Peter D. A. Schofield. Complete solution of the 'Eight‑Puzzle'. Machine Intelligence 1 (1967) 125‑133. This is the
3 x 3 version of the 15 Puzzle,
with the blank space in the centre.
Works with the corner twists which take the blank around a 2 x 2
corner in four moves. Shows that
the 5-puzzle, which is the 3 x 2 version, has every position reachable in at
most 20 moves, from which he shows that an upper bound for the 8-puzzle
is 48
moves. Since the blank is in the
middle, the 8!/2 = 20160 possible positions fall into 2572
equivalence classes. He also
considers having inverse permutations being equivalent, which reduces to 1439
classes, but this was too awkward to implement. An ATLAS program found that the maximum
number of moves required was 30 and
60 positions of 12
classes required this maximum number, but no example is given -- but see
previous entry.
A. L. Davies. Rotating the fifteen puzzle. MG 54 (No. 389) (Oct 1970) 237‑240. Studies versions where the numbers are printed
diagonally so one can make a 90o turn of the puzzle. Then any pattern can be brought to one of
two 'natural' patterns. He then asks
when this is true for an m x n board and obtains a complicated
solution. For an n x n
board, n must be divisible by 4.
R. M. Wilson. Graph puzzles, homotopy and the alternating
group. J. Combinatorial Thy., Ser. B,
16 (1974) 86‑96. Shows that a
sliding block puzzle, on any graph of n
+ 1 points which is non‑separable
and not a cycle, has at least An as its group -- except for one case on 7
points.
Alan G. & Dagmar R.
Henney. Systematic solutions of the
famous 15‑14 puzzles. Pi Mu
Epsilon J. 6 (1976) 197‑201. They
develop a test‑value which significantly prunes the search tree. Kraitchik gave a problem which took him 114
moves -- the authors show the best solution has 58
moves!
David Levy. Computer Gamesmanship. Century Publishing, London, 1983. [Most of the material appeared in Personal
Computer World, 1980‑1981.] Pp.
16‑29 discusses 8‑puzzle and uses the Henney's test‑value as
an evaluation function. Cites
Schofield.
Nigel Landon & Charles
Snape. A Way with Maths. CUP, 1984.
Cube moving, pp. 23 & 46. Consider
a 9-puzzle in the usual arrangement: 1
2 3, 4 5 6, 7 8 x. Move the 1 to
the blank position in the minimal number of moves, ignoring what happens to the
other pieces. Generalise. Their answer only says 13
is minimal for the 3 x 3 board.
My
student Tom Henley asked me the m x
n problem in 1993 and gave a
conjectural minimum, which I have corrected to: if m = n, then it can be done in 8m ‑ 11 moves;
but if n < m, then it can be done in 6m + 2n - 13 moves, using a straightforward method. However, I don't see how to show this is minimal, though it seems
pretty clear that it must be. I call
this a one-piece problem. See also
Ransom, 1993.
Len Gordon. Sliding the 15‑1 [sic, but 15‑14
must have been meant] puzzle to magic squares.
G&PJ 4 (Mar 1988) 56.
Reports on computer search to find minimal moves from either ordinary or
15‑14 forms to a magic square.
However, he starts with the blank before the 1, i.e. as a 0
rather than a 16.
Leonard J. Gordon. The 16‑15 puzzle or trapezeloyd. G&PJ 10 (1989) 164. Introduces his puzzle which has a
trapezoidal shape with a triangular wedge in the 2nd and 3rd row so the last
row can hold 5 pieces, while the other rows hold four pieces. Reversing the last two pieces can be done
in 85
moves, but this may not be minimal.
George T. Gilbert & Loren C.
Larson. A sliding block problem. CMJ 23:4 (Sep 1992) 315‑319. Essentially the same results as obtained by
R. Wilson (1974). Guy points this out
in 24:4 (Sep 1993) 355-356.
P. H. R. [Peter H. Ransom]. Adam's move. Mathematical Pie 128 (Spring 1993) 1017 & Notes, p. 3. Considers the one piece problem of Langdon
& Snape, 1984. Solution says the
minimal solution on a n x n board is
8n - 11, but doesn't give the
answer for the m x n board.
Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24-25
& 33 (Feb 1994) 32. In 1986, the German games company ASS
(Altenburg Stralsunder Spielkarten AG) produced a game called Vorsicht
(= Caution). Basically this is a 3 x 3
board considered as a doubly crossed square. It has pieces marked with
+ or x. The +
pieces can only move orthogonally;
the x pieces can only move diagonally.
The pieces are coloured and eight are placed on the board to be played
as a sliding piece puzzle from given starts to given ends. The diagonal moves are awkward to make and
Wiezorke suggests the board be spread out enough for diagonal moves to be
made. A note at the end says he has
received two similar games made by Y. A. D. Games in Israel.
Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of Computer
Science and Statistics, University of Rhode Island, Jan 1994. The Missing Link is a cylindrical form of
the Fifteen Puzzle, with four layers and four pieces in each layer. The middle two layers are rigidly joined,
but that makes little difference in solving the puzzle. After outlining the relevant group theory
and solving the Fifteen Puzzle, he shows the state space of the Missing Link
is S15.
Richard E. Korf &
Ariel Felner. Disjoint pattern
database heuristics. Artificial
Intelligence 134 (2002) 9-22. Discusses
heuristic methods of solving the Fifteen Puzzle, Rubik's Cube, etc. The authors applied their method to 1000
random positions of the Fifteen Puzzle.
The optimal solution length averaged 52.522 and the average time
required was 27 msec. They also did 50
random positions of the Twenty-Four Puzzle and found an average optimal
solution length of 100.78, with average time being two days on a 440MHz
machine.
S&B, pp. 130‑133, show
many versions.
See Kinsey, 1878, above, for
mention of triangular and diamond‑shaped pieces.
Henry Walton. US Patent 516,035 -- Puzzle. Applied: 14 Mar 1893; patented: 6 Mar 1894. 1p + 1p diagrams. Described in Hordern, pp. 27 & 68‑69,
C1. 4 x 4 area with five
1 x 2 & two
2 x 1 pieces.
Lorman P. Shriver. US Patent 526,544 -- Puzzle. Applied: 28 Jun 1894; patented: 25 Sep 1894, 2pp + 1p diagrams. Described in Hordern, p. 27. 4 x 5
area with two 2 x 1 &
15 1 x 1
pieces. Because there is only
one vacant space, the rectangles can only move lengthwise and so this is a dull
puzzle.
Frank E. Moss. US Patent 668,386 -- Puzzle. Applied: 8 Jun 1900; patented: 19 Feb 1901. 2pp + 1p diagrams. Described in Hordern, pp. 27‑28
& 75, C14. 4 x 4 area with six 1 x 1, two 1 x 2
&
two 2 x 1
pieces, allowing sideways movement of the rectangles.
William H. E. Wehner. US Patent 771,514 -- Game Apparatus. Applied: 15 Feb 1904; patented: 4 Oct 1904. 2pp + 1p diagrams. First to use L-shaped
pieces. Described in Hordern, pp. 28
& 107, D5.
Lewis W. Hardy. US Patent 1,017,752 -- Puzzle. Applied: 14 Dec 1907; patented: 20 Feb 1912. 3pp + 1p diagrams. Described in Hordern, pp. 29 & 89‑90, C43-45. 4 x 5
area with one 2 x 2, two
1 x 2, three 2 x 1
& four 1 x 1
pieces.
L. W. Hardy. Pennant Puzzle. Copyright 1909. Made by
OK Novelty Co., Chicago. No known
patent. Described in Gardner, SA (Feb
1964) = 6th Book, chap. 7 and in Hordern, pp. 28‑29 & 78-79,
C19. 4 x 5 area with one 2 x 2,
two 1 x 2, four 2 x 1, two
1 x 1 pieces.
Nob Yoshigahara designed Rush
Hour in the late 1970s and it was produced in Japan as Tokyo Parking Lot. Binary Arts introduced it to the US in 1996
and it became very popular.
Winning Ways. 1982.
Pp. 769-777: A trio of sliding block puzzles. This covers Dad's Puzzler (c19, with piece 8 moved two places to
the right), The Donkey (C27d, with all the central pieces moved down one
position) and The Century (C42), showing how one can examine partial problems
which allow one to consider many positions the same and much reduce the number
of positions to be studied. This allows
the graph to be written on a large sheet and solutions to be readily found.
Andrew N. Walker. Checkmate and other sliding-block
puzzles. Mathematics Preprint Series,
University of Nottingham, no. 95-32, 1995, 8pp + covers. Describes a version by W. G. H. [Wil]
Strijbos made by Pussycat. 4 x 4 with an extra position below the left
column. Pieces are alternately black
and white and have a black king, a white king and a white rook on them and the
object is to produce checkmate, but all positions must be legal in chess,
except that the black and white markings do not have to be correct in the
intermediate positions. However, one
soon finds that one tile is fixed in place and two other tiles are joined
together. He discusses general computer
solving techniques and finds there are five optimal solutions in 68 moves. He then discusses other problems, citing
Winning Ways, Hordern and my Sliding Block Puzzle Circulars. He gives the UNIX shell scripts that he
used.
Ivars Peterson. Simple puzzles can give computers an
unexpectedly strenuous workout. Science
News 162:7 (17 Aug 2002) 6pp PO from their website, http:''sciencenews.org . Reports on recent work by Gary W. Flake
& Eric B. Baum that Nob Yoshigahara's Rush Hour puzzle is PSPACE complete,
but is not polynomial time. Robert A.
Hearn and Erik D. Demaine have verified and extended this, showing other
sliding block puzzles are PSPACE complete, including the case where all pieces
are dominoes and can slide sideways as well as front and back.
5.A.2. THREE DIMENSIONAL VERSIONS
See Hordern, pp. 27, 156‑160
& plates IX & X.
P. G. Tait. Note on the Theory of the "15
Puzzle". Proc. Roy. Soc. Edin. 10
(1880) 664‑665. "...
conceivable, but scarcely realisable ..."
Charles I. Rice. US Patent 416,344 -- Puzzle. Applied: 9 Sep 1889; patented: 3 Dec 1889. 2pp + 1p diagrams. Described in Hordern, pp. 27 & 157‑158,
G2. 2 x 2 x 2 version with peepholes in the faces.
Ball. MRE, 1st ed., 1892, p. 78.
Mentions possibility.
Hoffmann. 1893.
Chap. X, No. 1: The John Bull political puzzle, pp. 331 &
357-358 = Hoffmann-Hordern, pp.
215-216. A 3 x 3 board in the form of
a cylinder, with an extra cell attached to one bottom cell. Pieces can move back and forth around each
level, but the connections from one level to the next are all parallel to one
of the diagonals -- though this isn't really a complication compared to having
vertical connections. The pieces have
two markings: three colours and three letters.
When they are randomly placed on the board, you have to move them so
they form a pair of orthogonal 3 x
3 Latin squares. Fortunately there are such arrangements
which differ by an odd permutation, so the puzzle can be solved from any random
starting point. Two examples done. Says the game is produced by Jaques &
Son.
Addison Coe. US Patent 785,665 -- Puzzle or Game
Apparatus. Applied: 17 Nov 1904; patented: 1 Mar 1905. 4pp + 3pp diagrams. Mentioned in Hordern, pp. 158‑159,
G3. Gives a 3 x 5 flat version and
a 3 x 3 x 3 cubical version with 3 x
3 arrays of holes in the six faces (in
order to push the pieces) and a
3 x 5 cylindrical version.
Burren Loughlin &
L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909.
The nine disks, pp. 29-34 & 60.
Same as Hoffmann except pieces have colour and shape.
Guy thinks Hein patented
Bloxbox, but I have not found any US patent of it -- ??CHECK.
Gardner. SA (Feb 1973). First mention of Hein's Bloxbox.
Daniel Kosarek. US Patent 3,845,959 -- Three-Dimensional
Block Puzzle. Filed: 14 Nov 1973; patented: 5 Nov 1974. 3pp + 1p diagrams (+ 1p abstract). Mentioned in Hordern, pp. 158‑159,
G3. 3 x 3 x 3 box with 3 x 3 array of portholes on each face. Mentions
4 x 4 x 4 and
larger versions.
Gabriel Nagorny. US Patent 4,428,581 -- Tri-dimensional
Puzzle. Filed: 16 Jun 1981; patented: 31 Jan 1984. Cover page + 3pp + 3pp diagrams. Three dimensional sliding cube puzzles with
central pieces joined together. A 3 x 3 x 3
version was made in Hungary and marketed as a Varikon Box. Inventor's address is in France and he cites
earlier French applications of 19 Jun 1980 and 19 Nov 1980. He also describes a 3 x 4 x 4 version with the central areas of each face
joined to a 1 x 2 x 2 block in the middle.
Here one has a set of solid pieces in
a tray and one tilts or rolls a piece into the blank space.
Thomas Henry Ward.
UK Patent 2,870 -- Apparatus for
Playing Puzzle or Educational Games.
Provisional: 8 Jun 1883;
Complete as: An Improved Apparatus to be Employed in Playing Puzzle or
Educational Games, 6 Dec 1883. 3pp + 1p
diagrams.
US Patent 287,352 -- Game
Apparatus. Applied: 13 Sep 1883; patented: 23 Oct 1883. 1p + 1p diagrams. Hexagonal board of 19 triangles with 18
tetrahedra to tilt.
George Mitchell &
George Springfield. UK Patent
6867 -- A novel puzzle, and improvements in the construction of apparatus
therefor. Applied: 16 Mar 1897; accepted: 5 Jun 1897. 2pp + 1p diagrams. Rolling cubes puzzle, where the cube faces are hollowed and fit
onto domes in the tray. Basic form has
four cubes in a row with two extra spaces above the middle cubes, but other
forms are shown.
Sven Bergling invented the
rolling ball labyrinth puzzle/game and they began to be produced in 1946. [Kenneth Wells; Wooden Puzzles and Games;
David & Charles, Newton Abbot, 1983, p. 114.]
Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London,
1963. Problem 3: Schwere Kiste, pp. 3-4
& 22-23 (= Heavy boxes, pp. 4-5
& 25-26). Three problems with 5
boxes some of which are so heavy that one has to tilt or roll them.
Gardner. SA (Dec 1963). = Sixth Book, chap. 8.
Gives Sprague's first problem.
Gardner. SA (Nov 1965). c= Carnival, chap. 9.
Prob. 1: The red-faced cube. Two
problems of John Harris involving one cube with one red face rolling on a
chessboard. Gardner says that the field
is new and that only Harris has made any investigations of the problem. The book chapter cites Harris's 1974 article,
below, and a 1971 board game called Relate with each player having four coloured
cubes on a 4 x 4 board.
Charles W. Trigg. Tetrahedron rolled onto a plane. JRM 3:2 (Apr 1970) 82-87. A tetrahedron rolled on the plane forms the
triangular lattice with each cell corresponding to a face of the
tetrahedron. He also considers rolling
on a mirror image tetrahedron and rolling octahedra.
John Harris. Single vacancy rolling cube problems. JRM 7:3 (1974) 220-224. This seems to be the first appearance of the
problem with one vacant space. He
considers cubes rolling on a chessboard.
Any even permutation of the pieces with the blank left in place is
easily obtained. From the simple
observation that each roll is an odd permutation of the pieces and an odd
rotation of the faces of a cube, he shows that the parity of the rotation of a
cube is the same as the parity of the number of spaces it has moved. He shows that any such rotation can be
achieved on a 2 x 3 board.
Rotating one cube 120o about a diagonal takes 32 moves. If the blank is allowed to move, the the
parity of the permutation of the pieces is the parity of the number of spaces
the blank moves, but each cube still has to have the parity of its rotation the
same as the parity of the number of spaces it has moved. If the identical pieces are treated as
indistinguishable, the parity of the permutation is only shown by the location
of the blank space. He suggests the use
of ridges on the board so that the cube will roll automatically -- this was
later used in commercial versions. He
gives a number of problems with different colourings of the cubes.
Gardner. SA (Mar 1975). = Time Travel, chap. 9.
Prob. 8: Rolling cubes. This is
the first of Harris's problems.
Computer analysis has found that it can be done in fewer moves than
Harris had. Gardner also reports on the
last of Harris's problems, which has also been resolved by computer.
A 3 x 3 array with 8
coloured cubes was available from Taiwan in the early 1980s. It was called Color Cube Mental Game -- I
called it 'Rolling Cubes'. The cubes
had thick faces, producing grooved edges which fit into ridges in the bottom of
the plastic frame, causing automatic rolling quite nicely. I wonder if this was inspired by Harris's
article.
John Ewing & Czes
Kośniowski. Puzzle it Out --
Cubes, Groups and Puzzles. CUP,
1982. The 8 Cubes Puzzle, pp.
58-59. Analysis of the Rolling Cubes
puzzle. The authors show how to rotate
a single cube about a diagonal in 36 moves.
Invented by Toshio Akanuma
(??SP). Manufactured by Tricks Co.,
Japan, in 1983. Described in Hordern,
pp. 144-145 & 220, E35, and in S&B, p. 135. This looks like a Tower of Hanoi (cf 7.M.2) with two differently
coloured piles of 10 pieces on the outside two tracks of three tracks of height
12 joined like a letter E. This is made
as a sliding block puzzle, but with blockages -- a piece cannot slide down a
track further than its original position.
Mark Manasse, Danny Sleator & Victor K. Wei. Some Results on the Panex Puzzle. Preprint sent by Jerry Slocum, 23pp, nd
[1983, but S&B gives 1985]. For
piles of size n, the minimum number of moves, T(n),
to move one pile to the centre track is determined by means of a 2nd
order, non-homogeneous recurrence which has different forms for odd and
even n. Compensating for this leads to a 2nd order non‑homogeneous
recurrence, giving T(10) =
4875 and T(n) ~ C(1 + Ö2)n. This solution doesn't ever move the other
pile. The minimum number of moves, X(n),
to exchange the piles is bounded above and below and determined exactly
for n £ 6 by computer search. X(5) = 343,
compared to bounds of 320 and
343. X(6) = 881, compared to the bounds of 796
and 881. For
n = 7, the bounds are 1944
and 2189, For
n = 10, the bounds are 27,564
and 31,537. The larger bounds are considered as probably
correct.
Christoph Hausammann. US Patent 5,261,668 -- Logic Game. Filed: 6 Aug 1992; patented: 16 Nov 1993. 1p
abstract + 2pp text + 3pp diagrams.
Essentially identical to Panex.
Vladimir Dubrovsky. Nesting Puzzles -- Part I: Moving oriental
towers. Quantum 6:3 (Jan/Feb 1996)
53-59 & 49-51. Says Panex was
produced by the Japanese Magic Company in the early 1980s. Discusses it and cites S&B for the
bounds given above. Sketches a number
of standard configurations and problems, leading to "Problem 9. Write out a complete solution to the Panex
puzzle." He says his method is
about 1700 moves longer than the upper bound given above.
Nick Baxter. Recent results for the Panex Puzzle. 4pp handout at G4G5, 2002. Describes the puzzle and its history. David Bagley wrote a program to implement
the Manasse, Sleator & Wei methods.
On 7 Feb 2002, this confirmed the conjecture that X(7) = 2189. On 26 Mar 2002, it obtained X(8) = 5359, compared to bounds of
4716 and 5359.
It is estimated that the cases
n = 9 and 10 will take 10
and 1200 years! If Moore's Law on the
increase of computing power continues for another 20 years, the latter answer
may be available by then. He gives a
simplified version of the algorithm for the upper bound, which gets 31,544
for n = 10. He has a Panex page: www.baxterweb.com/puzzles/panex/ and will be publishing an edited and
annotated version of the Manasse, Sleator & Wei paper on it.
See MUS I 1-13, Tropfke 658 and
also 5.N.
Wolf, goat and cabbages: Alcuin,
Abbot Albert, Columbia
Algorism, Munich 14684, Folkerts,
Chuquet, Pacioli, Tartaglia,
van Etten, Merry Riddles, Ozanam,
Dilworth, Wingate/Dodson, Jackson,
Endless Amusement II,
Boy's Own Book, Nuts to Crack, Taylor; The Riddler, Child,
Fireside Amusements,
Magician's Own Book,
Book of 500 Puzzles,
Boy's Own Conjuring Book, Secret Out (UK),
Mittenzwey, Carroll 1873, Kamp,
Carroll 1878, Berg, Lemon,
Hoffmann, Brandreth Puzzle
Book, Carroll 1899, King,
Voggenreiter, Stein, Stong,
Zaslavsky, Ascher, Weismantel (a film),
Verse version: Taylor,
Version with only one pair of
incompatibles: Voggenreiter
Extension to four items: Gori,
Phillips, M. Adams, Gibbs,
Ascher
Adults and children: Alcuin,
Kamp, Hoffmann, Parker?,
Voggenreiter, Gibbs
Three jealous husbands: Alcuin,
Abbot Albert, Columbia
Algorism, Munich 14684, Folkerts,
Rara, Chuquet, Pacioli,
Cardan, Tartaglia, H&S ‑ Trenchant, Gori,
Bachet, van Etten, Wingate/Kersey, Ozanam, Minguét, Dilworth,
Les Amusemens,
Wingate/Dodson, Jackson, Endless Amusement II, Nuts to Crack, Young Man's Book,
Family Friend,
Magician's Own Book,
The Sociable,
Book of 500 Puzzles,
Boy's Own Conjuring Book, Vinot, Secret Out
(UK), Lemon, Hoffmann, Fourrey, H. D. Northrop, Mr. X, Loyd, Williams,
Clark, Goodstein, O'Beirne,
Doubleday, Allen,
Verse mnemonic: Abbot
Albert, Munich 14684,
Verse solution:
Ozanam, Vinot,
Four or more jealous
husbands: Pacioli, Filicaia,
Tartaglia, Bachet, Delannoy,
Ball, Carroll-Collingwood, Dudeney,
O'Beirne
Jealous husbands, with island in
river: De Fontenay, Dudeney,
Ball, Loyd, Dudeney,
Pressman & Singmaster
Missionaries and cannibals: Jackson,
Mittenzwey, Cassell's, Lemon,
Pocock, Hoffmann, Brandreth Puzzle Book, H. D. Northrop, Schubert, Arbiter, H&S,
Abraham, Bile Beans,
Goodstein, Beyer, O'Beirne,
Pressman & Singmaster.
With only one cannibal who can row: Brandreth Puzzle Book,
Abraham, Beyer.
Bigger boats: Pacioli,
Filicaia?,
Bachet(-Labosne), Delannoy, Ball,
Dudeney, Abraham?, Goodstein,
Kaplan, O'Beirne,
Alcuin. 9C.
Prob.
17: Propositio de tribus fratribus singulas habentibus sorores. 3 couples, rather earthily expressed.
Prob.
18: Propositio de lupo et capra et fasciculo cauli. Wolf, goat, cabbages.
Prob.
19: Propositio de viro et muliere ponderantibus plaustrum. Man, wife and two small children.
Prob.
20: Propositio de ericiis. Rewording of
Prob. 19.
Ahrens. MUS II 315‑318, cites many sources,
mostly from folklore and riddle collections, with one from the 12C and several
from the 14C. ??NYS.
Abbot Albert. c1240.
Prob.
5, p. 333. Wolf, goat & cabbages.
Prob.
6, p. 334. 3 couples, with verse
mnemonic.
Columbia Algorism. c1350.
No.
122, pp. 130‑131 & 191: wolf, goat, bundle of greens. See also Cowley 402 & plate
opposite. P. 191 and the Cowley plate
are reproductions of the text with a crude but delightful illustration. P. 130 gives a small sketch of the
illustration. I have a colour slide
from the MS.
No.
124, p. 132: 3 couples. See also Cowley
403 & plate opposite. The plate
shows another crude but delightful illustration. I have a colour slide from the MS.
Munich 14684. 14C.
Prob.
XXVI, pp. 82‑83: 3 couples, with verse mnemonic.
Prob.
XXVII, p. 83: wolf, goat, cabbage.
Folkerts. Aufgabensammlungen. 13-15C.
11 sources with wolf, goat, cabbage.
12 sources with three jealous couples.
Rara, 459‑465, cites two
Florentine MSS of c1460 which include 'the jealous husbands'. ??NYS.
Chuquet. 1484.
Prob.
163: wolf, goat & cabbages. FHM 233
says that a 12C MS claims that every boy of five knows this problem.
Prob.
164: 3 couples. FHM 233.
Pacioli. De Viribus.
c1500.
Ff.
103v - 105v. LXI. C(apitolo). de .3.
mariti et .3. mogli gelosi (About 3 husbands and 3 wives). = Peirani 146-148. 3 couples. Says that
4 or 5 couples requires a 3
person boat.
F.
IIIv. = Peirani 6. The Index lists the above as Problem 66 and
lists a Problem 65: Del modo a salvare la capra el capriolo dal lupo al passar
de un fiume ch' non siano devorati (How to save the goat and the kid from the
wolf in crossing a river so they are not eaten).
Piero di Nicolao d'Antonio da
Filicaia. Libro dicto giuochi
mathematici. Early 16C -- ??NYS,
mentioned in Franci, op. cit. in 3.A.
Franci, p. 23, says Pacioli and Filicaia deal with the case of four or
five couples and that Pacioli considers bigger boats, but I'm not clear if
Filicaia also does so.
Cardan. Practica Arithmetice. 1539.
Chap. 66, section 73, f. FF.v.v (p. 157). (The 73 is not printed in the Opera Omnia). Three jealous husbands.
Tartaglia. General Trattato, 1556, art. 141‑143,
p. 257r‑ 257v.
Art.
141: wolf, goat and cabbages.
Art.
142: three couples.
Art.
143: four couples -- erroneously -- see Bachet.
H&S 51 says 3 couples occurs
in Trenchant (1566), ??NYS.
Gori. Libro di arimetricha.
1571.
Ff.
71r‑71v (p. 77). 3 couples.
F. 80v
(p. 77). Dog, wolf, sheep, horse to cross
river in boat which holds 2, but each cannot abide his neighbours in the
given list, so each cannot be alone with such a neighbour.
Bachet. Problemes.
1612. Addl. prob. IV: Trois
maris jaloux ..., 1612: 140-142; 1624: 212‑215; 1884: 148‑153. Three couples; four couples -- notes that Tartaglia is wrong by showing that one
can never get five persons on the far side.
Labosne gives a solution with a
3 person boat and does n
couples with an n‑1 person boat.
van Etten. 1624.
Prob.
14: Des trois maistres & trois valets, p. 14. 3 men and 3
valets. (The men hate the other
valets and will beat them if given a chance.)
(Not in English editions.)
Prob.
15: Du loup, de la chevre & du chou, pp. 14‑15. Wolf, goat & cabbages. (Not in English editions.)
Book of Merry Riddles. 1629
72 Riddle, pp. 43-44. "Over
a water I must passe, and I must carry a lamb, a woolfe, and a bottle of hay if
I carry any more than one at once my boat will sink." Tony Augarde; The Oxford Guide to Word
Games; OUP, 1984; p. 6 says wolf, goat, cabbage appears in the 1629 ed.
Wingate/Kersey. 1678?.
Prob. 6., p. 543. Three jealous
couples. Cf 1760 ed.
Ozanam. 1725.
Prob.
2, 1725: 3‑4. Prob. 18, 1778:
171; 1803: 171; 1814: 150.
Prob. 17, 1840: 77. Wolf, goat
and cabbage.
Prob.
3, 1725: 4‑5. Prob. 19, 1778:
171-172; 1803: 171-172; 1814: 150-151. Prob. 18, 1840: 77.
Jealous husbands. Latin verse
solution. He also discusses three
masters and valets: "none of the
the masters can endure the valets of the other two; so that if any one of them
were left with any of the other two valets, in the absence of his master, he
would infallibly cane him."
Minguet. 1733.
Pp. 158-159 (1755: 114-115; 1822: 175-176; 1864: 151). Three jealous couples.
Dilworth. Schoolmaster's Assistant. 1743.
Part IV: Questions: A short Collection of pleasant and diverting
Questions, p. 168.
Problem
6: Fox, goose and peck of corn. = D.
Adams; Scholar's Arithmetic; 1801, p. 200, no. 8.
Problem
7: Three jealous husbands. (Dilworth cites
Wingate for this -- but this is in Kersey's additions -- cf Wingate/Kersey,
1678? above.) = D. Adams; Scholar's
Arithmetic; 1801, p. 200, no. 9.
Les Amusemens. 1749.
Prob. 14, p. 136: Les Maris jaloux.
Solution is incorrect and has been corrected by hand in my copy.
Edmund Wingate (1596-1656). A Plain and Familiar Method for Attaining
the Knowledge and Practice of Common Arithmetic. .... 19th ed., previous
ed. by John Kersey (1616-1677) and George Shell(e)y, now by James Dodson. C. Hitch and L. Hawes, et al., 1760.
Art.
749. Prob. VI. P. 379.
Three jealous husbands. As in
1678? ed.
Art.
750. Prob. VII. P. 379.
Fox, goose and corn.
Jackson. Rational Amusement. 1821.
Arithmetical Puzzles.
No.
7, pp. 2 & 52. Fox, goose and corn. One solution.
No.
13, pp. 4 & 54. Three jealous
husbands.
No.
21, pp. 5 & 56. Three masters and
servants, where the servants will murder the masters if they outnumber them --
i.e. missionaries and cannibals. First
appearance of this type.
Endless Amusement II. 1826?
Prob.
17, pp. 198-199. Wolf, goat and
cabbage.
Prob.
25, pp. 201-202. Three jealous
husbands.
Boy's Own Book. The wolf, the goat and the cabbages. 1828: 418‑419; 1828-2: 423; 1829 (US): 214;
1855: 570; 1868: 670.
Nuts to Crack III (1834).
No.
209. Fox, goose and peck of corn.
No.
214. Three jealous husbands.
The Riddler. 1835.
The wolf, the goat and the cabbages, pp. 5-6. Identical to Boy's Own Book.
Young Man's Book. 1839.
Pp. 39-40. Three jealous
Husbands ..., identical to Wingate/Kersey.
Child. Girl's Own Book. 1842:
Enigma 49, pp. 237-238; 1876: Enigma
40, p. 200. Fox, goose and corn. Says it takes four trips instead of three --
but the solution has 7 crossings.
Walter Taylor. The Indian Juvenile Arithmetic, or Mental
Calculator; to which is added an appendix, containing arithmetical recreations
and amusements for leisure hours ....
For the author at the American Press, Bombay, 1849. [Quaritch catalogue 1224, Jun 1996, says
their copy has a note in French that Ramanujan learned arithmetic from this and
that it is not in BMC nor NUC. Graves
14.c.35.] P. 211, No. 8. Wolf, goat and cabbage in verse! No solution.
Upon
a river's brink I stand, it is both deep and wide;
With
a wolf, a goat, and cabbage, to take to the other side.
Tho'
only one each time can find, room in my little boat;
I
must not leave the goat and wolf, not the cabbage and the goat.
Lest
one should eat the other up, -- now how can it be done --
How
can I take them safe across without the loss of one?
Fireside Amusements. 1850: No. 24, pp. 111 & 181; 1890: No. 24, p. 100. Fox, goose and basket of corn.
Family Friend 3 (1850) 344 &
351. Enigmas, charades, etc. -- No. 17:
The three jealous husbands.
Magician's Own Book. 1857.
The
three jealous husbands, p. 251.
The
fox, goose, and corn, p. 253.
The Sociable. 1858.
Prob. 33: The three gentlemen and their servants, pp. 296 &
314-315. "None of the gentlemen shall
be left in company with any of the servants, except when his own servant is
present" -- so this is like the Jealous Husbands. = Book of 500 Puzzles, 1859, prob. 33, pp.
14 & 32-33. = Illustrated Boy's Own
Treasury, 1860, prob. 11, pp. 427-428 & 431.
Book of 500 Puzzles. 1859.
Prob.
33: The three gentlemen and their servants, pp. 14 & 32-33. As in The Sociable.
The
three jealous husbands, p. 65.
The
fox, goose and corn, p. 67.
Both identical to Magician's Own Book.
Boy's Own Conjuring Book. 1860.
The
three jealous husbands, pp. 222‑223.
The
fox, goose, and corn, pp. 225.
Both identical to Magician's Own Book.
Vinot. 1860. Art. XXXVII: Les
trois maris jaloux, pp. 56-57. Three
jealous husbands, with verse solution taken from Ozanam.
The Secret Out (UK). c1860.
A comical dilemma, p. 27.
Wolf, goat and cabbage. Varies
it as fox, goose and corn and then as gentlemen and servants, which is jealous
husbands, rather than the same problem.
Lewis Carroll. Letter of 15 Mar 1873 to Helen Feilden. Pp. 212-215 (Collins: 154-155). Fox, goose and bag of corn. "I rashly proposed to her to try the
puzzle (I daresay you know it) of "the fox, and goose, and bag of
corn."" Cf
Carroll-Collingwood, pp. 212-215 (Collins: 154-155); Carroll-Wakeling, prob. 28,
pp. 36-37 and Carroll-Gardner, p. 51. Cf Carroll, 1878.
Wakeling writes that this does not appear elsewhere in Carroll.
Bachet-Labosne. 1874.
For details, see Bachet, 1612.
Jens Kamp. Danske Folkeminder, Aeventyr, Folksagen,
Gaader, Rim og Folketro, Samlede fra Folkemende. R. Neilsen, Odense, 1877.
Marcia Ascher has kindly sent me a photocopy of the relevant material
with a translation by Viggo Andressen.
No.
18, pp. 326‑327: Fox, lamb and cabbage.
No.
19, p. 327: Husband, wife and two half‑size sons.
Lewis Carroll. Letter of 22 Jan 1878 to Jessie
Sinclair. Fox, goose and bag of
corn. Cf Carroll-Collingwood, pp.
205-207 (Collins: 150); Carroll-Wakeling, prob. 26: The fox, the goose and the
bag of corn, pp. 34 & 72. Cf
Carroll. 1872.
Mittenzwey. 1880.
Prob. 227-228, pp. 42 & 92;
1895?: 254-255, pp. 46 & 94;
1917: 254‑255, pp. 42 & 90. Bear, goat and cabbage, mentioning second
solution; three kings and three
servants, where the servants will rob the kings if they outnumber them, i.e.
like missionaries and cannibals.
Cassell's. 1881.
P. 105: The dishonest servants.
The servants are rogues who will murder masters if they outnumber them,
so this is equivalent to the missionaries and cannibals version.
Lucas. RM1. 1882. Pp. 1-18 is a general discussion of the
problem.
De Fontenay. Unknown source and date -- 1882?? Described in RM1, 1882, pp. 15‑18
(check 1st ed.??). n > 3 couples,
2 person boat, island in river,
can be done in 8n ‑ 8 passages.
Lucas says this was suggested at the Congrès de l'Association française
pour l'avancement des sciences at Montpellier in 1879, ??NYS. (De Fontenay is unclear -- sometimes he
permits bank to bank crossings, other times he only permits bank to island crossings. His argument really gives 8n - 6
if bank to bank crossings are prohibited. See Pressman & Singmaster, below, for clarification.)
Albert Ellery Berg, ed. Op. cit. in 4.B.1. 1883. P. 377: Fox, goose
& peck of corn.
Lemon. 1890.
Gentlemen
and their servants, no. 101, pp. 17‑18 & 101. This is the same as missionaries and
cannibals.
The
three jealous husbands, no. 151, pp. 24 & 103 (= Sphinx, no. 478, pp. 66
& 114.) The solution mentions
Alcuin.
Crossing
the river, no. 450, pp. 59 & 114.
English travellers and native servants
= missionaries and cannibals.
Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 136, no. 14. Fox,
goose and corn. No solution.
Herbert Llewelyn Pocock. UK Patent 15,358 -- Improvements in Toy
Puzzles. Applied: 29 Sep 1890; complete specification: 29 Jun 1891; accepted: 22 Aug 1891. 2pp + 1p diagrams. Three whites and three blacks and the blacks must never outnumber
the whites, i.e. same as missionaries and cannibals. He describes the puzzle as "well known".
Delannoy. Described in RM1, 1891, Note 1: Sur le jeu des traversées, pp. 221‑222. ??check 1882 ed. Shows n couples can cross in an x
person boat in N trips, for
n, x, N = 2, 2, 5; 3, 2, 11;
4, 3, 9; 5, 3, 11; n > 5, 4, 2n ‑ 3. (He has
2n ‑ 1 by mistake. Simple modification shows we also have 5, 4, 7;
6, 5, 9; 7, 6, 5; 8, 7, 7;
n > 8, n ‑ 1, 5.)
Ball. MRE, 1st ed., 1892, pp. 45‑47, says Lucas posed the problem
of minimizing x for a given
n and quotes the Delannoy
solution (with erroneous 2n ‑
1) and also gives De Fontenay's version
and solution. (He spells it De Fonteney
as does his French translator, though Ahrens gives De Fontenay and the famous
abbey in Burgundy is Fontenay -- ??)
The Ballybunnion and Listowel
Railway in County Kerry, Ireland, was a late 19C railway using the Lartigue
monorail system. This had a single
rail, about three feet off the ground, with a carriage hanging over both sides
of the rail. The principle job of the
conductor/guard to make sure the passengers and goods were equally distributed
on both sides. Kerry legend asserts
that a piano had to be sent on this railway and there were not enough
passengers or goods to balance it. So a
cow was sent on the other side. At the
far end, the piano was unloaded and replaced with two large calves and the
carriage sent back. The cow was then
unloaded and one calf moved to the other side, so the carriage could be sent
back to the far end and everyone was happy.
Hoffmann. 1893.
Chap. IV, pp. 157‑158 & 211‑213 = Hoffmann-Hordern, pp.
136-138, with photos.
No.
56: The three travellers. Masters and
servants, equivalent to missionaries and cannibals. Solution says Jaques & Son make a puzzle version with six
figures, three white and three black.
Photos in Hoffmann-Hordern, pp. 136 & 137 -- the latter shows Caught
in the Rain, 1880-1905, where Preacher, Deacon, Janitor and their wives have to
get somewhere using one umbrella.
No.
57: The wolf, the goat, and the cabbages.
Photo on p. 136 of La Chevre et le Chou. with box, by Watilliaux,
1874-1895. Hordern Collection, p. 72,
and S&B, p. 134, show the same puzzle.)
No.
58: The three jealous husbands.
No.
59: The captain and his company. This
is Alcuin's prop. 19 with many adults.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co.,
NY), nd [1895].
P. 7:
The wolf, the goat and the cabbages.
Identical to Hoffmann No. 57, with nice colour picture. No solution.
P. 9:
The missionaries' and cannibals' puzzle.
Usual form, with nice colour picture, but only one cannibal can
row. No solution. This seems to be the first to use the
context of missionaries and cannibals and the first to restrict the number of
rowers.
Lucas. L'Arithmétique Amusante.
1895. Les vilains maris jaloux,
pp. 125-144 & Note II, pp. 198-202.
Prob.
XXXVI: La traversée des trois ménages, pp. 125-130. 3 couples. Gives Bachet's 1624 reasoning for the
essentially unique solution -- but attributes it to 1613.
Prob.
XXXVII: La traversée des quatre ménages, pp. 130-132. 4 couples in a 3
person boat done in 9 crossings.
L'erreur
de Tartaglia, pp. 133-134. Discusses
Tartaglia's error and Bachet's notice of it and gives an easy proof that 4
couples cannot be done with a
2 person boat.
Prob.
XXXVIII: La station dans une île, pp. 135-140.
4 couples, 2
person boat, with an island.
Gives De Fontenay's solution in 24 crossings.
Prob.
XXXIX: La traversée des cinq ménages, pp. 141-143. 5 couples, 3
person boat in 11 crossings.
Énoncé
général du problème des traversées, pp. 143-144. n couples, x
person boat, can be done in N crossings as given by Delannoy above. He corrects
2n - 1 to 2n - 3
here.
Note
II: Sur les traversées, pp.
198-202. Gives Tarry's version with an
island and with n men having harems of size m, where the women are obviously unable to row. He gives solutions in various cases. For the ordinary case, i.e. m = 1, he finds a solution for
4 couples in 21
moves, using the basic ferrying technique that Pressman and Singmaster
found to be optimal, but the beginning and end take longer because the women
cannot row. He says this gives a
solution for n couples in
4n + 5 crossings. He then considers the case of n - 1
couples and a ménage with m wives and finds a solution in 8n + 2m + 7
crossings. I now see that this
solution has the same defects as those in Pressman & Singmaster, qv.
Ball. MRE, 3rd ed., 1896, pp. 61‑64, repeats 1st ed., but adds
that Tarry has suggested the problem for harems -- see above.
Dudeney. Problem 68: Two rural puzzles. Tit‑Bits 33 (5 Feb &
5 Mar 1898) 355 & 432.
Three men with sacks of treasure and a boat that will hold just two men
or a man and a sack, with additional restrictions on who can be trusted with
how much. Solution in 13
crossings.
Carroll-Collingwood. 1899.
P. 317 (Collins: 231 or 232 (missing in my copy)) Cf Carroll-Wakeling II, prob. 10:
Crossing the river, pp. 17 & 66.
Four couples -- only posed, no solution. Wakeling gives a solution, but this is incorrect. After one wife is taken across, he has
another couple coming across and from Bachet onward, this is considered
improper as the man could get out of the boat and attack the first, undefended,
wife.
E. Fourrey. Op. cit. in 4.A.1, 1899. Section 211: Les trois maîtres et les trois valets. Says a master cannot leave his valet with
the other masters for fear that they will intimidate him into revealing the
master's secrets. Hence this is the
same as the jealous couples.
H. D. Northrop. Popular Pastimes. 1901.
No. 5:
The three gentlemen and their servants, pp. 67 & 72. = The Sociable.
No.
12: The dishonest servants, pp. 68 & 73.
"... the servants on either side of the river should not outnumber
the masters", so this is the same as missionaries and cannibals.
Mr. X [cf 4.A.1]. His Pages.
The Royal Magazine 10:2 (Jun 1903) 140-141. A matrimonial difficulty.
Three couples. No answer given.
Dudeney. Problem 523. Weekly Dispatch (15
& 29 Nov 1903), both p. 10, (= AM, prob. 375, pp. 113 & 236‑237). 5
couples in a 3 person boat.
Johannes Bolte. Der Mann mit der Ziege, dem Wolf und dem
Kohle. Zeitschrift des Vereins für
Volkskunde 13 (1903) 95-96 & 311.
The first part is unaware of Alcuin and Albert. He gives a 12C Latin
solution: It capra, fertur olus, redit
hec, lupus it, capra transit [from Wattenbach; Neuen Archiv für ältere deutsche
Geschichtskunde 2 (1877) 402, from Vorauer MS 111, ??NYS] and a 14C solution: O natat, L sequitur, redit O, C navigat
ultra, / Nauta recurrit ad O, bisque natavit ovis (= ovis, lupus, ovis, caulis, ovis) [from Mone; Anzeiger für
Kunde der deutschen Vorzeit 45 (No. 105) (1838), from Reims MS 743,
??NYS]. Cites Kamp and several other
versions, some using a fox, a sheep, or a lamb. The addendum cites and quotes Alcuin and Albert as well as
relatively recent French and Italian versions.
H. Parker. Ancient Ceylon. Op. cit. in 4.B.1.
1909. Crossing the river, p.
623.
A
King, a Queen, a washerman and a washerwoman have to cross a river in a boat
that holds two. However the King and
Queen cannot be left on a bank with the low caste persons, though they can be
rowed by the washerperson of the same sex.
Solution in 7 crossings.
Ferry-man
must transport three leopards and three goats in a boat which holds himself and
two others. If leopards ever outnumber
goats, then the goats get eaten. So
this is like missionaries and cannibals, but with a ferry-man. Solution in 9 crossings.
H. Schubert. Mathematische Mussestunde. Vol. 2, 3rd ed., Göschen, Leipzig,
1909. Pp. 160‑162: Der drei
Herren und der drei Sklaven. (Same as
missionaries and cannibals.)
Arbiter Co. (Philadelphia). 1910.
Capital and Labor Puzzle. Shown
in S&B, p. 134. Equivalent to
missionaries and cannibals.
Ball. MRE, 5th ed., 1911, pp. 71-73, repeats 3rd ed., but omits the
details of De Fonteney's solution in
8(n-1) crossings.
Loyd. Cyclopedia, 1914.
Summer
tourists, pp. 207 & 366. 3 couples,
2 person boat, with additional
complications -- the women cannot row and there have been some arguments. Solution in
17 crossings.
The
four elopements, pp. 266 & 375.
4 couples, 2
person boat, with an island and the stronger constraint that no man is
to get into the boat alone if there is a girl alone on either the island or the
other shore. "The [problem]
presents so many complications that the best or shortest answer seems to have
been overlooked by mathematicians and writers on the subject." "Contrary to published answers, ... the
feat can be performed in 17 trips, instead of 24."
Ball. MRE, 6th ed., 1914, pp. 71-73, repeats 5th ed., but adds
that 6n ‑ 7 trips suffices for n couples with an island,
though he gives no reference.
Williams. Home Entertainments. 1914.
Alcuin's riddle, pp. 125-126.
"This will be recognized as perhaps the most ancient British riddle
in existence, though there are several others conceived on the same
lines." Three jealous
couples.
Clark. Mental Nuts. 1916, no.
67. The men and their wives. "... no man shall be left alone with
another's wife."
Dudeney. AM.
1917. Prob. 376: The four
elopements, pp. 113 & 237. 4 couples,
2 person boat, with island, can
be done in 17 trips and that this cannot be improved. This is the same solution as given by Loyd. (See Pressman and Singmaster, below.)
Ball. MRE, 8th ed., 1919, pp. 71-73 repeats 6th ed. and adds a citation
to Dudeney's AM prob. 376 for the solution in
6n ‑ 7 trips for n
couples.
Hummerston. Fun, Mirth & Mystery. 1924.
Crossing the river puzzles, Puzzle no, 52, pp. 128 & 180. 'Puzzles of this type ... interested people
who lived more than a thousand years ago'.
No. 1:
The eight travellers. Six men and two
boys who weigh half as much.
No. 2:
White and black. = Missionaries and
cannibals.
No. 3:
The fox, the goose, and the corn.
No. 4:
the jealous husbands.
H&S, 1927, p. 51 says
missionaries and cannibals is 'a modern variant'.
King. Best 100. 1927. No. 10, pp. 10 & 40. Dog, goose and corn.
Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930.
P.
106: Der Wolf, die Ziege und der Kohlkopf.
Usual wolf, goat, cabbage.
Pp.
106-107: Die 100 Pfund-Familie. Parents
weigh 100 pounds; the two children weigh 100 pounds together.
P.
107: Der Landjäger and die Strolche [The policeman and the vagabonds]. Two of the vagabonds hate each other so much
that they cannot be left together. As
far as I recall, this formulation is novel and I was surprised to realise that
it is essentially equivalent to the wolf, goat and cabbage version.
Phillips. Week‑End. 1932. Time tests of
intelligence, no. 41, pp. 22 & 194.
Rowing explorer with 4 natives:
A, B, C, D, who cannot abide
their neighbours in this list. A can row.
They get across in seven trips.
Abraham. 1933.
Prob. 54 -- The missionaries at the ferry, pp. 18 & 54 (14 &
115). 3 missionaries and 3 cannibals.
Doesn't specify boat size, but says 'only one cannibal can row'. 1933 solution says 'eight double journeys',
1964 says 'seven crossings'. This seems
to assume the boat holds 3. (For a
2 man boat, it takes 11
crossings with one missionary and two cannibals who can row or 13
crossings with one missionary and one cannibal who can row.)
The Bile Beans Puzzle Book. 1933.
No. 34: Missionaries & cannibals.
Three of each but only one of each can row. Done in 13 crossings.
Phillips. Brush.
1936. Prob. L.2: Crossing the
Limpopo, pp. 39‑40 & 98. Same
as in Week‑End, 1932.
M. Adams. Puzzle Book. 1939. Prob. C.63: Going
to the dance, pp. 139 & 178. Same
as Week‑End, 1932, phrased as travelling to a dance on a motorcycle which
carries one passenger.
R. L. Goodstein. Note 1778:
Ferry puzzle. MG 28 (No. 282)
(1944) 202‑204. Gives a graphical
way of representing such problems and considers m soldiers and m
cannibals with an n person boat, 3 jealous husbands and
how many rowers are required.
David Stein. Party and Indoor Games. P. M. Productions, London, nd [c1950?]. P. 98, prob. 5: Man with cat, parrot
and bag of seeds.
C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960.
A
puzzle-solving machine, pp. 377-384.
Describes how Paul Bezold made a logic machine from relays to solve the
fox, goose, corn problem.
How to
design a "Pircuit" or Puzzle circuit, pp. 388-394. On pp. 391-394, Harry Rudloe describes relay
circuits for solving the three jealous couples problem, which he attributes to
Tartaglia, and the missionaries and cannibals problem.
Nathan Altshiller Court. Mathematics in Fun and in Earnest. Mentor (New American Library), NY,
1961. [John Fauvel sent some pages from
a different printing which has much different page numbers than my copy.] "River crossing" problems,
pp. 168‑171. Discusses
various forms of the problem and adds a problem with two parents weighing 160,
two children weighing 80 and a dog weighing 12, with a boat
holding 160.
E. A. Beyer, proposer; editorial solution. River‑crossing dilemma. RMM 4 (Aug 1961) 46 &
5 (Oct 1961) 59. Explorers and
natives (= missionaries and cannibals), with all the explorers and one native
who can row. Solves in 13
crossings, but doesn't note that only one rowing explorer is
needed. (See note at Abraham, 1933,
above.)
Philip Kaplan. Posers.
(Harper & Row, 1963);
Macfadden Books, 1964. Prob. 36,
pp. 41 & 91. 5 men and a
3 person boat on one side, 5
women on the other side. One man
and one woman can row. Men are not
allowed to outnumber women on either side nor in the boat. Exchange the men and the women in 7
crossings.
T. H. O'Beirne. Puzzles and Paradoxes, 1965, op. cit. in
4.A.4, chap. 1, One more river to cross, pp. 1‑19. Shows
2n ‑ 1 couples (or 2n ‑ 1 each of missionaries and cannibals ?) can cross in a n
person boat in 11 trips.
2n ‑ 2 can cross in 9
trips. He also considers
variants on Gori's second version.
Doubleday - 2. 1971.
Family outing, pp. 49-50. Three
couples, but one man has quarrelled with the other men and his wife has
quarrelled with the other women, so this man and wife cannot go in the boat nor
be left on a bank with others of their sex.
Further men cannot be outnumbered by women on either bank. Gives a solution in 9
crossings, but I find the conditions unworkable -- e.g. the initial position
is prohibited!
Claudia Zaslavsky. Africa Counts. Prindle, Weber & Schmidt, Boston, 1973. Pp. 109‑110 says that leopard, goat
and pile of cassava leaves is popular with the Kpelle children of Liberia. However, Ascher's Ethnomathematics (see
below), p. 120, notes that this is based on an ambiguous description and that
an earlier report of a Kpelle version has the form described below.
Ball. MRE, 12th ed., 1974, p. 119, corrects Delannoy's 2n ‑ 1 to 2n ‑ 3 and corrects De Fontenay's 8n ‑ 8 to 8n ‑ 6, but still gives the solution for n = 4
with 24 crossings.
W. Gibbs. Pebble Puzzles -- A Source Book of Simple
Puzzles and Problems. Curriculum
Development Unit, Solomon Islands, 1982.
??NYS, o/o??. Excerpted in: Norman K. Lowe, ed.; Games and Toys in the
Teaching of Science and Technology; Science and Technology Education, Document
Series No. 29, UNESCO, Paris, 1988, pp. 54‑57. On pp. 56‑57 is a series of river crossing problems. E.g. get people of weights 1, 2, 3
across with a boat that holds a weight of at most 3.
Also people numbered 1, 2, 3, 4,
5 such that no two consecutive people
can be in the boat or left together.
In about 1986, James Dalgety
designed interactive puzzles for Techniquest in Cardiff. Their version has a Welshman with a dragon,
a sheep and a leek!
Ian Pressman & David
Singmaster. Solutions of two river
crossing problems: The jealous husbands and the missionaries and the
cannibals. Extended Preprint, April
1988, 14pp. MG 73 (No. 464) (Jun 1989)
73‑81. (The preprint contains
historical and other detail omitted from the article as well as some further
information.) Observes that De Fontenay
seems to be excluding bank to bank crossings and that Lucas' presentation is
cryptic. Shows that De Fontenay's
method should be 8n ‑ 6 crossings for n > 3 and that this is
minimal. If bank to bank crossings are
permitted, as by Loyd and Dudeney, a computer search revealed a solution
with 16 crossings for n = 4, using an ingenious move that Dudeney could
well have ignored. For n > 4,
there is a simple solution in 4n
+ 1 crossings, and these numbers are
minimal. [When this was written, I had
forgotten that Loyd had done the problem for
4 couples in 17
moves, which changes the history somewhat. However, I now see that Loyd was copying from Dudeney's Weekly
Dispatch problem 270 of 23 Apr 1899 & 11 Jun 1899. Loyd states what appears to be a stronger
constraint but all the methods in our article do obey the stronger
constraint. However, one could make the
constraints stronger -- e.g. our solutions have a husband taking the boat from
bank to bank while his wife and another wife are on the island -- the solution
of Loyd & Dudeney avoids this and may be minimal in this case --??.]
For
the missionaries and cannibals problem, the
16 crossing solution reduces
to 15
and gives a general solution in
4n ‑ 1 crossings, which is
shown to be minimal. If bank to bank
crossings are not permitted, then De Fontenay's amended 8n ‑ 6 solution is still optimal.
Marcia Ascher. A river‑crossing problem in cultural
perspective. MM 63 (1990) 26‑28. Describes many appearances in folklore of
many cultures. Discusses African
variants of the wolf, goat and cabbage problem in which the man can take two of
the items in the boat. This is much easier,
requiring only three crossings, but some versions say that the man cannot
control the items in the boat, so he cannot have the wolf and goat or the goat
and cabbage in the boat with him. This
still only takes three crossings.
Various forms of these problems are mentioned: fox, fowl and corn;
tiger, sheep and reeds; jackal,
goat and hay; caged cheetah, fowl and
rice; leopard, goat and leaves -- see
below for more details.
She
also discusses an Ila (Zambia) version with leopard, goat, rat and corn which
is unsolvable!
Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10.
1991. Section 4.8, pp.
109-116 & Note 8, pp. 119-121. Good
survey of the problem and numerous references to the folklore and ethnographic
literature. Amplifies the above
article. A version like the Wolf, goat
and cabbage is found in the Cape Verde Islands, in Cameroon and in
Ethiopia. The African version is found
as far apart as Algeria and Zanzibar, but with some variations. An Algerian version with jackal, goat and hay
allows one to carry any two in the boat, but an inefficient solution is presented
first. A Kpelle (Liberia) version with
cheetah, fowl and rice adds that the man cannot keep control while rowing so he
cannot take the fowl with either the cheetah or the rice in the boat. A Zanzibar version with leopard, goat and
leaves adds instead that no two items can be left on either bank together. (A similar version occurs among
African-Americans on the Sea Islands of South Carolina.) Ascher notes that Zaslavsky's description is
based on an ambiguous report of the Kpelle version and probably should be like
the Algerian or Kpelle version just described.
Liz Allen. Brain Sharpeners. New English Library (Hodder & Stoughton), London, 1991. Crossing the river, pp. 62 & 125. Three mothers and three sons. The sons are unwilling to be left with
strange mothers, so this is a rephrasing of the jealous husbands.
Yuri B. Chernyak & Robert S.
Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 1, probs. 4-6: The knights and the
pages; More knights and pages; Yet more knights and pages: no man is an island,
pp. 4-5 & 100-102. Equivalent to
the jealous couples. Prob. 4 is three
couples, solved in 11 crossings. Prob.
5 is four couples -- "There is no solution unless one of the four pages is
sacrificed. (In medieval times, this
was not a problem.)" Prob. 6 is
four couples with an island in the river, solved in general by moving all pages
to the island, then having the pages go back and accompany his knight to other
side, then return to the island. After
the last knight is moved, the pages then move from the island to the other
side. This takes 7n - 6
steps in general. It satisfies the
jealousy conditions used by Pressman & Singmaster, but not those of Loyd
& Dudeney.
John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for all
Ages. Keystone Agencies, Radnor, Ohio,
1997. P. 16: The missionaries and the
pirates. Politically correct rephrasing
of the missionaries and the cannibals version.
All the missionaries, but only one pirate, can row. Solves in 13 crossings.
Prof. Dr. Robert Weismantel,
Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, PSF 3120,
D-39016 Magdeburg, Germany; tel: 0391/67-18745; email:
weismantel@imo.math.uni-magdeburg.de; has produced a 45 min. film: "Der
Wolf, die Ziege und Kohlköpfe
Transportprobleme von Karl dem Grossen bis heute", suitable for the
final years of school.
5.B.1. LOWERING FROM TOWER PROBLEM
The problem is for a collection of
people (and objects or animals) to lower themselves from a window using a rope
over a pulley, with baskets at each end.
The complication is that the baskets cannot contain very different
weights, i.e. there is a maximum difference in the weights, otherwise they go
too fast. This is often attributed to
Carroll.
Carroll-Collingwood. 1899.
P. 318 (Collins: 232-233 (232 is lacking in my copy)). = Carroll-Wakeling II, prob. 4: The
captive queen, pp. 8 & 65-66.
3 people of weights 195, 165, 90 and a weight of 75, with difference at most 15.
He also gives a more complex form.
No solutions. Although the text
clearly says 165, the prevalence of the
exact same problem with 165 replaced by
105 makes me wonder if this was
a misprint?? Wakeling says there is no
explicit evidence that Carroll invented this, and neither book assigns a date,
but Carroll seems a more original source than the following and he was more
active before 1890 than after.
An
addition is given in both books: add
three animals, weighing 60, 45,
30.
Lemon. 1890. The prisoners in
the tower, no. 497, pp. 65 & 116.
c= Sphinx, The escape, no. 113, pp. 19 & 100‑101. Three people of weights 195, 105, 90 with a weight of 75. The difference in weights cannot be more
than 15.
Hoffmann. 1893.
Chap. IV, no. 28: The captives in the tower, pp. 150 & 196
= Hoffmann‑Hordern, p. 123.
Same as Lemon.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co.,
NY), nd [1895]. P. 3: The captives in
the tower. Same as Lemon. Identical to Hoffmann. With colour picture. No solution.
Loyd. The fire escape puzzle.
Cyclopedia, 1914, pp. 71 & 348.
c= MPSL2, prob. 140, pp. 98‑99 & 165. = SLAHP: Saving the family, pp. 59 &
108. Simplified form of Carroll's
problem. Man, wife, baby & dog,
weighing a total of 390.
Williams. Home Entertainments. 1914.
The escaping prisoners, pp. 126-127.
Same as Lemon.
Rudin. 1936. No. 92, pp. 31-32
& 94. Same as Lemon.
Haldeman-Julius. 1937.
No. 150: Fairy tale, pp. 17 & 28.
Same as Lemon, except the largest weight is printed as 196, possibly an
error.
Kinnaird. Op. cit. in 1 -- Loyd. 1946.
Pp. 388‑389 & 394.
Same as Lemon.
Simon Dresner. Science World Book of Brain Teasers. Scholastic Book Services, NY, 1962. Prob. 61: Escape from the tower, pp. 29
& 99‑100. Same as Lemon.
Robert Harbin [pseud. of Ned
Williams]. Party Lines. Oldbourne, London, 1963. Escape, p. 29. As in Lemon.
Howard P. Dinesman. Superior Mathematical Puzzles. Allen & Unwin, London, 1968. No. 60: The tower escape, pp. 78 &
118. Same as Carroll. Answer in
15 stages. He cites Carroll, noting that Carroll did
not give a solution and he asks if a shorter solution can be found.
F. Geoffrey Hartswick. In:
H. O. Ripley & F. G. Hartswick; Detectograms and Other Puzzles;
Scholastic Book Services, NY, 1969. No.
15: Stolen treasure puzzle, pp. 54‑55 & 87. Same as Lemon.
5.B.2. CROSSING A BRIDGE WITH A TORCH
New
section.
Four
people have to get across a bridge which is dark and needs to be lit with the
torch. The torch can serve for at most
two people and the gap is too wide to throw the torch across, so the torch has
to be carried back and forth. The
various people are of different ages and require 5, 10, 20, 25 minutes to
cross and when two cross, they have to go at the speed of the slower. But the torch (= flashlight) battery will
only last an hour. Can it be done? I heard this about 1997, when it was claimed
to be used by Microsoft in interviewing candidates. I never found any history of it, until I recently found a
discussion on Torsten Sillke's site: Crossing the bridge in an hour
(www.mathematik.uni‑bielefeld.de/~sillke/PUZZLES/crossing-bridge),
starting in Jun 1997 and last updated in Sep 2001. This cites the 1981 source and the other references below. Denote the problem with speeds a, b, c, d
and total time t by
(a,
b, c, d; t), etc. t is
sometimes given, sometimes not.
Saul X. Levmore &
Elizabeth Early Cook. Super
Strategies for Puzzles and Games.
Doubleday, 1981, p. 3 -- ??NYS.
(5, 10, 20, 25; 60), as in the
introduction to this section..
Heinrich Hemme. Das Problem des Zwölf-Elfs. Vandenhoeck & Ruprecht, 1998. Prob. 81: Die Flucht, pp. 40 & 105-106,
citing a web posting by Gunther Lientschnig on 4 Dec 1996. (2, 4, 8, 10; t).
Dick Hess. Puzzles from Around the World. Apr 1997.
Prob. 107: The Bridge.
(1,
2, 5, 10; 17). Poses versions with more
people: (1, 3, 4, 6, 8, 9; 31) and, with a three-person bridge, (1, 2, 6, 7, 8, 19, 10; 25).
Quantum (May/Jun 1997) 13. Brainteaser B 205: Family planning. Problem (1, 3, 8, 10; 20).
Karen Lingel. Email of 17 Sep 1997 to rec.puzzles. Careful analysis, showing that the 'trick' solution is better
than the 'direct' solution if and only if
a + c > 2b.
[Indeed, a + c - 2b is the time saved by the 'trick' solution.] She cites
(2, 3, 5, 8; 19) and (2, 2, 3, 3; 11) to Sillke and (1, 3, 6,
8, 12; 30), from an undated
website. Expressing the solution for
more people seems to remain an open question.
5.C. FALSE COINS WITH A BALANCE
See
5.D.3 for use of a weighing scale.
There
are several related forms of this problem.
Almost all of the items below deal with 12 coins with one false, either
heavy or light, and its generalizations, but some other forms occur, including
the following.
8 coins, £1 light: Schell, Dresner
26
coins, £1 light: Schell
8 coins,
1 light: Bath (1959)
9 coins,
1 light: Karapetoff, Meyer
(1946), Meyer (1948), M. Adams, Rice
I have been sent an article by Jack Sieburg; Problem
Solving by Computer Logic; Data Processing Magazine, but the date is cut off --
??
E. D. Schell, proposer; M. Dernham, solver. Problem E651 -- Weighed and found
wanting. AMM 52:1 (Jan 1945) 42 &
7 (Aug/Sep 1945) 397. 8 coins,
at most one light -- determine the light one in two weighings.
Benjamin L. Schwartz. Letter:
Truth about false coins. MM 51
(1978) 254. States that Schell told
Michael Goldberg in 1945 that he had originated the problem.
Emil D. Schell. Letter of 17 Jul 1978 to Paul J.
Campbell. Says he did NOT originate the
problem, nor did he submit the version published. He first heard of it from Walter W. Jacobs about
Thanksgiving 1944 in the form of finding at most one light coin among 26 good
coins in three weighings. He submitted
this to the AMM, with a note disclaiming originality. The AMM problem editor published the simpler version described
above, under Schell's name. Schell says
he has heard Eilenberg describe the puzzle as being earlier than Sep 1939. Campbell wrote Eilenberg, but had no
response.
Schell's
letter is making it appear that the problem derives from the use of 1, 3, 9, ... as weights.
This usage leads one to discover that a light coin can be found in 3n coins using n weighings.
This is the problem mentioned by Karapetoff. If there is at most one light coin, then n
weighings will determine it among
3n ‑ 1 coins,
which is the form described by Schell.
The problem seems to have been almost immediately converted into the
case with one false coin, either heavy or light.
Walter W. Jacobs. Letter of 15 Aug 1978 to Paul J.
Campbell. Says he heard of the problem
in 1943 (not 1944) and will try to contact the two people who might have told
it to him. However, Campbell has had no
further word.
V. Karapetoff. The nine coin problem and the mathematics of
sorting. SM 11 (1945) 186‑187. Discusses 9 coins, one light, and asks for a
mathematical approach to the general problem.
(?? -- Cites AMM 52, p. 314, but I cannot find anything relevant in the
whole volume, except the Schell problem.
Try again??)
Dwight A. Stewart,
proposer; D. B. Parkinson & Lester
H. Green, solvers. The counterfeit
coin. In: L. A. Graham, ed.; Ingenious Mathematical Problems and Methods;
Dover, 1959; pp. 37‑38 & 196‑198. 12 coins. First appeared
in Oct 1945. Original only asks for the
counterfeit, but second solver shows how to tell if it is heavy or light.
R. L. Goodstein. Note 1845:
Find the penny. MG 29 (No. 287)
(Dec 1945) 227‑229. Non‑optimal
solution of general problem.
Editorial Note. Note 1930:
Addenda to Note 1845. Ibid. 30
(No. 291) (Oct 1946) 231. Comments on how to extend to optimal solution.
Howard D. Grossman. The twelve‑coin problem. SM 11:3/4 (Sep/Dec 1945) 360‑361. Finds counterfeit and extends to 36 coins.
Lothrop Withington, Jr. Another solution of the 12‑coin
problem. Ibid., 361‑362. Finds also whether heavy or light.
Donald Eves, proposer; E. D. Schell & Joseph Rosenbaum,
solvers. Problem E712 -- The extended
coin problem. AMM 53:3 (Mar 1946) 156 &
54:1 (Jan 1947) 46‑48.
12 coins.
Jerome S. Meyer. Puzzle Paradise. Crown, NY, 1946. Prob.
132: The nine pearls, pp. 94 & 132.
Nine pearls, one light, in two weighings.
N. J. Fine, proposer &
solver. Problem 4203 -- The generalized
coin problem. AMM 53:5 (May 1946)
278 & 54:8 (Oct 1947) 489‑491.
General problem.
H. D. Grossman. Generalization of the twelve‑coin
problem. SM 12 (1946) 291‑292. Discusses Goodstein's results.
F. J. Dyson. Note 1931:
The Problem of the Pennies. MG
30 (No. 291) (Oct 1946) 231‑234.
General solution.
C. A. B. Smith. The Counterfeit Coin Problem. MG 31 (No. 293) (Feb 1947) 31‑39.
C. W. Raine. Another approach to the twelve‑coin
problem. SM 14 (1948) 66‑67. 12 coins only.
K. Itkin. A generalization of the twelve‑coin
problem. SM 14 (1948) 67‑68. General solution.
Howard D. Grossman. Ternary epitaph on coin problems. SM 14 (1948) 69‑71. Ternary solution of Dyson & Smith.
Jerome S. Meyer. Fun-to-do.
A Book of Home Entertainment.
Dutton, NY, 1948. Prob. 40: Nine
pearls, pp. 41 & 188. Nine pearls,
one light, in two weighings.
Blanche Descartes [pseud. of
Cedric A. B. Smith]. The twelve coin
problem. Eureka 13 (Oct 1950) 7 &
20. Proposal and solution in verse.
J. S. Robertson. Those twelve coins again. SM 16 (1950) 111‑115. Article indicates there will be a
continuation, but Schaaf I 32 doesn't cite it and I haven't found it yet.
E. V. Newberry. Note 2342:
The penny problem. MG 37 (No.
320) (May 1953) 130. Says he has made a
rug showing the 120 coins problems and makes comments similar to Littlewood's,
below.
J. E. Littlewood. A Mathematician's Miscellany. Methuen, London, 1953; reprinted with minor corrections, 1957
(& 1960). [All the material cited
is also in the later version:
Littlewood's Miscellany, ed. by B. Bollobás, CUP, 1986, but on different
pages. Since the 1953 ed. is scarce, I
will also cite the 1986 pages in ( ).] Pp. 9 & 135 (31 & 114).
"It was said that the 'weighing‑pennies' problem wasted
10,000 scientist‑hours of war‑work, and that there was a proposal
to drop it over Germany."
John Paul Adams. We Dare You to Solve This! Berkley Publishing, NY, nd [1957?]. [This is apparently a collection of problems
used in newspapers. The copyright is
given as 1955, 1956, 1957.] Prob. 18:
Weighty problem, pp. 13 & 46. 9
equal diamonds but one is light, to be found in 2 weighings.
Hubert Phillips. Something to Think About. Revised ed., Max Parrish, London, 1958. Foreword, p. 6 & prob. 115: Twelve
coins, pp. 81 & 127‑128. Foreword says prob. 115 has been added to this edition and
"was in oral circulation during the war.
So far as I know, it has only appeared in print in the Law Journal,
where I published both the problem and its solution." This may be an early appearance, so I should
try and track this down. ??NYS
Dan Pedoe. The Gentle Art of Mathematics. (English Universities Press, 1958); Pelican (Penguin), 1963. P. 30:
"We now come to a problem which is said to have been planted over
here during the war by enemy agents, since Operational Research spent so many
man‑hours on its solution."
Philip E. Bath. Fun with Figures. The Epworth Press, London, 1959.
No. 7: No weights -- no guessing, pp. 8 & 40. 8
balls, including one light, to be determined in two weighings. Method actually works for £ 1 light.
M. R. Boothroyd &
J. H. Conway. Problems drive,
1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 9. Five boxes of sugar, but some has been taken from one box and put
in another. Determine which in least
number of weighings. Does by weighing
each division of A, B, C, D into two pairs.
Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. The "False
Coin" problem, pp. 178-182.
Sketches history and solution.
Simon Dresner. Science World Book of Brain Teasers. 1962.
Op. cit. in 5.B.1. Prob. 46: Dud
reckoning, pp. 21 & 94. Find one
light among eight in two weighings.
Philip Kaplan. More Posers. (Harper & Row, 1964);
Macfadden-Bartell Books, 1965.
Prob. 55, pp. 57 & 98.
Six identical appearing coins, three of which are identically
heavy. In two weighings, identify two
of the heavy coins.
Charlie Rice. Challenge!
Hallmark Editions, Kansas City, Missouri, 1968. Prob. 7, pp. 22 & 54-55. 9 pearls, one light.
Jonathan Always. Puzzling You Again. Tandem, London, 1969. Prob. 86: Light‑weight contest, pp. 51‑52
& 106‑107. 27 weights of
sizes 1, 2, ..., 27, except one is light.
Find it in 3 weighings. He
divides into 9 sets of three having equal weights. Using two weighings, one locates the light weight in a set of
three and then weighing two of these with good weights reveals the light
one. [3 weights 1, 2, 3 cannot be done
in one weighing, but 9 weights 1, 2, ..., 9 can be done in two weighings.]
Robert H. Thouless. The 12‑balls problem as an
illustration of the application of information theory. MG 54 (No. 389) (Oct 1970) 246‑249. Uses information theory to show that the
solution process is essentially determined.
Ron Denyer. Letter.
G&P, No. 37 (Jun 1975) 23. Asks
for a mnemonic for the 12 coins puzzles.
He notes that one can use three predetermined weighings and find the
coin from the three answers.
Basil Mager & E. Asher. Letters:
Coining a mnemonic. G&P, No.
40 (Sep 1975) 26. One mnemonic for a
variable method, another for a predetermined method.
N. J. Maclean. Letter:
The twelve coins. G&P, No.
45 (Feb 1976) 28-29. Exposits a ternary
method for predetermined weighings for
(3n-3)/2 in n
weighings. Each weighing
determines one ternary digit and the resulting ternary number gives both the
coin and whether it is heavy or light.
Tim Sole. The Ticket to Heaven and Other Superior
Puzzles. Penguin, 1988. Weighty problems -- (iii), pp. 124 &
147. Nine equal pies, except someone
has removed some filling from one and inserted it in a pie, possibly the same
one. Determine which, if any, are the
heavy and light ones in 4 balancings.
Calvin T. Long. Magic in base 3. MG 76 (No. 477) (Nov 1992) 371-376. Good exposition of the base 3 method for 12 coins.
Ed Barbeau. After Math.
Wall & Emerson, Toronto, 1995.
Problems for an equal-arm balance, pp. 137-141.
1. Six balls, two of each of three
colours. One of each colour is lighter
than normal and all light weights are equal.
Determine the light balls in three weighings.
2. Five balls, three normal, one heavy, one
light, with the differences being equal, i.e. the heavy and the light weigh as
much as two normals. Determine the
heavy and light in three weighings.
3. Same problem with nine balls and seven
normals, done in four weighings.
5.C.1RANKING COINS WITH A BALANCE
If
one weighs only one coin against another, this is the problem of sorting except
that we don't actually put the objects in order. If one weighs pairs, etc., this is a more complex problem.
J. Schreier. Mathesis Polska 7 (1932) 154‑160. ??NYS -- cited by Steinhaus.
Hugo Steinhaus. Mathematical Snapshots. Not in Stechert, NY, 1938, ed. OUP, NY:
1950: pp. 36‑40 & 258;
1960: pp. 51‑55 & 322;
1969 (1983): pp. 53‑56 & 300. Shows n objects can be ranked in M(n) = 1 + kn ‑ 2k steps where
k = 1 + [log2 n].
Gets M(5) = 8.
Lester R. Ford Jr. & Selmer
M. Johnson. A tournament problem. AMM 66:5 (May 1959) 387‑389. Note that
élog2 n!ù = L(n) is a lower bound from information
theory. Obtain a better upper bound
than Steinhaus, denoted U(n), which is too complex to state here. For convenience, I give the table of these
values here.
n 1 2
3 4 5 6 7
8 9 10 11 12
13
M(n) 0
1 3 5 8 11
14 17 21 25 29
33 37
U(n) 0
1 3 5 7 10
13 16 19 22 26
30 34
L(n) 0 1 3 5
7 10 13 16 19
22 26 29 33
U(n)
= L(n) also holds at n = 20 and 21.
Roland Sprague. Unterhaltsame Mathematik. Op. cit. in 4.A.1. 1961. Prob. 22: Ein noch
ungelöstes Problem, pp. 16 & 42‑43.
(= A still unsolved problem, pp. 17 & 48‑49.) Sketches Steinhaus's method, then does 5
objects in 7 steps. Gives the lower
bound L(n) and says the case n =
12 is still unsolved.
Kobon Fujimura, proposer; editorial comment. Another balance scale problem.
RMM 10 (Aug 1962) 34 & 11 (Oct 1962) 42. Eight coins of different weights and a balance. How many weighings are needed to rank the
coins? In No. 11, it says the solution
will appear in No. 13, but it doesn't appear there or in the last issue,
No. 14. It also doesn't appear in the
proposer's Tokyo Puzzles.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 6: In the
balance, pp. 18 & 85-86. Rank five
balls in order in seven weighings.
John Cameron. Establishing a pecking order. MG 55 (No. 394) (Dec 1971) 391‑395. Reduces Steinhaus's M(n)
by 1 for n ³
5, but this is not as good as Ford
& Johnson.
W. Antony Broomhead. Letter:
Progress in congress? MG 56 (No.
398) (Dec 1972) 331. Comments on
Cameron's article and says Cameron can be improved. States the values
U(9) and U(10),
but says he doesn't know how to do
9 in 19 steps. Cites Sprague for numerical values, but
these don't appear in Sprague -- so Broomhead presumably computed L(9)
and L(10). He gets
10 in 23 steps, which is better
than Cameron.
Stanley Collings. Letter:
More progress in congress. MG 57
(No. 401) (Oct 1973) 212‑213. Notes
the ambiguity in Broomhead's reference to Sprague. Improves Cameron by
1 (or more??) for n ³ 10, but still not as good as Ford & Johnson.
L. J. Upton, proposer; Leroy J. Myers, solver. Problem 1138. CM 12 (1986) 79 & 13 (1987) 230‑231. Rank coins weighing 1, 2, 3, 4
with a balance in four weighings.
See
MUS I 105-124, Tropfke 659.
NOTATION: I-(a, b, c)
means we have three jugs of sizes
a, b, c with a
full and we want to divide
a in half using b
and c. We normally assume a ³ b ³
c and
GCD(a, b, c) = 1.
Halving a is clearly impossible if GCD(b, c)
does not divide a/2 or if
b+c < a/2, unless
one has a further jug or one can drink some.
If a ³
b+c ³ a/2 and GCD(b, c) divides a/2, then the problem is solvable.
More
generally, the question is to determine what amounts can be produced, i.e.
given a, b, c as above, can one measure out an amount d? We denote this by II-(a, b, c; d). Since this also produces
a-d, we can assume that d £ a/2. Then we must have d £ b+c for a solution. When a ³ b+c ³
d, the condition GCD(b, c) ½ d guarantees that d can be produced. This also holds for a =
b+c‑1 and a = b+c‑2. The simplest impossible cases are I‑(4, 4, 3) = II-(4, 4, 3; 2) and
II‑(5, 5, 3; 1). Case I‑(a,
b, c) is the same as II-(a, b, c; a/2).
If a is
a large source, e.g. a stream or a big barrel, we have the problem of
measuring d using b and
c without any constraint on a
and we denote this II-(¥,
b, c; d). However, the solution may not
use the infiniteness of the source and such a problem may be the same as II‑(b+c, b, c; d).
The
general situation when a < b+c is more complex and really requires us to
consider the most general three jug problem:
III‑(A; a, b, c; d) means
we have three jugs of sizes
a, b, c, containing a
total amount of liquid A (in some initial configuration) and we wish
to measure out d. In our previous problems, we had A = a.
Clearly we must have a+b+c ³ A. Again, producing d also produces A-d,
so we can assume d £
A/2. By considering the amounts of
empty space in the containers, the problem
III-(A; a, b, c; d) is
isomorphic to III‑(a+b+c‑A; a, b, c; d') for several possible d'.
NOTES. I have been re-examining this problem and I
am not sure if I have reached a final interpretation and formulation. Also, I have recently changed to the above
notation and I may have made some errors in so doing. I have long had the problem in my list of projects for students,
but no one looked at it until 1995-1996 when Nahid Erfani chose it. She has examined many cases and we have have
discovered a number of properties which I do not recall seeing. E.g. in case I-(a,b,c) with a ³ b ³
c and
GCD(b,c) = 1, there are two ways
to obtain a/2. If we start by pouring into b,
it takes b + c - 1 pourings; if we start by pouring into c,
it takes b + c pourings; so it is always best to start
pouring into the larger jug. A number
of situations II-(a,b,c;d) are solvable for all values of d,
except a/2. E.g.
II‑(a,b,c;a/2) with b+c > a
and c > a/2 is unsolvable.
From about the mid 19C, I have not
recorded simple problems.
I-( 8, 5, 3): almost
all the entries below
I-(10, 6, 4): Pacioli, Court
I-(10, 7, 3): Yoshida
I-(12, 7, 5): Pacioli, van Etten/Henrion, Ozanam, Bestelmeier, Jackson,
Manuel des Sorciers,
Boy's Own Conjuring Book
I-(12, 8, 4): Pacioli
I-(12, 8, 5): Bachet, Arago
I-(16, 9, 7): Bachet-Labosne
I-(16,11, 6): Bachet-Labosne
I-(16,12, 7): Bachet-Labosne
I-(20,13, 9): Bachet-Labosne
I-(42,27,12): Bachet-Labosne
II-(10,3,2;6) Leacock
= II(10,3,2;4)
II-(11,4,3;9): McKay
= II(11,4,3;2)
II-( ¥,5,3;1): Wood, Serebriakoff, Diagram Group
II-( ¥,5,3;4): Chuquet, Wood,
Fireside Amusements,
II-( ¥,7,4;5): Meyer, Stein,
Brandes
II-( ¥,8,5;11): Young World,
III-(20;19,13,7;10): Devi
General problem, usually form I,
sometimes form II: Bachet‑Labosne, Schubert,
Ahrens, Cowley, Tweedie,
Grossman, Buker, Goodstein,
Browne, Scott, Currie,
Sawyer, Court, O'Beirne,
Lawrence, McDiarmid & Alfonsin.
Versions with 4 or more
jugs: Tartaglia, Anon: Problems drive (1958), Anon (1961), O'Beirne.
Impossible versions: Pacioli,
Bachet, Anon: Problems drive
(1958).
Abbot Albert. c1240.
Prob. 4, p. 333. I-(8,5,3) -- one solution.
Columbia Algorism. c1350.
Chap. 123: I-(8,5,3). Cowley 402‑403 & plate opposite
403. The plate shows the text and three
jars. I have a colour slide of the
three jars from the MS.
Munich 14684. 14C.
Prob. XVIII & XXIX, pp. 80 & 83. I-(8,5,3).
Folkerts. Aufgabensammlungen. 13-15C.
16 sources with I-(8,5,3).
Pseudo-dell'Abbaco. c1440.
Prob. 66, p.62. I-(8,5,3) -- one solution. "This problem is of little utility ...." I have a colour slide of this.
Chuquet. 1484.
Prob. 165. Measure 4 from a cask
using 5 and 3. You can pour back into
the cask, i.e. this is II-(¥,5,3;4). FHM 233 calls this the tavern-keeper's
problem.
HB.XI.22. 1488.
P. 55 (= Rath 248). Same as
Abbot Albert.
Pacioli. De Viribus.
c1500.
Ff.
97r - 97v. LIII. C(apitolo). apartire
una botte de vino fra doi (To divide a bottle of wine between two). = Peirani 137-138. I-(8,5,3). One solution.
Ff.
97v - 98v. LIIII. C(apitolo). a partire
unaltra botte fra doi (to divide another bottle between two). = Peirani 138-139. I-(12,7,5). Dario Uri
points out that the solution is confused and he repeats himself so it takes
him 18
pourings instead of the usual
11. He then says one can
divide 18 among three brothers who have containers of sizes 5, 6, 7,
which he does by filling the
6 and then the problem is
reduced to the previous problem. [He
could do it rather more easily by pouring the
6 into the 7
and then refilling the 6!]
Ff.
98v - 99r. LV. (Capitolo) de doi altri
sotili divisioni. de botti co'me se dira (Of two other subtle divisions of
bottles as described). = Peirani
139-140. I‑(10,6,4) and
I-(12,8,4). Pacioli suggests
giving these to idiots.
Ghaligai. Practica D'Arithmetica. 1521.
Prob. 20, ff. 64v-65r. I‑(8,5,3). One solution.
Cardan. Practica Arithmetice. 1539.
Chap. 66, section 33, f. DD.iiii.v (p. 145). I-(8,5,3). Gives one
solution and says one can go the other way.
H&S 51 says I-(8,5,3)
case is also in Trenchant (1566).
??NYS
Tartaglia. General Trattato, 1556, art. 132 & 133,
p. 255v‑256r.
Art.
132: I-(8,5,3).
Art.
133: divide 24 in thirds, using 5, 11, 13.
Buteo. Logistica. 1559. Prob. 73, pp. 282-283. I-(8,5,3).
Gori. Libro di arimetricha.
1571. Ff. 71r‑71v (p.
76). I-(8,5,3).
Bachet. Problemes.
1612. Addl. prob. III: Deux bons
compagnons ont 8 pintes de vin à partager entre eux également, ..., 1612: 134-139; 1624: 206-211; 1884: 138‑147. I‑(8,5,3) -- both solutions; I-(12,8,5)
(omitted by Labosne). Labosne
adds I‑(16,9,7); I‑(16,11,6); I‑(42,27,12);
I-(20,13,9); I-(16,12,7) (an impossible case!) and discusses general case. (This seems to be the first discussion of
the general case.)
van Etten. 1624.
Prob. 9 (9), pp. 11 & fig. opp. p. 1 (pp. 22‑23). I‑(8,5,3) -- one solution.
Henrion's Nottes, 1630, pp. 11‑13, gives the second solution and
poses and solves I‑(12,7,5).
Hunt. 1631 (1651). P. 270
(262). I-(8,5,3). One solution.
Yoshida (Shichibei) Kōyū (= Mitsuyoshi Yoshida)
(1598-1672). Jinkō‑ki. 2nd ed., 1634 or 1641??. ??NYS
The recreational problems are discussed in Kazuo Shimodaira; Recreative
Problems on "Jingōki", a 15 pp booklet sent by Shigeo
Takagi. [This has no details, but Takagi
says it is a paper that Shimodaira read at the 15th International Conference
for the History of Science, Edinburgh, Aug 1977 and that it appeared in
Japanese Studies in the History of Science 16 (1977) 95-103. I suspect this is a copy of a preprint.] This gives both Jingōki and
Jinkōki as English versions of the title and says the recreational
problems did not appear in the first edition, 4 vols., 1627, but did appear in
the second edition of 5 vols. (which may be the first use of coloured wood cuts
in Japan), with the recreational problems occurring in vol. 5. He doesn't give a date, but Mikami, p. 179,
indicates that it is 1634, with further editions in 1641, 1675, though an
earlier work by Mikami (1910) says 2nd ed. is 1641. Yoshida (or Suminokura) is the family name. Shimodaira refers to the current year as the
350th anniversary of the edition and says copies of it were published
then. I have a recent transcription of
some of Yoshida into modern Japanese and a more recent translation into English,
??NYR, but I don't know if it is the work mentioned by Shimodaira.
Shimodaira
discusses a jug problem on p. 14:
I-(10,7,3) -- solution in 10
moves. Shimodaira thinks Yoshida heard
about such puzzles from European contacts, but without numerical values, then
made up the numbers. I certainly can
see no other example of these numbers.
The recent transcription includes this material as prob. 7 on pp. 69-70.
Wingate/Kersey. 1678?.
Prob. 7, pp. 543-544.
I-(8,5,3). Says there is a
second way to do it.
Witgeest. Het Natuurlyk Tover-Boek. 1686.
Prob. 38, p. 308. I-(8,5,3).
Ozanam. 1694.
Prob.
36, 1696: 91-92; 1708: 82‑83. Prob. 42, 1725: 238‑240. Prob. 21, 1778: 175‑177; 1803: 174-176; 1814: 153-154. Prob. 20,
1840: 79. I-(8,5,3) -- both solutions.
Prob.
43, 1725: 240‑241. Prob. 22,
1778: 177-178; 1803: 176-177; 1814: 154-155. Prob. 21, 1840: 79‑80.
I-(12,7,5) -- one solution.
Dilworth. Schoolmaster's Assistant. 1743.
Part IV: Questions: A short Collection of pleasant and diverting
Questions, p. 168. Problem 8. I-(8,5,3).
(Dilworth cites Wingate for this -- cf in 5.B.) = D. Adams; Scholar's Arithmetic; 1801, p.
200, no. 10.
Les Amusemens. 1749.
Prob. 17, p. 139: Partages égaux avec des Vases inégaux. I-(8,5,3)
-- both solutions.
Bestelmeier. 1801.
Item 416: Die 3 Maas‑Gefäss.
I-(12,7,5).
Badcock. Philosophical Recreations, or, Winter
Amusements. [1820]. Pp. 48-49, no. 75: How to part an eight
gallon bottle of wine, equally between two persons, using only two other bottles,
one of five gallons, and the other of three.
Gives both solutions.
Jackson. Rational Amusement. 1821.
Arithmetical Puzzles.
No.
14, pp. 4 & 54. I-( 8,5,3).
One solution.
No.
52, pp. 12 & 67. I-(12,7,5). One solution.
Rational Recreations. 1824.
Exer. 10, p. 55. I-(8,5,3) one way.
Manuel des Sorciers. 1825.
??NX
Pp.
55-56, art. 27-28. I-(8,5,3) two ways.
P. 56,
art. 29. I-(12,7,5).
Endless Amusement II. 1826?
Prob. 7, pp. 193-194.
I-(8,5,3). One solution. = New Sphinx, c1840, p. 133.
Nuts to Crack III (1834), no.
212. I-(8,5,3). 8 gallons of spirits.
Young Man's Book. 1839.
Pp. 43-44. I-(8,5,3). Identical to Wingate/Kersey.
The New Sphinx. c1840.
P. 133. I-(8,5,3). One solution.
Boy's Own Book. 1843 (Paris): 436 & 441, no. 7. The can of ale: 1855: 395; 1868:
432. I‑(8,5,3). One solution. The 1843 (Paris) reads as though the owners of the 3 and 5 kegs
both want to get 4, which would be a problem for the owner of the 3. = Boy's Treasury, 1844, pp. 425 & 429.
Fireside Amusements. 1850.
Prob. 9, pp. 132 & 184. II-(¥,5,3;4). One solution.
Arago. [Biographie de] Poisson (16 Dec 1850). Oeuvres, Gide & Baudry, Paris, vol. 2, 1854, pp. 593‑??? P. 596 gives the story of Poisson's being
fascinated by the problem I‑(12,8,5). "Poisson résolut à l'instant cette
question et d'autres dont on lui donna l'énoncé. Il venait de trouver sa véritable vocation." No solution given by Arago.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Arithmetical puzzles, no. 8, pp. 174-175
(1868: 185-186). I-(8,5,3). Milkmaid with eight quarts of milk.
Magician's Own Book. 1857.
P.
223-224: Dividing the beer: I-(8,5,3).
P.
224: The difficult case of wine:
I-(12,7,5).
Pp.
235-236: The two travellers:
I-(8,5,3) posed in verse.
Each problem gives just one solution.
Boy's Own Conjuring Book. 1860.
P.
193: Dividing the beer: I-(8,5,3).
P.
194: The difficult case of wine:
I-(12,7,5).
Pp.
202‑203: The two travellers:
I-(8,5,3) posed in verse.
Each problem gives just one solution.
Illustrated Boy's Own
Treasury. 1860. Prob. 21, pp. 428-429 & 433. I‑(8,5,3). "A man coming from the Lochrin distillery with an 8-pint jar
full of spirits, ...."
Vinot. 1860. Art. XXXVIII: Les
cadeaux difficiles, pp. 57-58.
I-(8,5,3). Two solutions.
The Secret Out (UK). c1860.
To divide equally eight pints of wine ..., pp. 12-13.
Bachet-Labosne. 1874.
For details, see Bachet, 1612.
Labosne adds a consideration of the general case which seems to be the
first such.
Kamp. Op. cit. in 5.B.
1877. No. 17, p. 326: I-(8,5,3).
Mittenzwey. 1880.
Prob. 106, pp. 22 & 73-74;
1895?: 123, pp. 26 & 75-76;
1917: 123, pp. 24 & 73-74.
I-(8,5,3). One solution.
Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 135, no. 1.
I-(8,5,3). No solution.
Loyd. Problem 11: "Two thieves of Damascus". Tit‑Bits 31 (19 Dec 1896 &
16 Jan 1897) 211 & 287.
Thieves found with 2 &
2 quarts in pails of size 3 & 5.
They claim the merchant measured the amounts out from a fresh
hogshead. Solution is that this could
only be done if the merchant drained the hogshead, which is unreasonable!
Loyd. Problem 13: The Oriental problem. Tit‑Bits 31 (19 Jan,
30 Jan & 6 Feb 1897) 269, 325 & 343.
= Cyclopedia, 1914, pp. 188 & 364: The merchant of Bagdad. Complex problem with hogshead of water,
barrel of honey, three 10 gallon jugs to be filled with 3 gallons of water, of
honey and of half and half honey & water.
There are a 2 and a 4 gallon measure and also 13 camels to receive 3
gallons of water each. Solution
takes 521 steps. 6 Feb reports
solutions in 516 and
513 steps. Cyclopedia gives solution in 506
steps.
Dudeney. The host's puzzle. London Magazine 8 (No. 46) (May 1902) 370 &
8 (No. 47) (Jun 1902) 481‑482 (= CP, prob. 6, pp. 28‑29
& 166‑167). Use 5 and 3
to obtain 1 and 1 from a cask. One must drink some!
H. Schubert. Mathematische Mussestunden, 3rd ed.,
Göschen, Leipzig, 1907. Vol. 1, chap.
6, Umfüllungs‑Aufgaben, pp. 48‑56.
Studies general case and obtains some results. (The material appeared earlier in Zwölf Geduldspiele, 1895, op.
cit. in 5.A, Chap. IX, pp. 110-119.
The 13th ed. (De Gruyter, Berlin, 1967), Chap. 9, pp. 62‑70, seems
to be a bit more general (??re-read).)
Ahrens. MUS I, 1910, chap. 4, Umfüllungsaufgaben,
pp. 105‑124. Pp. 106‑107 is
Arago's story of Poisson and this problem.
He also extends and corrects Schubert's work.
Dudeney. Perplexities: No. 141: New measuring
puzzle. Strand Magazine 45 (Jun 1913)
710 & 46 (Jul 1913) 110. (= AM,
prob. 365, pp. 110 & 235.) Two 10
quart vessels of wine with 5 and 4 quart measures. He wants 3 quarts in each measure. (Dudeney gives numerous other versions in AM.)
Loyd. Cyclopedia. 1914. Milkman's puzzle, pp. 52 & 345. (= MPSL2, prob. 23, pp. 17 & 127‑128 = SLAHP: Honest John, the milkman, pp. 21
& 90.) Milkman has two full 40
quart containers and two customers with 5 and 4 quart pails, but both want 2
quarts. (Loyd Jr. says "I first published [this] in
1900...")
Williams. Home Entertainments. 1914.
The measures puzzle, p. 125.
I-(8,5,3).
Hummerston. Fun, Mirth & Mystery. 1924.
A shortage of milk, Puzzle no. 75, pp. 164 & 183. I-(8,5,3),
one solution.
Elizabeth B. Cowley. Note on a linear diophantine equation. AMM 33 (1926) 379‑381. Presents a technique for resolving I-(a,b,c),
which gives the result when a =
b+c. If a < b+c, she only
seems to determine whether the method gets to a point with A
empty and neither B nor
C full and it is not clear to me
that this implies impossibility. She
mentions a graphical method of Laisant (Assoc. Franç. Avance. Sci, 1887, pp.
218-235) ??NYS.
Wood. Oddities. 1927.
Prob.
15: A problem in pints, pp. 16-17.
Small cask and measures of size 5 and 3, measure out 1 in each measure. Starts by filling the 5 and the 3 and then emptying the cask, so
this becomes a variant of II-(¥,5,3;1).
Prob.
26: The water-boy's problem, pp. 28-29.
II-(¥;,5,3;4).
Ernest K. Chapin. Scientific Problems and Puzzles. In:
S. Loyd Jr.; Tricks and Puzzles,
Vol. 1 (only volume to appear);
Experimenter Publishing Co., NY, nd [1927] and Answers to Sam Loyd's
Tricks and Puzzles, nd [1927]. [This
book is a selection of pages from the Cyclopedia, supplemented with about 20
pages by Chapin and some other material.]
P. 89 & Answers p. 8. You
have a tablet that has to be dissolved in
7½ quarts of water, though you
only need 5 quarts of the resulting mixture.
You have 3 and 5 quart measures and a tap.
Stephen Leacock. Model Memoirs and Other Sketches from Simple
to Serious. John Lane, The Bodley Head,
1939, p. 298. "He's trying to
think how a farmer with a ten-gallon can and a three-gallon can and a
two-gallon can, manages to measure out six gallons of milk." II-(10,3,2;6) = II-(10,3,2;4).
M. C. K. Tweedie. A graphical method of solving Tartaglian
measuring puzzles. MG 23 (1939) 278‑282. The elegant solution method using triangular
coordinates.
H. D. Grossman. A generalization of the water‑fetching
puzzle. AMM 47 (1940) 374‑375. Shows
II-(¥,b,c;d) with GCD(b,c) = 1 is solvable.
McKay. Party Night. 1940.
No.
18, p. 179. II-(11,4,3;9).
No.
19, pp. 179-180. I-(8,5,3).
Meyer. Big Fun Book. 1940. No. 10, pp. 165 & 753. II-(¥,7,4,5).
W. E. Buker, proposer. Problem E451. AMM 48 (1941) 65.
??NX. General problem of what
amounts are obtainable using three jugs, one full to start with, i.e. I-(a,b,c).
See Browne, Scott, Currie below.
Eric Goodstein. Note 153:
The measuring problem. MG 25
(No. 263) (Feb 1941) 49‑51.
Shows II-(¥,b,c;d) with
GCD(b,c) = 1 is solvable.
D. H. Browne & Editors. Partial solution of Problem E451. AMM 49 (1942) 125‑127.
W. Scott. Partial solution of E451 -- The generalized
water‑fetching puzzle. AMM 51
(1944) 592. Counterexample to
conjecture in previous entry.
J. C. Currie. Partial solution of Problem E451. AMM 53 (1946) 36‑40. Technical and not complete.
W. W. Sawyer. On a well known puzzle. SM 16 (1950) 107‑110. Shows that
I-(b+c,b,c) is solvable if b & c
are relatively prime.
David Stein. Party and Indoor Games. Op. cit. in 5.B. c1950. Prob. 13, pp. 79‑80. Obtain 5 from a spring using measures 7 and
4, i.e. II-(¥,7,4,5).
Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 8.
Given an infinite source, use:
6, 10, 15 to obtain 1, 6, 7
simultaneously; 4, 6, 9, 12 to obtain
1, 2, 3, 4 simultaneously; 6, 9, 12, 15, 21 to obtain 1, 3, 6, 8,
9 simultaneously. Answer simply says the first two are possible
(the second being easy) and the third is impossible.
Young World. c1960.
P. 58: The 11 pint problem. II-(¥,8,5;11). This is the same as II‑(13,8,5;11) or
II-(13,8,5,2).
Anonymous. Moonshine sharing. RMM 2 (Apr 1961) 31
& 3 (Jun 1961) 46. Divide
24 in thirds using cylindrical
containers holding 10, 11, 13. Solution in No. 3 uses the cylindricity of a
container to get it half full.
Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. "Pouring"
problems -- The "robot" method.
General description of the problem.
Attributes Tweedie's triangular 'bouncing ball' method to Perelman, with
no reference. Does I‑(8,5,3) two ways, also
I-(12,7,5) and I-(16,9,7),
then considers type II questions.
Considers the problem with
II-(10,6,4;d) and extends
to II-(a,6,4;d) for
a > 10, leaving it to
the reader to "try to formulate some rule about the results." He then considers II‑(7,6,4;d),
noting that the parallelogram has a corner trimmed off. Then considers II-(12,9,7;d) and II-(9,6,3;d).
Lloyd Jim Steiger. Letter.
RMM 4 (Aug 1961) 62. Solves the
RMM 2 problem by putting the 10 inside the 13 to measure 3.
Irving & Peggy Adler. The Adler Book of Puzzles and Riddles. Or Sam Loyd Up-To-Date. John Day, NY, 1962. Pp. 32 & 46. Farmer has two full 10-gallon cans. Girls come with 5-quart and 4-quart cans and each wants 2 quarts.
Philip Kaplan. More Posers. (Harper & Row, 1964);
Macfadden-Bartell Books, 1965.
Prob. 80, pp. 81 & 109.
Tavern has a barrel with 15 pints of beer. Two customers, with 3 pint and 5 pint jugs appear and ask for 1
pint in each jug. Bartender finds it
necessary to drink the other 13 pints!
T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965.
Chap. 4: Jug and bottle department, pp. 49‑75. This gives an extensive discussion of
Tweedie's method and various extensions to four containers, a barrel of unknown
size, etc.
P. M. Lawrence. An algebraic approach to some pouring
problems. MG 56 (No. 395) (Feb 1972) 13‑14. Shows
II-(¥,b,c,d) with d £ b+c and GCD(b,c) = 1 is possible and extends to more jugs.
Louis Grant Brandes. The Math. Wizard. revised ed., J. Weston Walch, Portland, Maine, 1975. Prob. 5: Getting five gallons of water: II‑(¥,7,4,5).
Shakuntala Devi. Puzzles to Puzzle You. Orient Paperbacks (Vision Press), Delhi,
1976.
Prob.
53: The three containers, pp. 57 & 110.
III-(20;19,13,7;10). Solution in
15 steps. Looking at the triangular
coordinates diagram of this, one sees that it is actually isomorphic to II-(19,13,7;10) and this can be seen by considering the amounts of empty space in
the containers.
Prob.
132: Mr. Portchester's problem, pp. 82 & 132. Same as Dudeney (1913).
Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A Second Mensa Puzzle
Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London,
1991.) Problem T.16: Pouring puttonos,
part b, pp. 19-20 (1991: 37-38) & Answer 19, pp. 102-103 (1991:
118-119). II-( ¥,5,3;1).
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex,
1984. Problem 161, with Solution at the
back of the book. II-(¥,5,3;1),
which can be done as II-(8,5,3;1).
D. St. P. Barnard. 50 Daily Telegraph Brain Twisters. 1985.
Op. cit. in 4.A.4. Prob. 4:
Measure for measure, pp. 15, 79‑80, 103.
Given 10 pints of milk, an 8 pint bowl, a jug and a flask. He describes how he divides the milk in
halves and you must deduce the size of the jug and the flask.
Colin J. H. McDiarmid &
Jorge Ramirez Alfonsin. Sharing
jugs of wine. Discrete Mathematics 125
(1994) 279-287. Solves I-(b+c,b,c)
and discusses the problem of getting from one state of the problem to
another in a given number of steps, showing that GCD(b,c) = 1 guarantees the
graph is connected. indeed essentially cyclic.
Considers GCD(b,c) ¹
1. Notes that the work done easily
extends to a > b + c. Says the second author's PhD at Oxford,
1993, deals with more cases.
John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for all
Ages. Keystone Agencies, Radnor, Ohio,
1997. P. 11: The spoon and the
bottle. Given a 160 ml bottle and a 30
ml spoon, measure 230 ml into a bucket.
5.D.2. RULER WITH MINIMAL NUMBER OF MARKS
Dudeney. Problem 518: The damaged measure. Strand Mag. (Sep 1920) ??NX.
Wants a minimal ruler for 33 inches total length. (=? MP 180)
Dudeney. Problem 530: The six cottagers. Strand Mag. (Jan 1921) ??NX.
Wants 6 points on a circle to give all arc distances 1, 2, ..., 20. (=? MP 181)
Percy Alexander MacMahon. The prime numbers of measurement on a
scale. Proc. Camb. Philos. Soc. 21
(1922‑23) 651‑654. He
considers the infinite case, i.e. a(0)
= 0,
a(i+1) = a(i) + least integer which is not yet
measurable. This gives:
0, 1, 3, 7, 12, 20, 30, 44, ....
Dudeney. MP.
1926.
Prob.
180: The damaged measure, pp. 77 & 167.
(= 536, prob. 453, pp. 173, 383‑384.) Mark a ruler of length 33 with 8 (internal)
marks. Gives 16 solutions.
Prob.
181: The six cottagers, pp. 77‑78 & 167. = 536, prob. 454, pp. 174 & 384.
A. Brauer. A problem of additive number theory and its
application in electrical engineering.
J. Elisha Mitchell Sci. Soc. 61 (1945) 55‑56. Problem arises in designing a resistance box.
Л.
Редеи
& А. Реньи [L. Redei & A. Ren'i
(Rényi)]. О
представленин
чисел
1, 2, ..., N
лосредством
разностей [O
predstavlenin chisel 1, 2, ... , N losredstvom raznosteĭ (On the
representation of 1, 2, ..., N by differences)]. Мат.
Сборник [Mat. Sbornik] 66
(NS 24) (1949) 385‑389.
Anonymous. An unsolved problem. Eureka 11 (Jan 1949) 11 & 30. Place as few marks as possible to permit
measuring integers up to n. For
n = 13, an example is: 0, 1, 2, 6, 10, 13. Mentions some general results for a circle.
John Leech. On the representation of 1, 2, ..., n by differences. J. London
Math. Soc. 31 (1956) 160‑169.
Improves Redei & Rényi's results.
Gives best examples for small n.
Anon. Puzzle column: What's your potential? MTg 19 (1962) 35
& 20 (1962) 43. Problem posed in terms of transformer
outputs -- can we arrange 6 outputs to give every integral voltage up
through 15? Problem also asks for the general case. Solution asserts, without real proof, that the optimum occurs
with 0, 1, 4, 7, 10, ..., n‑11, n‑8,
n‑5, n‑2 or its complement.
T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965.
Chap. 6 discusses several versions of the problem.
Gardner. SA (Jan 1965) c= Magic Numbers, chap.
6. Describes 1, 2, 6, 10 on a ruler 13 long. Says 3
marks are sufficient on 9 and
4 marks on 12
and asks for proof of the latter and for the maximum number of distances
that 3
marks on 12 can produce. How can you mark a ruler
36 long? Says Dudeney, MP prob. 180, believed that 9
marks were needed for a ruler longer than 33, but Leech managed to
show 8
was sufficient up to 36.
C. J. Cooke. Differences. MTg 47 (1969) 16. Says
the problem in MTg 19 (1962) appears in H. L. Dorwart's The Geometry of
Incidence (1966) related to perfect difference sets but with an erroneous
definition which is corrected by references to H. J. Ryser's Combinatorial
Mathematics. However, this doesn't
prove the assertions made in MTg 20.
Jonathan Always. Puzzles for Puzzlers. Tandem, 1971. Prob. 22: Starting and stopping, pp. 18 & 66. Circular track, 1900 yards around. How can one place marker posts so every
multiple of 100 yards up to 1900 can be run.
Answer: at 0, 1, 3, 9, 15.
Gardner. SA (Mar 1972) = Wheels, Chap. 15.
5.D.3 FALSE COINS WITH A WEIGHING SCALE
H. S. Shapiro, proposer; N. J. Fine, solver. Problem E1399 -- Counterfeit coins. AMM 67 (1960) 82 & 697‑698. Genuines weigh 10, counterfeits
weigh 9. Given 5 coins and a scale, how many weighings are
needed to find the counterfeits? Answer
is 4.
Fine conjectures that the ratio of weighings to coins decreases to 0.
Kobon Fujimura & J. A. H.
Hunter, proposers; editorial
solution. There's always a way. RMM 6 (Dec 1961) 47 &
7 (Feb 1962) 53.
(c= Fujimura's The Tokyo Puzzles (Muller, London, 1979), prob. 29:
Pachinko balls, pp. 35 & 131.) Six
coins, one false. Determine which is
false and whether it is heavy or light in three weighings on a scale. In fact one also finds the actual weights.
K. Fujimura, proposer; editorial solution. The 15‑coin puzzle. RMM 9 (Jun 1962) & 10 (Aug 1962) 40‑41. Same problem with fifteen coins and four
weighings.
5.D.4. TIMING WITH HOURGLASSES
I
have just started these and they are undoubtedly older than the examples here. I don't recall ever seeing a general approach
to these problems.
Simon Dresner. Science World Book of Brain Teasers. 1962.
Op. cit. in 5.B.1. Prob. 17: Two‑minute
eggs, pp. 9 & 87. Time 2
minutes with 3 & 5 minute timers.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 21: The sands
of time, pp. 35 & 93. Time 9
minutes with 4 & 7 minute timers.
David B. Lewis. Eureka!
Perigee (Putnam), NY, 1983. Pp.
73‑74. Time 9
minutes with 4 & 7 minute timers.
Yuri B. Chernyak & Robert S.
Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 1, prob. 8: Grandfather's breakfast,
pp. 6 & 102. Time 15
minutes with 7 & 11 minute timers.
I
have just started this and there must be much older examples.
Benson. 1904.
The water‑glass puzzle, p. 254.
Dudeney. AM.
1917. Prob. 364: The barrel
puzzle, pp. 109-110 & 235.
King. Best 100. 1927. No. 1, pp. 7 & 38.
Collins. Fun with Figures. 1928. The dairymaid's
problem, pp. 29-30.
William A. Bagley. Puzzle Pie.
Vawser & Wiles, London, nd [BMC gives 1944]. [There is a revised edition, but it only
affects material on angle trisection.]
No. 14: 'Arf an' 'arf, p. 15.
Anon. The Little Puzzle Book.
Peter Pauper Press, Mount Vernon, NY, 1955. P. 52: The cider barrel.
Jonathan Always. Puzzles for Puzzlers. Tandem, London, 1971. Prob. 87: But me no butts, pp. 42 & 88.
Richard I. Hess. Email Christmas message to NOBNET, 24 Nov
2000. Solution sent by Nick Baxter on
the same day. You have aquaria (assumed
cuboidal) which hold 7 and 12 gallons and a water supply. The 12 gallon aquarium has dots accurately
placed in the centre of each side face.
How many steps are required to get 8 gallons into the 12 gallon
aquarium? Fill the 12 gallon aquarium
and tilt it on one corner so the water level passes through the centres of the
two opposite faces. This leaves 8
gallons! Nick says this is two steps.
Euler
circuits have been used in primitive art, often as symbols of the passage of
the soul to the land of the dead. [MTg
110 (Mar 1985) 55] shows examples from Angola and New Hebrides. See Ascher (1988 & 1991) for many other
examples from other cultures.
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Above
is the 'five‑brick pattern'.
See: Clausen, Listing,
Kamp, White, Dudeney,
Loyd Jr, Ripley, Meyer,
Leeming, Adams, Anon.,
Ascher. Prior to Loyd Jr, the
problem asked for the edges to be drawn in three paths, but about 1920 the
problem changed to drawing a path across every wall.
Trick
solutions: Tom Tit, Dudeney (1913), Houdini, Loyd Jr, Ripley,
Meyer, Leeming, Adams,
Gibson, Anon. (1986).
Non-crossing
Euler circuits: Endless Amusement
II, Bellew, Carroll 1869, Mittenzwey, Bile Beans,
Meyer, Gardner (1964), Willson,
Scott, Singmaster.
Kn denotes the complete graph on n
vertices.
Matthäus Merian the Elder. Engraved map of Königsberg. Bernhard Wiezorke has sent me a coloured
reproduction of this, dated as 1641. He
used an B&W version in his article: Puzzles und Brainteasers; OR News,
Ausgabe 13 (Nov 2001) 52-54. BLW use a
B&W version on their dust jacket and on p. 2 which they attribute to M.
Zeiller; Topographia Prussiae et Pomerelliae; Frankfurt, c1650. I have seen this in a facsimile of the
Cosmographica due to Merian in the volume on Brandenburg and Pomerania, but it
was not coloured. There seem to be at
least two versions of this picture --??CHECK.
L. Euler. Solutio problematis ad geometriam situs
pertinentis. (Comm. Acad. Sci.
Petropol. 8 (1736(1741)) 128‑140.)
= Opera Omnia (1) 7 (1923) 1‑10.
English version: Seven Bridges
of Königsberg is in: BLW, 3‑8; SA 189 (Jul 1953) 66‑70;
World of Mathematics, vol. 1, 573‑580; Struik, Source Book, 183‑187.
My late colleague Jeremy Wyndham
became interested in the seven bridges problem and made inquiries which turned
up several maps of Königsberg and a list of all the bridges and their dates of
construction (though there is some ambiguity about one bridge). The first bridge was built in 1286 and until
the seventh bridge of 1542, an Euler path was always possible. No further bridge was built until a railway
bridge in 1865 which led to Saalschütz's 1876 paper -- see below. In 1905 and later, several more bridges were
added, reaching a maximum of ten bridges in 1926 (with 4512 paths from the
island), then one was removed in 1933.
Then a road bridge was added, but it is so far out that it does not show
on any map I have seen. Bombing and
fighting in 1944-1945 apparently destroyed all the bridges and the Russians
have rebuilt six or seven of them. I
have computed the number of paths in each case -- from 1865 until 1935 or 1944,
there were always Euler paths.
L. Poinsot. Sur les polygones et les polyèdres. J. École Polytech. 4 (Cah. 10) (1810) 16‑48. Pp. 28‑33 give Euler paths on K2n+1 and Euler's criterion.
Discusses square with diagonals.
Endless Amusement II. 1826?
Prob. 34, p. 211. Pattern of two
overlapping squares has a non-crossing Euler circuit.
Th. Clausen. De linearum tertii ordinis
propietatibus. Astronomische
Nachrichten 21 (No. 494) (1844), col. 209‑216. At the very end, he gives the five‑brick pattern and says
that its edges cannot be drawn in three paths.
J. B. Listing. Vorstudien zur Topologie. Göttinger Studien 1 (1847) 811‑875. ??NYR.
Gives five brick pattern as in Clausen.
?? Nouv. Ann. Math. 8
(1849?) 74. ??NYS. Lucas says this poses the problem of finding
the number of linear arrangements of a set of dominoes. [For a double N set, N = 2n,
this is (2n+1)(n+1) times the number of circular arrangements,
which is n2n+1 times the number of Euler circuits on K2n+1.]
É. Coupy. Solution d'un problème appartenant a la
géométrie de situation, par Euler.
Nouv. Ann. Math. 10 (1851) 106‑119. Translation of Euler.
Translator's note on p. 119 applies it to the bridges of Paris.
The Sociable. 1858.
Prob. 7: Puzzle pleasure garden, pp. 288 & 303. Large maze-like garden and one is to pass
over every path just once -- phrased in verse.
= Book of 500 Puzzles, 1859, prob. 7, pp. 6 & 21. = Illustrated Boy's Own Treasury, 1860,
prob. 49, pp. 405 & 443. In fact,
if one goes straight across every intersection, one finds the path, so this is
really almost a unicursal problem.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 587, pp. 297
& 410: Ariadnerätsel. Three
diagrams to trace with single lines. No
attempt to avoid crossings.
Frank Bellew. The Art of Amusing. Carleton, NY (& Sampson Low & Co., London), 1866
[C&B list a 1871]; John Camden
Hotten, London, nd [BMC & NUC say 1870] and John Grant, Edinburgh, nd
[c1870 or 1866?], with slightly different pagination. 1866: pp. 269-270;
1870: p. 266. Two
overlapping squares have a non-crossing Euler circuit.
Lewis Carroll. Letter of 22 Aug 1869 to Isabel
Standen. Taken from: Stuart Dodgson
Collingwood; The Life and Letters of Lewis Carroll; T. Fisher Unwin, London,
(Dec 1898), 2nd ed., Jan 1899, p. 370: "Have you succeeded in drawing the
three squares?" On pp. 369-370,
the recipient is identified as Isabel Standen and she is writing Collingwood,
apparently sending him the letter.
Collingwood interpolates:
"This puzzle was, by the way, a great favourite of his; the problem
is to draw three interlaced squares without going over the same lines twice, or
taking the pen off the paper". But
no diagram is given.
Dudeney;
Some much‑discussed puzzles; op. cit. in 2; 1908, quotes Collingwood,
gives the diagram and continues: "This is sometimes ascribed to him [i.e.
Carroll] as its originator, but I have found it in a little book published in
1835." This was probably a
printing of Endless Amusement II, qv above and in Common References, though
this has two interlaced squares. John
Fisher; The Magic of Lewis Carroll; op. cit. in 1; pp. 58‑59, says
Carroll would ask for a non‑crossing Euler circuit, but this is not
clearly stated in Collingwood. Cf
Carroll-Wakeling, prob. 29: The three squares, pp. 38 & 72, which
clearly states that a non-crossing circuit is wanted and notes that there is
more than one solution. Cf Gardner
(1964). Carroll-Gardner, pp. 52-53.
Mittenzwey. 1880.
Prob. 269-279, pp. 47-48 & 98-100;
1895?: 298-308, pp. 51-52 & 100‑102; 1917: 298-308, pp. 46-48 & 95-97. Straightforward unicursal patterns. The first is K5, but one of the diagonals was missing in my
copy of the 1st ed. -- the path is not to use two consecutive outer edges. The third is the 'envelope' pattern. The fourth is three overlapping squares,
where the two outer squares just touch in the middle. The last is a simple maze with no dead ends and the path is not
to cross itself. See also the entry for
Mittenzwey in 5.E.1, below.
M. Reiss. Évaluation du nombre de combinaisons
desquelles les 28 dés d'un jeu de dominos sont susceptibles d'après la règle de
ce jeu. Annali di Matematica Pura ed
Applicata (2) 5 (1871) 63‑120.
Determines the number of linear arrangements of a double‑6 set of
dominoes, which gives the number of Euler circuits on K7.
L. Saalschütz. [Report of a lecture.] Schriften der Physikalisch‑Ökonomischen
Gesellschaft zu Königsberg 16 (1876) 23‑24. Sketches Euler's work, listing the seven bridges. Says that a recent railway bridge, of 1865,
connecting regions B and
C on Euler's diagram, can be
considered within the walkable region.
He shows there are 48 x 2 x
4 = 384 possible paths -- the
48 are the lists of regions
visited starting with A; the
2 corresponds to reversing these
lists; the 4 (= 2 x 2) corresponds
to taking each of the two pairs of bridges connecting the same regions in
either order, He lists the 48
sequences of regions which start at
A. I wrote a program to compute
Euler paths and I tested it on this situation.
I find that Saalschütz has omitted two cases, leading to four sequences
or 16
paths starting at A or
32 paths considering both
directions. That is, his 48
should be 52 and his
384 should be 416.
Kamp. Op. cit. in 5.B. 1877. Pp. 322‑327 show several unicursal
problems.
No. 8
is the five‑brick pattern as in Clausen.
No. 10
is two overlapping squares.
No. 11
is a diagram from which one must remove some lines to leave an Eulerian figure.
C. Hierholzer. Ueber die Möglichkeit, einen Linienzug ohne
Wiederholung und ohne Unterbrechung zu umfahren. Math. Annalen 6 (1873) 30‑32. (English is in BLW, 11‑12.)
G. Tarry. Géométrie de situation: Nombre de manières
distinctes de parcourir en une seule course toutes les allées d'un labyrinthe
rentrant, en ne passant qu'une seule fois par chacune des allées. Comptes Rendus Assoc. Franç. Avance. Sci.
15, part 2 (1886) 49‑53 & Plates I & II. General technique for the number of Euler
circuits.
Lucas. RM2. 1883. Le jeu de dominos -- Dispositions
rectilignes, pp. 63‑77
& Note 1: Sur le jeu de dominos, p. 229.
RM4. 1894.
La géométrie des réseaux et le problème des dominos, pp. 123‑151.
Cites
Reiss's work and says (in RM4) that it has been confirmed by Jolivald. The note in RM2 is expanded in RM4 to
explain the connection between dominoes and
K2n+1. There are
obviously 2 Euler circuits on K3. He sketches Tarry's method and uses it to
compute that K5 has
88 Euler circuits and K7 has 1299 76320. [This gives
28 42582 11840 domino rings for
the double-6 set.] He says Tarry has
found that K9 has
911 52005 70212 35200.
Tom Tit, vol. 3. 1893.
Le rectangle et ses diagonales, pp. 155-156. = K, no. 16: The rectangle and its diagonals, pp. 46‑48. = R&A, The secret of the rectangle, p.
100. Trick solutions by folding the
paper and making an arc on the back.
Hoffmann. 1893.
Chap. X, no. 9: Single‑stroke figures, pp. 338 & 375 =
Hoffmann-Hordern, pp. 230-231. Three
figures, including the double crescent 'Seal of Mahomet'. Answer states Euler's condition.
Dudeney. The shipman's puzzle. London Mag. 9 (No. 49) (Aug 1902) 88‑89 &
9 (No. 50) (Sep 1902) 219 (= CP, prob. 18, pp. 40‑41 &
173). Number of Euler circuits on K5.
Benson. 1904.
A geometrical problem, p. 255. Seal
of Mahomet.
William F. White. A Scrap‑Book of Elementary
Mathematics. Open Court, 1908. [The 4th ed., 1942, is identical in content
and pagination, omitting only the Frontispiece and the publisher's catalogue.] Bridges and isles, figure tracing, unicursal
signatures, labyrinths, pp. 170‑179.
On p. 174, he gives the five‑brick puzzle, asking for a route
along its edges.
Dudeney. Perplexities: No. 147: An old three‑line
puzzle. Strand Magazine 46 (Jul 1913)
110 & (Aug 1913) 221. c= AM,
prob. 239: A juvenile puzzle, pp. 68‑69 & 197. Five‑brick form to be drawn or rubbed
out on a board in three strokes. Either
way requires doing two lines at once, either by folding the paper as you draw
or using two fingers to rub out two lines at once. "I believe Houdin, the conjurer, was fond of showing this to
his child friends, but it was invented before his time -- perhaps in the Stone
Age."
Loyd. Problem of the bridges.
Cyclopedia, 1914, pp. 155 & 359‑360. = MPSL1, prob. 28, pp. 26‑27 & 130‑131. Eight bridges. Asks for number of routes.
Loyd. Puzzle of the letter carrier's route. Cyclopedia, 1914, pp 243 & 372. Asks for a circuit on a 3
x 4 array with a minimal length of
repeated path.
Dudeney. AM.
1917.
Prob.
242: The tube inspector's puzzle, pp. 69 & 198. Minimal route on a 3 x
4 array.
Prob.
261: The monk and the bridges, pp. 75-76 & 202-203. River with one island. Four bridges from island, two to each side
of the river, and another bridge over the river. How many Euler paths from a given side of the river to the
other? Answer: 16.
Collins. Book of Puzzles. 1927. The fly on the
octahedron, pp. 105-108. Asserts there
are 1488 Euler circuits on the edges of an octahedron. He counts the reverse as a separate circuit.
Harry Houdini [pseud. of Ehrich
Weiss] Houdini's Book of Magic. 1927 (??NYS); Pinnacle Books, NY, 1976, p.
19: Can you draw this? Take a square
inscribed in a circle and draw both diagonals.
"The idea is to draw the figure without taking your pencil off the
paper and without retracing or crossing a line. There is a trick to it, but it can be done. The trick in drawing the figure is to fold
the paper once and draw a straight line between the folded halves; then, not
removing your pencil, unfold the paper.
You will find that you have drawn two straight lines with one
stroke. The rest is simple." This perplexed me for some time, but I
believe the idea is that holding the pencil between the two parts of the folded
sheet and moving the pencil parallel to the fold, one can draw a line, parallel
to the fold, on each part.
Loyd Jr. SLAHP.
1928. Pp. 7‑8. Discusses what he calls the "Five‑brick
puzzle", the common pattern of five rectangles in a rectangle. He says that the object was to draw the
lines in four strokes -- which is easily done -- but that it was commonly
misprinted as three strokes, which he managed to do by folding the paper. He says "a similar puzzle ... some ten
or fifteen years ago" asked for a path crossing each of the 16 walls once,
which is also impossible.
The Bile Beans Puzzle Book. 1933.
No.
32. Draw the triangular array of three
on an edge without crossing.
No.
36. Draw the five-brick pattern in
three lines. Folds paper and draws two
lines at once.
R. Ripley. Believe It Or Not! Book 2. (Simon &
Schuster, 1931); Pocket Books, NY,
1948, pp. 70‑71. = Omnibus
Believe It Or Not! Stanley Paul, London, nd [c1935?], p. 270. Gives the five‑brick problem of
drawing a path crossing each wall once, with the trick solution having the path
going along a wall. Asserts "This
unicursal problem was solved thus by the great Euler himself." and cites
the Euler paper above!!
Meyer. Big Fun Book. 1940.
Tryangle,
pp. 98 & 731. Triangle subdivided
into triangles, with three small triangles along each edge. Draw an Euler circuit without crossings.
Cutting
the walls, pp. 637 & 794.
Five-brick problem. Solution has
line crossing through a vertex.
Ern Shaw. The Pocket Brains Trust - No. 2. W. H. Allen, London, nd but inscribed
1944. Prob. 29: Five bricks teaser, pp.
10 & 39.
Leeming. 1946.
Chap. 6, prob. 2: Through the walls, pp. 70 & 184. Five‑brick puzzle, with trick solution
having the path go through an intersection.
John Paul Adams. We Dare You to Solve This!. Op. cit. in 5.C. 1957? Prob. 49: In just
one line, pp. 30 & 48-49.
Five-brick puzzle, with answer having the path going along a wall, as in
Ripley. Asserts Euler invented this
solution.
Gibson. Op. cit. in 4.A.1.a. 1963.
Pp. 70 & 75: The "impossible" diagram. Same as Tom Tit.
Gardner. SA (Apr 1964) = 6th Book, chap. 10. Says Carroll knew that a planar Eulerian
graph could be drawn without crossings.
Gives a method of O'Beirne for doing this -- two colour the regions and
then make a path which separates the colours into simply connected regions.
Ripley's Puzzles and Games. 1966.
P. 39. Euler paths on the
'envelope', i.e. a rectangle with its diagonals drawn and an extra connection
between the top corners, looking like an unfolded envelope. Asserts the envelope has 50 solutions, but
it is not clear if the central crossing is a further vertex. I did this by hand but did not get 50, so I
wrote a program to count Euler paths.
If the central crossing is not a vertex, then I find 44 paths from one
of the odd vertices to the other, and of course 44 going the other way -- and I
had found this number by hand. However,
if the central crossing is a vertex, then my hand solution omitted some cases
and the computer found 120 paths from one odd vertex to the other.
Pp.
40-43 give many problems of drawing non-crossing Euler paths or circuits.
W. Wynne Willson. How to abolish cross‑roads. MTg 42 (Spring 1968) 56‑59. Euler circuit of a planar graph can be made
without crossings.
[Henry] Joseph & Lenore
Scott. Master Mind Brain Teasers. Tempo (Grosset & Dunlap), NY, 1973
(& 1978?? -- both dates are given -- I'm presuming the 1978 is a 2nd ptg or
a reissue under a different imprint??).
One line/no crossing, pp. 85-86.
Non-crossing Euler circuits on the triangular array of side 3 and
non-crossing Euler paths on the 'envelope' -- cf under Ripley's, above. Asserts the envelope has 50 solutions. I adapted the program mentioned above to
count the number of non-crossing Euler paths -- one must rearrange the first case
as a planar graph -- and there are
16 in the first case and 26
in the second case. Taking the
reversals doubles these numbers so it is possible that the Scotts meant the
second case and missed one path and its reversal.
David Singmaster, proposer; Jerrold W. Grossman & E. M. Reingold,
solvers. Problem E2897 -- An Eulerian
circuit with no crossings. AMM 88:7
(Aug 1981) 537-538 & 90:4 (Apr 1983) 287-288. A planar Eulerian graph can be drawn with no
crossings. Solution cites some previous
work.
Anon. [probably Will Shortz ??check with Shortz]. The impossible file. No. 2: In just one line. Games (Apr 1986) 34 & 64 & (Jul 1986) 64. Five brick pattern -- draw a line crossing
each wall once. Says it appeared in a
1921 newspaper [perhaps by Loyd Jr??].
Gives the 1921 solution where the path crosses a corner, hence two walls
at once. Also gives a solution with the
path going along a wall. In the July
issue, Mark Kantrowitz gives a solution by folding over a corner and also a
solution on a torus.
Marcia Ascher. Graphs in cultures: A study in
ethnomathematics. HM 15 (1988) 201‑227. Discusses the history of Eulerian circuits
and non-crossing versions and then exposits many forms of the idea in many
cultures.
Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10.
1991. Chapter Two: Tracing
graphs in the sand, pp. 30-65. Sketches
the history of Eulerian graphs with some interesting references -- ??NYS. Describes graph tracing in three cultures:
the Bushoong and the Tshokwe of central Africa and the Malekula of Vanuatu
(ex-New Hebrides). Extensive references
to the ethnographic literature.
This
section is mainly concerned with the theory.
The history of mazes is sketched first, with references to more detailed
sources. There is even a journal,
Caerdroia (53 Thundersley Grove, Thundersley, Essex, SS7 3EB, England), devoted
to mazes and labyrinths, mostly concentrating on the history. It is an annual, began in 1980 and issue 31
appeared in 2000.
Mazes
are considered under Euler Circuits, since the method of Euler Circuits is
often used to find an algorithm.
However, some mazes are better treated as Hamiltonian Circuits -- see
5.F.2.
A
maze can be considered as a graph formed by the nodes and paths -- the path
graph. For the usual planar maze, one
can also look at the graph formed by the walls -- the wall graph, which is a
kind of dual to the path graph. In
later mazes, the walls do not form a connected whole, and an isolated part of the
wall appears as a region or 'face' in the path graph. Such isolated bits of walling are sometimes called islands, but
they are the same as the components of the wall graph, with the outer wall
being one component, so the number of components is one more than the number of
islands. The 'hand-on-wall' method will
solve a maze if and only if the goals are adjacent to walls in the component of
the outer wall.
A
'ring maze' is a plate with holes and raised areas with an open ring which must
be removed by moving it from hole to hole.
I have put these in 11.K.5 as they are a kind of mechanical or
topological puzzle, though there are versions with a simple two legged spacer.
HISTORICAL
SOURCES
W. H. Matthews. Mazes & Labyrinths: A General Account of Their History and
Developments. Longmans, Green and Co.,
London, 1922. = Mazes and
Labyrinths: Their History and
Development. Dover, 1970. (21 pages of references.) [For more about the book and the author,
see: Zeta Estes; My Father, W. H.
Matthews; Caerdroia (1990) 6-8.]
Walter Shepherd. For Amazement Only. Penguin, 1942; Let's go amazing, pp. 5-12.
Revised as: Mazes and Labyrinths
-- A Book of Puzzles. Dover, 1961; Let's go a‑mazing, pp. v‑xi. (Only a few minor changes are made in the
text.) Sketch of the history.
Sven Bergling invented the
rolling ball labyrinth puzzle/game and they began being produced in 1946. [Kenneth Wells; Wooden Puzzles and Games;
David & Charles, Newton Abbot, 1983, p. 114.]
Walter Shepherd. Big Book of Mazes and Labyrinths. Dover, 1973, More amazement,
pp. vii-x. Extends the historical
sketch in his previous book, arguing that mazes with multiple choices perhaps
derive from Iron Age hill forts whose entrances were designed to confuse an
enemy.
Janet Bord. Mazes and Labyrinths of the World. Latimer, London, 1976. (Extensively illustrated.)
Nigel Pennick. Mazes and Labyrinths. Robert Hale, London, 1990.
Adrian Fisher [& Georg
Gerster (photographer)]. The Art of the
Maze. Weidenfeld and Nicolson, London,
1990. (Also as: Labyrinth; Solving the Riddle of the Maze;
Harmony (Crown Publishers), NY, 1990.)
Origins and History occupies pp. 11-56, but he also describes many
recent developments and innovations. He
has convenient tables of early examples.
Adrian Fisher & Diana Kingham. Mazes.
Shire Album 264. Shire,
Aylesbury, 1991.
Adrian Fisher & Jeff
Saward. The British Maze Guide. Minotaur Designs, St. Alban's, 1991.
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HISTORICAL
SKETCH
Up
to about the 16C, all mazes were unicursal, i.e. with no decision points. The word labyrinth is sometimes used to
distinguish unicursal mazes from others, but this distinction is not made consistently. Until about 1000, all mazes were of the
classical 'Cretan' seven-ring type shown above. (However, see Shepherd's point in his 1973 book, above.) The oldest examples are rock carvings, the
earliest being perhaps that in the Tomba del Labirinto at Luzzanas, Sardinia, c‑2000
[Fisher, pp. 12, 25, 26, with photo on p. 12].
(In fact, Luzzanas is a local name for an uninhabited area of fields, so
does not appear on any ordinary map. It
is near Benetutti. See my A
Mathematical Gazetteer or Mazing in Sardinia (Caerdroia 30 (1999) 17-21). Jeff Saward writes that current
archaeological feeling is that the maze is Roman, though the cave is probably
c-2000.) On pottery, there are
labyrinths on fragments, c‑1300, from Tell Rif'at, Syria [the first photos
of this appeared in [Caerdroia 30 (2000) 54-55]), and on tablets, c‑1200,
from Pylos. Fisher [p. 26] lists
the early examples. Staffen Lundán; The
labyrinth in the Mediterranean; Caerdroia 27 (1996) 28-54, catalogues all known
'Cretan' labyrinths from prehistory to the end of antiquity, c250, excluding
the Roman 'spoked' form. All these
probably had some mystical significance about the difficulty of reaching a
goal, often with substantial mythology -- e.g. Theseus in the Labyrinth or,
later, the Route to Jerusalem.
Roman
mosaics were unicursal but essentially used the Cretan form four times over in
the four corners. Lundán, above, calls
these 'spoked'. Most of the extant
examples are 2C‑4C, but some BC examples are known -- the earliest seems
to be c-110 at Selinunte, Sicily.
Fisher [pp. 36-37] lists all surviving examples. Saward says the earliest Roman example is at
Pompeii, so £ 79.
In
the medieval period, the Christians developed a quite different unicursal
maze. See Fisher [pp. 60-67] for
detailed comparison of this form with the Roman and Cretan forms. The earliest large Christian example is the
Chemin de Jerusalem of 1235 on the floor of Chartres Cathedral. Fisher [pp. 41 & 48] lists early and
later Christian examples.
The
legendary Rosamund's Bower was located in Woodstock Park, Oxfordshire, and its
purported site is marked by a well and fountain. It was some sort of maze to conceal Rosamund Clifford, the
mistress of Henry II (1133‑1189), from the Queen, Eleanor of
Aquitaine. Legend says that about 1176,
Eleanor managed to solve the maze and confronted Rosamund with the choice of a
dagger or poison -- she drank the poison and Henry never smiled again. [Fisher, p. 105]. Historically, Henry had imprisoned Eleanor for fomenting rebellion
by her sons and Rosamund was his acknowledged mistress. Rosamund probably spent her last days at a
nunnery in Godstow, near Oxford. The
legend of the bower dates from the 14C and her murder is a later addition
[Collins, Book of Puzzles, 1927, p. 121.]
In the 19C, many puzzle collections had a maze called Rosamund's Bower.
The
earliest record of a hedge maze is of one destroyed in a siege of Paris in
1431.
Non-unicursal
mazes and islands in the wall graph start to appear in the late 16C. Matthews [p. 96] says that: "A simple
"interrupted-circle" type of labyrinth was adopted as a heraldic
device by Gonzalo Perez, a Spanish ecclesiastic ... and published ... in 1566
..." in his translation of the Odyssey.
Matthews doesn't show this, but he then [pp. 96-97] describes and
illustrates a simple maze used as a device by Bois-dofin de Laval, Archbishop
of Embrum. He copies it from Claude
Paradin; Devises Héroiques et Emblèmes of the early 17C. It has four entrances and possibly three
goals, with walls having 8 components, two being part of the outer wall. The central goals is accessible from two of
the entrances, but the two minor goals are each accessible from just one of the
other entrances. Presumably this sort
of thing is what Matthews meant as an "interrupted circle".
However,
Saward has found a mid 15C anonymous English poem, The Assembly of Ladies,
which describes the efforts of a group of ladies to reach the centre of a maze,
which, as he observes, implies there must be some choices involved.
[Matthews,
p. 114] has three examples from a book by Androuet du Cerceau; Les Plus
Excellents Bastiments de France of 1576.
Fig. 82 was in the gardens at Charleval and has four entrances, only one
of which goes to the central goal.
There are four minor goals. The
N entrance connects to the NE and SE goals, with several dead ends. The E entrance is a dead end. The S entrance goes to the SW goal. The W entrance goes to the central goal, but
the NW goal is on an island, though 'left-hand-on-wall' goes past it. Figs. 83 and 84 are essentially identical
and seem to be corruptions of unicursal examples so that most of the maze is
bypassed. In fig. 84, one has to walk
around to the back of the maze to find the correct entrance to get to the
central goal, which is an interesting idea.
A small internal change in both cases and moving the entrances converts
them to a standard unicursal pattern.
Matthews'
Chap. XIII [pp. 100-109] is on floral mazes and reproduces some from Jan
Vredeman De Vries; Hortorum Viridariorumque Formae; Antwerp, 1583. Fig. 74 is one of these and has two
components and a short dead-end, but the 'hand-on-wall' rule solves it. Fig. 73 is another of De Vries's, but
it is not all shown. It appears to have
two entrances and there is certainly a decision point by the far gate, but one
route goes to the apparent exit at the bottom of the page. There is a small dead end near the central
goal. Fig. 78 shows a maze from a 17C
manuscript book in the Harley Manuscripts at the BL, identified on p. 224 as
BM Harl. 5308 (71, a, 12). This
has two components with the central goal in the inner component, so the
'hand-on-wall' rule fails, but it brings you within sight of the centre and
Matthews describes it as unicursal!
Fig. 79 is from Adam Islip; The Orchard and the Garden, compiled from
continental sources and published in 1602.
It has 5 components, but four of these are small enclosures which could
be considered as minor goals, especially if they had seats in them. The 'hand-on-wall' rule gets to the central
goal. There is a lengthy dead end which
goes to two of the inner islands. Fig.
80 is from a Dutch book: J. Commelyn; Nederlantze Hesperides of 1676. It has two components, a central goal and
four minor goals. The 'hand-on-wall'
gets you to the centre and passes two minor goals. One minor goal is on a dead end so 'left-hand-on-wall' gets to
it, but 'right-hand-on-wall' does not.
The fourth minor goal is on the island.
At
Versailles, c1675, André Le Nôtre built a Garden Maze, but the objective was to
visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at least one
fountain. Some fountains were not at
path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian
problem, except that one fountain was located at the end of a short dead
end. [Fisher, pp. 49, 79, 130 &
144-145, with contemporary map on p. 144.
Fisher says there are 39 fountains, and the map has 40. Close examination shows that the map counts
two statues at the entrance but omits to count a fountain between numbers 37
and 38. Matthews, pp. 117‑121,
says it was built by J. Hardouin-Mansart and his map has 39 fountains.] It has a main entrance and exit but there is
another exit, so the perimeter wall already has three components, and there are
14 other components. Sadly, it was
destroyed in 1775.
Several
other mazes, of increasing complexity, occur in the second half of the 17C
[Matthews, figs. 93-109, opp. p. 120 - p. 127]. Several of these could be from 20C maze books. Fig. 94, designed for Chantilly by Le Nôtre,
is surprisingly modern in that there are eight paths spiralling to the
centre. The entrance path takes you
directly to the centre, so the real problem is getting back out! One of the mazes presently at Longleat has
this same feature.
The
Hampton Court Maze, planted c1690, is the oldest extant hedge maze and one of
the earliest puzzle mazes.
([Christopher Turner; Hampton Court, Richmond and Kew Step by Step;
(As part of: Outer London Step by Step, Faber, 1986); Revised and published in
sections, Faber, 1987, p. 16] says the present shape was laid out in 1714,
replacing an earlier circular shape, but I haven't seen this stated
elsewhere.) Matthews [p. 128] says it
probably replaced an older maze. It has
dead ends and one island, i.e. the graph has two components, though the 'hand
on wall' rule will solve it.
The
second Earl Stanhope (1714-1786) is believed to be the first to design mazes
with the goal (at the centre) surrounded by an island, so that the 'hand on
wall' rule will not solve it. It has
seven components and only a few short dead ends.. The fourth Earl planted one of these at Chevening, Kent, in c1820
and it is extant though not open to the public. [Fisher, p. 71, with photo on p. 72 and diagram on p. 73.] However, investigation in Matthews revealed
the earlier examples above. Further
Bernhard Wiezorke (below at 2001) has found a hedge maze in Germany, dating
from c1730, which is not solved by the 'hand on wall' rule. This maze has 12 components.
In
1973, Stuart Landsborough, an Englishman settled at Wanaka, South Island, New
Zealand, began building his Great Maze.
This was the first of the board mazes designed by Landsborough which
were immensely popular in Japan. Over
200 were built in 1984-1987, with 20 designed by Landsborough. Many of these were three dimensional -- see
below. About 60 have been demolished
since then. [Fisher, pp. 78‑79
& 118-121 has 6 colour photos, pp. 156-157 lists Landsborough's designs.]
If Minos' labyrinth ever really
existed, it may have been three dimensional and there may have been garden
examples with overbridges, but I don't know of any evidence for such early
three dimensional mazes. Lewis Carroll
drew mazes which had paths that crossed over others making a simple three
dimensional maze, in his Mischmasch of c1860, see below. John Fisher [The Magic of Lewis Carroll;
(Nelson, 1973), Penguin, 1975, pp. 19-20] gives this and another example. Are there earlier examples? Boothroyd & Conway, 1959, seems to be
the earliest cubical maze. Much more
complex versions were developed by Larry Evans from about 1970 and published in
a series of books, starting with 3-Dimensional Mazes (Troubador Press, San
Francisco, 1976). His 3‑Dimensional
Maze Art (Troubador, 1980) sketches some general history of the maze and
describes his development of pictures of three dimensional mazes. The first actual three dimensional maze
seems to be Greg Bright's 1978 maze at Longleat House, Warminster. [Fisher, pp. 74, 76, 94-95 & 152-153,
with colour photos on pp. 94-95.]
Since then, Greg Bright, Adrian Fisher, Randoll Coate, Stuart
Landsborough and others have made many innovations. Bright seems to have originated the use of colour in mazes c1980
and Fisher has extensively developed the idea.
[Fisher, pp. 73-79.]
Abu‘l-Rayhan Al‑Biruni (=
’Abû-alraihân [the h should have an underdot] Muhammad ibn ’Ahmad
[the h
should have an underdot] Albêrûnî).
India. c1030. Chapter XXX. IN: Al‑Beruni's India, trans. by E. C. Sachau, 2 vols.,
London, 1888, vol. 1, pp. 306-307
(= p. 158 of the Arabic ed., ??NYS). In
describing a story from the fifth and sixth books of the Ramayana, he says that
the demon Ravana made a labyrinthine fortress, which in Muslim countries
"is called Yâvana-koti, which has been frequently explained as
Rome." He then gives "the
plan of the labyrinthine fortress", which is the classical Cretan
seven-ring form. Sachau's notes do not
indicate whether this plan is actually in the Ramayana, which dates from
perhaps -300.
Pliny. Natural History.
c77. Book 36, chap. 19. This gives a brief description of boys
playing on a pavement where a thousand steps are contained in a small
space. This has generally been
interpreted as referring to a maze, but it is obviously pretty vague. See: Michael Behrend; Julian and Troy names;
Caerdroia 27 (1996) 18-22, esp. note 5 on p. 22.
Pacioli. De Viribus.
c1500. Part II: Cap.
(C)XVII. Do(cumento). de saper fare
illa berinto con diligentia secondo Vergilio, f. 223v = Peirani 307-308. A
sheet (or page) of the MS has been lost.
Cites Vergil, Æneid, part six, for the story of Pasiphæ and the
Minotaur, but the rest is then lost.
Sebastiano Serlio. Architettura, 5 books, 1537-1547. The separate books had several editions
before they were first published together in 1584. The material of interest is in Book IV which shows two unicursal
mazes for gardens. I have seen the
following.
Tutte
l'Opere d'Architetture et Prospetiva, ....
Giacomo de'Franceschi, Venice, 1619;
facsimile by Gregg Press, Ridgewood, New Jersey, 1964. F. 199r shows the designs and f. 197v has
some text, partly illegible in my photocopy.
[Cf Caerdroia 30 (1999) 15.]
Sebastiano
Serlio on Architecture Volume One Books I-V of 'Tutte l'Opere d'Architettura
et Prospetiva'. Translated and edited
by Vaughan Hart and Peter Hicks. Yale
Univ. Press, New Haven, 1996. P. 388
shows the designs and p. 389 has the text, saying these 'are for the
compartition of gardens'. The sidenotes
state that these pages are ff. LXXVr and LXXIIIIr of the 3rd ed. of 1544 and
ff. 198v-199r and 197v-198r of the 1618/19 ed.
William Shakespeare. A Midsummer Night's Dream. c1610.
Act II, scene I, lines 98-100:
"The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton
green For lack of tread are
undistinguishable." Fiske 126
opines that the latter two lines may indicate that the board was made in the
turf, though he admits that they may refer just to dancers' tracks, but to me
it clearly refers to turf mazes.
John Cooke. Greene's Tu Quoque; or the Cittie Gallant; a
Play of Much Humour. 1614. ??NYS -- quoted by Matthews, p. 135. A challenge to a duel is given by Spendall
to Staines.
Staines. I accept it ; the meeting place?
Spendall. Beyond the maze in Tuttle.
This
refers to a maze in Tothill Fields, close to Westminster Abbey.
Lewis Carroll. Untitled maze. In: Mischmasch, the last of his youthful MS magazines, with
entries from 1855 to 1862. Transcribed
version in: The Rectory Umbrella and
Mischmasch; Cassell, 1932; Dover, 1971; p. 165 of the Dover ed. John Fisher [The Magic of Lewis Carroll;
(Nelson, 1973), Penguin, 1975, pp. 19-20] gives this and another example. Cf Carroll-Wakeling, prob. 35: An amazing
maze, pp. 46-47 & 75 and Carroll-Gardner, pp. 80-81 for the Mischmasch
example. I don't find the other example
elsewhere, but it was for Georgina "Ina" Watson, so probably c1870.
Mittenzwey. 1880.
Prob. 281, pp. 50 & 100;
1895?: 310, pp. 53-54 & 102;
1917: 310, pp. 49 & 97.
The garden of a French place has a maze with 31 points to see. Find a path past all of them with no
repeated edges and no crossings. The
pattern is clearly based on the Versailles maze of c1675 mentioned in the Historical
Sketch above, but I don't recall the additional feature of no crossings
occurring before.
C. Wiener. Ueber eine Aufgabe aus der Geometria
situs. Math. Annalen 6 (1873) 29‑30. An algorithm for solving a maze. BLW asserts this is very complicated, but it
doesn't look too bad.
M. Trémaux. Algorithm.
Described in Lucas, RM1, 1891, pp. 47‑51. ??check 1882 ed. BLW assert Lucas' description is faulty. Also described in MRE, 1st ed., 1892, pp. 130‑131; 3rd ed., 1896, pp. 155-156; 4th ed., 1905, pp. 175-176 is vague; 5th-10th ed., 1911‑1922, 183; 11th ed., 1939, pp. 255‑256 (taken
from Lucas); (12th ed. describes
Tarry's algorithm instead) and in Dudeney, AM, p. 135 (= Mazes, and how to
thread them, Strand Mag. 37 (No. 220) (Apr 1909) 442‑448, esp. 446‑447).
G. Tarry. Le problème des labyrinthes. Nouv. Annales de Math. (3) 4 (1895) 187‑190. ??NYR
Collins. Book of Puzzles. 1927. How to thread any
maze, pp. 122-124. Discusses right hand
rule and its failure, then Trémaux's method.
M. R. Boothroyd &
J. H. Conway. Problems drive,
1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 2. 5 x 5 x 5 cubical
maze. Get from a corner to an antipodal
corner in a minimal number of steps.
Anneke Treep. Mazes... How to get out! (part I).
CFF 37 (Jun 1995) 18-21. Based
on her MSc thesis at Univ. of Twente.
Notes that there has been very little systematic study. Surveys the algorithms of Tarry, Trémaux,
Rosenstiehl. Rosenstiehl is greedy on
new edges, Trémaux is greedy on new nodes and Trémaux is a hybrid of
these. ??-oops-check. Studies probabilities of various routes and
the expected traversal time. When the
maze graph is a tree, the methods are equivalent and the expected traversal
time is the number of edges.
Bernhard Wiezorke. Puzzles und Brainteasers. OR News, Ausgabe 13 (Nov 2001) 52-54. This reports his discovery of a hedge maze
in Germany -- the first he knew of. It
is in Altjessnitz, near Dessau in Sachsen-Anhalt. (My atlas doesn't show such a place, but Jessnitz is about 10km
south of Dessau.) This maze dates from
1720 and has 12 components, with the goal completely separated from the outside
so that the 'hand on wall' rule does not solve it. Torsten Silke later told Wiezorke of two other hedge mazes in
Germany. One, in Probststeierhagen,
Schleswig-Holstein, about 12km NE of Kiel, is in the grounds of the restaurant
Zum Irrgarten (At the Labyrinth) and is an early 20C copy of the Altjessnitz
example. The other, in Kleinwelka,
Sachsen, about 50km NE of Dresden, was made in 1992 and is private. Though it has 17 components, the 'hand on
wall' method will solve it. He gives
plans of both mazes. He discusses the
Seven Bridges of Königsberg, giving a B&W print of the 1641 plan of the
city mentioned at the beginning of Section 5.E -- he has sent me a colour
version of it. He also describes
Tremaux's solution method.
5.E.2. MEMORY WHEELS = CHAIN CODES
These
are cycles of 2n 0s
and 1s such that each n‑tuple
of 0s
and 1s appears just once. They
are sometimes called De Bruijn sequences, but they have now been traced back to
the late 19C. An example for n = 3
is 00010111.
Émile Baudot. 1884.
Used the code for 25 in telegraphy. ??NYS -- mentioned by Stein.
A. de Rivière, proposer; C. Flye Sainte-Marie, solver. Question no. 58. L'Intermédiare des Mathématiciens 1 (1894) 19-20 & 107-110. ??NYS -- described in Ralston and
Fredricksen (but he gives no. 48 at one point). Deals with the general problem of a cycle of kn symbols such that every n‑tuple
of the k basic symbols occurs just once.
Gives the graphical method and shows that such cycles always exist and
there are k!g(n)/ kn of them,
where g(n) = kn‑1. This work was unknown to the following authors until about 1975.
N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch. Proc. 49 (1946) 758‑764. ??NYS -- described in Ralston and
Fredricksen. Gives the graphical method
for finding examples and finds there are
2f(n) solutions,
where f(n) = 2n-1
- n.
I. J. Good. Normal recurring decimals. J. London Math. Soc. 21 (1946) 167-169. ??NYS -- described in Ralston and
Fredricksen. Shows there are solutions
but doesn't get the number.
R. L. Goodstein. Note 2590:
A permutation problem. MG 40
(No. 331) (Feb 1956) 46‑47.
Obtains a kind of recurrence for consecutive n‑tuples.
Sherman K. Stein. Mathematics: The Man‑made Universe.
Freeman, 1963. Chap. 9: Memory
wheels. c= The mathematician as
explorer, SA (May 1961) 149‑158.
Surveys the topic. Cites the
c1000 Sanskrit word: yamátárájabhánasalagám used as the mnemonic for 01110100(01) giving all triples of short and long beats in Sanskrit poetry and
music. Describes the many reinventions,
including Baudot (1882), ??NYS, and the work of Good (1946), ??NYS, and de
Bruijn (1946), ??NYS. 15 references.
R. L. Goodstein. A generalized permutation problem. MG 54 (No. 389) (Oct 1970) 266‑267. Extends his 1956 note to find a cycle
of an symbols such that the n‑tuples are distinct.
Anthony Ralston. De Bruijn sequences -- A model example of
the interaction of discrete mathematics and computer science. MM 55 (1982) 131‑143 & cover. Deals with the general problem of cycles
of kn symbols such that every n‑tuple of the k
basic symbols occurs just once.
Discusses the history and various proofs and algorithms which show that
such cycles always exist. 27
references.
Harold Fredricksen. A survey of full length nonlinear shift
register cycle algorithms. SIAM Review
24:2 (Apr 1982) 195-221. Mostly about
their properties and their generation, but includes a discussion of the door
lock connection, a mention of using the
23 case as a switch
for three lights, and gives a good history.
The door lock connection is that certain push button door locks will
open when a four digit code is entered, but they open if the last four buttons
pressed are the correct code, so using a chain code reduces the number of
button pushes required by a burglar to
1/4 of the number required if he
tries all four digit combinations. 58
references.
At G4G2, 1996, Persi Diaconis
spoke about applications of the chain code in magic and mentioned uses in
repeated measurement designs, random number generators, robot location, door
locks, DNA comparison.
They
were first used in card tricks by Charles T. Jordan in 1910. Diaconis' example had a deck of cards which
were cut and then five consecutive cards were dealt to five people in a
row. He then said he would determine
what cards they had, but first he needed some help so he asked those with red
cards to step forward. The position of
the red cards gives the location of the five cards in a cycle of 32
(which was the size of the deck)!
Further, there are simple recurrences for the sequence so it is fairly
easy to determine the location. One can
code the binary quintuples to give the suit and value of the first card and
then use the succeeding quintuples for the succeeding cards.
Long
versions of the chain code are printed on factory floors so that a robot can
read it and locate itself.
In Jan 2000, I discussed the
Sanskrit chain code with a Sanskrit scholar, Dominik Wujastyk, who said that
there is no known Sanskrit source for it.
He has asked numerous pandits who did not know of it and he said there
is is a forthcoming paper on it, but that it did not locate any Sanskrit
source.
Haubrich's
1995-1996 surveys, op. cit. in 5.H.4, include this.
B. Astle. Pantactic squares. MG 49 (No. 368) (May 1965) 144‑152. This is a two‑dimensional version of
the memory wheel. Take a 5 x 5
array of cells marked 0 or
1 (or Black or White). There are
16 ways to take a 2 x 2
subarray from the 5 x 5 array.
If these give all 16 2 x 2
binary patterns, the array is called pantactic. The author shows a number of properties and
some types of such squares.
C. J. Bouwkamp, P. Janssen &
A. Koene. Note on pantactic
squares. MG 54 (No. 390) (Dec 1970) 348‑351. They find 800 such squares, forming 50
classes of 16 forms.
[Surprisingly, neither paper
considers a 4 x 4 array viewed toroidally, which is the
natural generalization of the memory wheel.
Precisely two of the fifty classes, namely nos. 25 & 41, give such a
solution and these are the same pattern on the torus. One can also look at the
4 x 4 subarrays of a 131 x 131
or a 128 x 128 array, etc., as well as 3 and higher
dimensional arrays. I submitted the
question of the existence and numbers of these as a problem for CM, but it was
considered too technical.]
Ivan Moscovich. US Patent 3,677,549 -- Board Game
Apparatus. Applied: 14 Jun 1971; patented: 18 Jul 1972. Front page, 1p diagrams, 2pp text. Reproduced in Haubrich, About ..., 1996, op.
cit. in 5.H.4. 2pp + 2pp diagrams. This uses the 16 2 x 2 binary patterns as game pieces. He allows the pieces to be rotated, scoring
different values according to the orientation.
No mention of reversing pieces or of the use of the pieces as a puzzle.
John Humphries. Review of Q-Bits. G&P 54 (Nov 1976) 28.
This is Moscovich's game idea, produced by Orda. Though he mentions changing the rules to
having non-matching, there is no mention of two-sidedness.
Pieter van Delft & Jack
Botermans. Creative Puzzles of the
World. (As: Puzzels uit de hele wereld; Spectrum Hobby, 1978); Harry N. Abrams, NY, 1978. The colormatch square, p. 165. See Haubrich,1994, for description.
Jacques Haubrich. Pantactic patterns and puzzles. CFF 34 (Oct 1994) 19-21. Notes the toroidal property just mentioned. Says Bouwkamp had the idea of making the 16
basic squares in coloured card and using them as a MacMahon-type puzzle, with
the pieces double-sided and such that when one side had MacMahon matching, the
other side had non-matching. There are
two different bijections between matching patterns and non-matching patterns,
so there are also 800 solutions in 50 classes for the non-matching
problem. Bouwkamp's puzzle appeared in
van Delft & Botermans, though they did not know about and hence did not
mention the double-sidedness. [In an
email of 22 Aug 2000, Haubrich says he believes Bouwkamp did tell van Delft and
Botermans about this, but somehow it did not get into their book.] The idea was copied by two manufacturers
(Set Squares by Peter Pan Playthings and Regev Magnetics) who did not
understand Bouwkamp's ideas -- i.e. they permitted pieces to rotate. Describes Verbakel's puzzle of 5.H.2.
Jacques Haubrich. Letter: Pantactic Puzzles = Q-Bits. CFF 37 (Jun 1995) 4. Says that Ivan Moscovich has responded that
he invented the version called "Q-Bits" in 1960-1964, having the same
tiles as Bouwkamp's (but only one-sided [clarified by Haubrich in above
mentioned email]). His US Patent
3,677,549 (see above) is for a game version of he idea. The version produced by Orda Ltd. was
reviewed in G&P 54 (Nov 1976) (above).
So it seems clear that Moscovich had the idea of the pieces before
Bouwkamp's version was published, but Moscovich's application was to use them
in a game where the orientations could be varied.
For
queen's, bishop's and rook's tours, see 6.AK.
A
tour is a closed path or circuit.
A
path has end points and is sometimes called an open tour.
5.F.1. KNIGHT'S TOURS AND PATHS
GENERAL
REFERENCES
Antonius van der Linde. Geschichte und Literatur des
Schachspiels. (2 vols., Springer, Berlin,
1874); one vol. reprint, Olms, Zürich,
1981. [There are two other van der
Linde books: Quellenstudien zur
Geschichte des Schachspiels, Berlin, 1881, ??NYS; and Das Erste Jartausend
[sic] der Schachlitteratur (850‑1880),
(Berlin, 1880); reprinted with some notes and corrections, Caissa Limited
Editions, Delaware, 1979, which is basically a bibliography of little use
here.]
Baron Tassilo von Heydebrand und
von der Lasa. Zur Geschichte und
Literatur des Schachspiels.
Forschungen. Leipzig, 1897. ??NYS.
Ahrens. MUS I.
1910. Pp. 319-398.
Harold James Ruthven
Murray. A History of Chess. OUP, 1913;
reprinted by Benjamin Press, Northampton, Massachusetts, nd [c1986]. This has many references to the problem,
which are detailed below.
Reinhard Wieber. Das Schachspiel in der arabischen Literatur
von den Anfängen bis zur zweiten Hälfte des 16.Jahrhunderts. Verlag für Orientkunde Dr. H. Vorndran,
Walldorf‑Hessen, 1972.
George P. Jelliss.
Special
Issue: Notes on the Knight's Tour. Chessics 22 (Summer 1985) 61‑72.
Further
notes on the knight's tour. Chessics 25
(Spring 1986) 106‑107.
Notes
on Chessics 22 continued. Chessics 29
& 30 (1987) 160.
This is a progress report on his
forthcoming book on the knight's tour.
I will record some of his comments at the appropriate points below. He also studies the 3 x n
board extensively.
Two
problems with knights on a 3 x 3 board are generally treated here, but cf
5.R.6.
The
4 knights problem has two W and two B knights at the corners (same colours at
adjacent corners) and the problem is to exchange them in 16 moves. The graph of knight's connections is an
8-cycle with the pieces at alternate nodes.
[Putting same colours at opposite corners allows a solution in 8 moves.]
The
7 knights problem is to place 7 knights on a
3 x 3 board in the 4 corners and
3 of the sides so each is a knight's move from the previously placed one. This is equivalent to the octagram puzzle of
5.R.6.
4
knights problem -- see: at‑Tilimsâni,
1446; Civis Bononiae, c1475;
7
knights problem -- see: King's Library
MS.13, A.xviii, c1275; "Bonus
Socius", c1275; at‑Tilimsâni,
1446;
Al‑Adli (c840) and as‑Suli
(c880‑946) are the first two great Arabic chess players. Although none of their works survive, they
are referred to by many later writers who claim to have used their material.
Rudraţa:
Kāvyālaʼnkāra [NOTE:
ţ denotes a t
with an underdot and ʼn denotes an
n with an overdot.]. c900.
??NYS -- described in Murray 53‑55, from an 1896 paper by Jacobi,
??NYS. The poet speaks of verses which
have the shapes of "wheel, sword, club, bow, spear, trident, and plough,
which are to be read according to the chessboard squares of the chariot [=
rook], horse [= knight], elephant [c= bishop], &c." According to Jacobi, the poet placed
syllables in the cells of a half chessboard so that it reads the same straight
across as when following a piece's path.
With help from the commentator Nami, of 1069, the rook's and knight's
path's are reconstructed, and are given on Murray 54. Both are readily extended to full board paths, but not
tours. The elephant's path is confused.
Kitâb ash‑shatranj mimma’l‑lafahu’l‑‘Adli
waş‑Şûlî wa ghair‑huma [Book of the Chess; extracts from
the works of al‑'Adlî, aş‑Şûlî and others]. [NOTE:
ş, Ş denote
s, S with underdot.] Copied by
Abû Ishâq [the h should have an underdot] Ibrâhîm ibn al‑Mubârak
ibn ‘Alî al‑Mudhahhab al‑Baghdâdî.
Murray 171‑172 says it is MS ‘Abd‑al‑Hamid
[the H
should have an underdot] I, no. 560, of 1140, and denotes it AH. Wieber 12‑15 says it is now MS Lala
Ismail Efendi 560, dates it July‑August 1141, and denotes it L. Both cite van der Linde, Quellenstudien, no.
xviii, p. 331+, ??NYS. The author is
unknown. This MS was discovered in
1880. Catalogues in Istanbul listed it
as Risâla fi’sh‑shaţranj by Abû’l‑‘Abbâs Ahmad [the h
should have an underdot] al‑‘Adlî. It is sometimes attributed to al‑Lajlâj who wrote one short
section of this book. Murray, van der
Linde and Wieber (p. 41) cite another version:
MS Khedivial Lib., Cairo, Mustafa Pasha, no. 8201, copied c1370, which
Murray denotes as C and Wieber lists as unseen.
Murray
336 gives two distinct tours: AH91 & AH92.
The solution of AH91 is a numbered diagram, but AH92 is 'solved' four times
by acrostic poems, where the initial letters of the lines give the tour in an
algebraic notation. Wieber 479‑480
gives 2 tours from ff. 74a‑75b: L74a = AH91 and L74b = reflection of
AH92. [Since the 'solutions' of AH92
are poetic, it is not unreasonable to consider the reflection as
different.] Also AH94 = L75b is a
knight/bishop tour, where moves of the two types alternate. These tours may be due to as‑Suli. AH196 is a knight/queen tour.
Arabic MS Atif Efendi 2234
(formerly Vefa (‘Atîq Efendî) 2234), Eyyub, Istanbul. Copied by Muhammad [the
h should have an underdot] ibn
Hawâ (or Rahwâr -- the MS is obscure) ibn ‘Othmân al‑Mu’addib in
1221. Murray 174‑175 describes it
as mostly taken from the above book and denotes it V. A tour is shown on p. 336 as V93 = AH92. Wieber 20‑24 denotes it A. On p. 479, he shows the tour from f. 68b
which is the same as L74b, the reflection of AH92.
King's Library MS.13, A.xviii,
British Museum, in French, c1275. Described in van der Linde I 305‑306. Described and transcribed in Murray 579‑582
& 588‑600, where it is denoted as K.
Van der Linde discusses the knight's path on I 295, with diagram no. 244
on p. 245. Murray 589 gives the text
and a numbered diagram of a knight's path as K1. The path splits into two half board paths: a1 to d1 and e3 to h1, so the first half and
the whole are corner to corner. The
first half is also shown as diagram K2 with the half board covered with pieces
and the path described by taking of pieces.
K3 is the 7 knights problem
"Bonus Socius"
[perhaps Nicolas de Nicolaï]. This is
the common name of a collection of chess problems, assembled c1275, which was
copied and translated many times. See
Murray 618‑642 for about 11 MSS.
Some of these are given below.
Fiske 104 & 110‑111 discusses some MSS of this collection.
MS
Lat. 10286, Nat. Lib., Paris.
c1350. Van der Linde I 293‑295
describes this but gives the number as 10287 (formerly 7390). Murray 621 describes it and denotes it
PL. Van der Linde describes a half board
knight's path, with a diagram no. 243 shown on p. 245. The description indicates a gap in the path
which can only be filled in one way.
This is a path from a8 to h8 which cannot be extended to the full
board. Murray 641 says that PL275
is the same as problems in two similar MSS and as CB244, diagrammed on p.
674. However, this is not the same as
van der Linde's no. 243, though cells 1‑19 and 31‑32 are the same
in both paths, so this is also an a8 to h8 path which does not extend to a full
board.
Murray
620 mentions a path in a late Italian MS version of c1530 (Florence, Nat. Lib.
XIX.7.51, which he denotes It) which may be the MS described by van der Linde
I 284 as no. 4 and the half board path described on I 295 with diagram no.
245 on I 245. Fiske 210-211 describes
this and says von der Lasa 163-165 (??NYS) describes it as early 16C, but
Murray does not mention von der Lasa.
Fiske says it contains a tour on f. 28b, which von der Lasa claims
is "das älteste beispiel eines vollkommenen rösselsprunges", but
Murray does not detail the problems so I cannot compare these citations. Fiske also says it also contains the 7
knights problem.
Dresden MS 0/59, in French,
c1400. Murray describes this on pp. 607‑613
and denotes it D. On p. 609, Murray
describes D57 which asks for a knight's path on a 4 x 4 board. No solution is given -- indeed this is
impossible, cf Persian MS 211 in the RAS.
Ibid. is D62 which asks for a half board tour, but no answer is
provided.
Persian MS 211 in Royal Asiatic
Society. Early 15C. ??NYS.
Extensively
described as MS 250 bequeathed by Major David Price in: N. Bland; On the Persian game of chess; J.
Royal Asiatic Soc. 13 (1852) 1‑70.
He dates it as 'at least 500 years old' and doesn't mention the knight's
tour.
Described,
as MS No. 260, and partially translated in Duncan Forbes; The History of Chess;
Wm. H. Allen, London, 1860. Forbes says
Bland's description is "very detailed but unsatisfactory". On p. 82 is the end of the translation of
the preface: '"Finally I will show
you how to move a Knight from any individual square on the board, so that he
may cover each of the remaining squares in as many moves and finally come to
rest on that square whence he started.
I will also show how the same thing may be done by limiting yourself
only to one half, or even to one quarter (1) of the board." -- Here the
preface abruptly terminates, the following leaf being lost.' Forbes's footnote (1) correctly doubts that
a knight's tour (or even a knight's path) is possible on the 4 x 4
board.
Murray
177 cites it as MS no. 211 and denotes it RAS.
He says that it has been suggested that this MS may be the work of
‘Alâ'addîn Tabrîzî = ‘Alî ash‑Shatranjî, late 14C, described on Murray
171. Murray mentions the knight's tour
passage on p. 335. This may be in van
der Linde, ??NX. Wieber 45 mentions the
MS.
Abû Zakarîyâ Yahya [the h
should have an underdot] ibn Ibrâhîm al‑Hakîm[the H
should have an underdot]. Nuzhat
al‑arbâb al‑‘aqûl fî’sh‑shaţranj [NOTE: ţ
denotes a t with an underdot] al‑manqûl (The
delight of the intelligent, a description of chess). Arabic MS 766, John Rylands Library, Manchester.
Bland,
loc. cit., pp. 27‑28, describes this as no. 146 of Dr. Lee's catalogue
and no. 76 of the new catalogue.
Forbes, loc. cit., says that Dr. Lee had loaned his two MSS to someone
who had not yet returned them, so Forbes copies Bland's descriptions (on
pp. 27‑31) as his Appendix C, with some clarifying notes. (The other of Dr. Lee's MSS is described
below.) Van der Linde I 107ff (??NX)
seems to copy Bland & Forbes.
Murray
175‑176 describes it as Arab. 59 at John Rylands Library and denotes it
H. He says it was Bland who had
borrowed the MSS from Dr. Lee and Murray traces their route to Dr. Lee and to
Manchester. Murray says it is late 15C,
is based on al‑Adli and as‑Suli and he also describes a later
version, denoted Z, late 18C. Wieber 32‑35
cites it as MS 766(86) at John Rylands, dates it 1430 and denotes it Y1.
Murray
336 gives three paths. H73 = H75 are
the same tour, but with different keys, one poetic as in Rudraţa
[NOTE: ţ denotes a t with an underdot.], one numeric. H74 is a path attributed to Ali Mani with
similar poetic solution. Wieber 480
shows two diagrams. Y1‑39a, Y1‑39b,
Y1‑41b are the same tour as H73, but with different descriptions, the
latter two being attributed to al‑Adli.
Y1‑39a (second diagram) = H74 is attributed to ‘Ali ibn Mani.
Shihâbaddîn Abû’l‑‘Abbâs Ahmad [the h should have an
underdot] ibn Yahya [the h should have an underdot] ibn Abî Hajala
[the H
should have an underdot] at‑Tilimsâni alH‑anbalî [the H
should have an underdot]. Kitâb
’anmûdhaj al‑qitâl fi la‘b ash‑shaţranj [NOTE: ţ
denotes a t with an underdot] (Book of the examples of
warfare in the game of chess). Copied
by Muhammed ibn ‘Ali ibn Muhammed al‑Arzagî in 1446.
Bland,
loc. cit., pp. 28‑31, describes this as the second of Dr. Lee's MSS, old
no. 147, new no. 77. Forbes copies
this and adds notes. Van der Linde I
105‑107 seems to copy from Bland and Forbes. Murray 176‑177 says the author died in 1375, so this might
be c1370. He says it is Dr. Lee's on
175‑176, that it is MS Arab. 93 at the John Rylands Library and denotes
it Man. Wieber 29‑32 cites it as
MS 767(59) at the Rylands Library and denotes it H. On p. 481, he shows a half‑board path which cannot be
extended to the full board.
This
MS also gives the 4 knights and 7 knights problems. Murray 337, 673 (CB236) & 690 and Wieber 481 show these
problems.
Risâlahĭ Shatranj. Persian poem of unknown date and
authorship. A copy was sent to Bland by
Dr. Sprenger of Delhi. See Bland, loc.
cit., pp. 43‑44. [Bland uses á
for â.] Bland says it has the problem
of the knight's tour or path. [I think
this is the poem mentioned on Murray 182-183 and hence on Wieber 42.]
Şifat mal ‘ûb al‑faras
fî gamî abyât aš‑šaţranğ [NOTE: Ş, ţ. denote
S, t with underdot.] MS Gotha 10, Teil 6; ar. 366; Stz. Hal. 408. Date unknown. Wieber 37 & 480 describes this and gives a path from h8 to e4
which occurs on ff. 70 & 68.
Civis Bononiae [Citizen of
Bologna]. Like Bonus Socius, this is a
collection of chess problems, from c1475, which exists in several MSS and
printings. All are in Latin, from Italy,
and give essentially the same 288 problems.
See Murray 643‑703 for description of about 10 texts and
transcription of the problems. Many of
the texts are not in van der Linde.
Murray 643 cites MS Lasa, in the library of Baron von der Lasa, c1475,
as the most accurate and complete of the texts. Two other well known versions are described below.
Paulo
Guarino (di Forli) (= Paulus Guarinus).
No real title, but the end has 'Explicit liber de partitis scacorum'
with the writer's name and the date 4 Jan 1512. This MS was in the Franz Collection and is now (1913) in the John
G. White Collection in Cleveland, Ohio.
This version only contains 76 problems.
Van der Linde I 295‑297 describes the MS and on p. 294 he
describes a half board path and says Guarino's 74 is a reflection of his no.
243. Murray 645 describes the MS but
doesn't list the individual problems.
He implies that CB244, on p. 674, is the tour that appears in all of the
Civis Bononiae texts, but this is not the same as van der Linde's no. 243. CB236, pp. 673 & 690, is the 4 knights
problem, which is Guarino's 42 [according to Lucas, RM4, p. 207], but I don't
have a copy of van der Linde's no. 215 to check this, ??NX.
Anon.
Sensuit Jeux Partis des eschez: composez nouvellement Pour recreer tous nobles
cueurs et pour eviter oysivete a ceulx qui ont voulente: desir et affection de
le scavoir et apprendre et est appelle ce Livre le jeu des princes et
damoiselles. Published by Denis Janot,
Paris, c1535, 12 ff. ??NYS. (This is the item described by von der Lasa
as 'bei Janot gedrucktes Quartbändchen' (MUS #195).) This a late text of 21 problems, mostly taken from Civis
Bononiae. Only one copy is known, now
(1913) in Vienna. See van der Linde I
306‑307 and Murray 707‑708 which identify no. 18 as van der Linde's
no. 243 and with CB244, as with the Guarino work. I can't tell but van der Linde may identify no. 11 as the 4
knights problem (??NX).
Murray
730 gives another half board path, C92, of c1500 which goes from a8 to g5. Murray 732 notes that a small rearrangement
makes it extendable to the whole board.
Horatio Gianutio della
Mantia. Libro nel quale si tratta della
Maniera di giuocar' à Scacchi, Con alcuni sottilissimi Partiti. Antonio de' Bianchi, Torino, 1597. ??NYS.
Gives half board tours which can be assembled into to a full tour. (Not in the English translation: The Works of Gianutio and Gustavus Selenus,
on the game of Chess, Translated and arranged by J. H. Sarratt; J. Ebers,
London, 1817, vol. 1. -- though the copy I saw didn't say vol. 1. Van der Linde, Erste Jartausend ... says
there are two volumes.)
Bhaţţa
Nīlakaņţha. [NOTE: ţ,
ņ denote t,
n with underdot.] Bhagavantabhāskara. 17C.
End of 5th book. ??NYS,
described by Murray 63‑66. The
author gives three tours, in the poetic form of Rudraţa [NOTE: ţ
denotes a t with an underdot.], which are the same tour
starting at different points. The tour
has 180 degree rotational symmetry.
Ozanam. 1725.
Prob. 52, 1725: 260‑269.
Gives solutions due to Pierre Rémond de Montmort, Abraham de Moivre,
Jean‑Jacques d'Ortous de Mairan (1678-1771). Surprisingly, these are all distinct and different from the
earlier examples. Ozanam says he had
the problem and the solution from de Mairan in 1722. Says the de Moivre is the simplest. Kraitchik, Math. des Jeux, op. cit. in 4.A.2, p. 359, dates the
de Montmort as 1708 and the de Moivre as 1722, but gives no source for
these. Montmort died in 1719. Ozanam died in 1717 and this edition was
edited by Grandin. Van der Linde and
Ahrens say they can find no trace of these solutions prior to Ozanam
(1725). See Ozanam-Montucla, 1778.
Ball,
MRE, 1st ed., 1892, p. 139, says the earliest examples he knows are the De
Montmort & De Moivre of the late 17C, but he only cites them from
Ozanam-Hutton, 1803, & Ozanam-Riddle, 1840. In the 5th ed., 1911, p. 123, he adds that
"They were sent by their authors to Brook Taylor who seems to have
previously suggested the problem."
He gives no reference for the connection to Taylor and I have not seen
it mentioned elsewhere. This note is
never changed and may be the source of the common misconception that knight's
tours originated c1700!
Les Amusemens. 1749.
Prob. 181, p. 354. Gives de
Moivre's tour. Says one can imagine
other methods, but this is the simplest and most interesting.
L. Euler. Letter to C. Goldbach, 26 Apr 1757. In:
P.‑H. Fuss, ed.; Correspondance Mathématique et Physique de
Quelques Célèbres Géomètres du XVIIIème Siècle; (Acad. Imp. des Sciences, St.
Pétersbourg, 1843) = Johnson Reprint,
NY, 1968, vol. 1, pp. 654‑655.
Gives a 180o symmetric tour.
L. Euler. Solution d'une question curieuse qui ne
paroit soumise à aucune analyse. (Mém.
de l'Académie des Sciences de Berlin, 15 (1759 (1766)), 310‑337.) = Opera Omnia (1) 7 (1923) 26‑56. (= Comm. Arithm. Coll., 1849, vol. 1, pp.
337‑355.) Produces many
solutions; studies 180o
symmetry, two halves, and other size boards.
[Petronio dalla Volpe]. Corsa del Cavallo per tutt'i scacchi dello
scacchiere. Lelio della Volpe, Bologna,
1766. 12pp, of which 2 and 12 are
blanks. [Lelio della Volpe is sometimes
given as the author, but he died c1749 and was succeeded by his son
Petronio.] Photographed and printed by
Dario Uri from the example in the Libreria Comunale Archiginnasio di Bologna,
no. 17 CAPS XVI 13. The booklet is
briefly described in: Adriano Chicco;
Note bibliografiche su gli studi di matematica applicata agli scacchi,
publicati in Italia; Atti del Convegno Nazionale sui Giochi Creative, Siena, 11‑14
Jun 1981, ed. by Roberto Magari; Tipografia Senese for GIOCREA (Società
Italiana Giochi Creativi), 1981; p. 155.
The
Introduction by the publisher cites Ozanam as the originator of this 'most
ingenious' idea and says he gives examples due to Montmort, Moivre and
Mairan. He also says this material has
'come to hand' but doesn't give any source, so it is generally thought he was
the author. He gives ten paths,
starting from each of the 10 essentially distinct cells. He then gives the three cited paths from
Ozanam. He then gives six tours. Each path is given as a numbered board and a
line diagram of the path, which led Chicco to say there were 38 paths. The line drawing of the first tour is also
reproduced on the cover/title page.
Ozanam-Montucla. 1778.
Prob. 23, 1778: 178-182; 1803:
177-180; 1814: 155-157. Prob. 22, 1840: 80‑81. Drops the reference to de Mairan as the
source of the problem and adds a fourth tour due to "M. de W***, capitaine
au régiment de Kinski". All of
these have a misprint of 22 for 42 in the right hand column of De Moivre's
solution.
H. C. von Warnsdorff. Des Rösselsprunges einfachste und
allgemeinste Lösung. Th. G. Fr.
Varnhagenschen Buchhandlung, Schmalkalden, 1823, 68pp. ??NYS -- details from Walker. Rule to make the next move to the cell with
the fewest remaining neighbours. Lucas,
L'Arithmétique Amusante, p. 241, gives the place of publication as Berlin.
Boy's Own Book. Not in 1828. 1828-2: 318 states a knight's tour can be made.
George Walker. The Art of Chess-Play: A New Treatise on the
Game of Chess. (1832, 80pp. 2nd ed., Sherwood & Co, London, 1833,
160pp. 3rd ed., Sherwood & Co.,
London, 1841, 300pp. All ??NYS --
details from 4th ed.) 4th ed., Sherwood,
Gilbert & Piper, London, 1846, 375pp.
Chap. V -- section: On the knight, p. 37. "The problem respecting the Knight's covering each square of
the board consecutively, has attracted, in all ages, the attention of the first
mathematicians." States
Warnsdorff's rule, without credit, but gives the book in his bibliography on p.
375, and asserts the rule will always give a tour. No diagram.
Family Friend 2 (1850) 88 &
119, with note on 209. Practical Puzzle
-- No. III. Find a knight's path. Gives one answer. Note says it has been studied since 'an early period' and cites
Hutton, who copies some from Montucla, an article by Walker in Frasers Magazine
(??NYS) which gives Warnsdorff's rule and an article by Roget in Philosophical
Magazine (??NYS) which shows one can start and end on any two squares of
opposite colours. Describes using a
pegged board and a string to make pretty patterns.
Boy's Own Book. Moving the knight over all the squares
alternately. 1855: 511-512; 1868: 573; 1881 (NY): 346-347. 1855
says the problem interested Euler, Ozanam, De Montmart [sic], De Moivre, De
Majron [sic] and then gives Warnsdorff's rule, citing George Walker's 'Treatise
on Chess' for it -- presumably 'A New Treatise', London, 1832, with 2nd ed.,
1833 & 3rd ed., 1841, ??NYS.
Walker also wrote On Moving the Knight, London, 1840, ??NYS. 1868 drops all the names, but the NY ed. of
1881 is the same as the 1855. Gives a
circuit due to Euler.
Magician's Own Book. 1857.
Art. 46: Moving the knight over all the squares alternately,
pp. 283-287. Identical to Boy's
Own Book, 1855, but adds Another Method.
= Book of 500 Puzzles; 1859, art. 46, pp. 97-101. = Boy's Own Conjuring Book, 1860, prob. 45,
pp. 246‑251.
Landells. Boy's Own Toy-Maker. 1858.
Moving the knight over all the squares alternately, p. 143. This is the Another Method of Magician's Own
Book, 1857. Cf Illustrated Boy's Own
Treasury, 1860.
Illustrated Boy's Own
Treasury. 1860. Prob. 47: Practical chess puzzle, pp. 404
& 443. Knight's tour. This is the Another Method of Magician's Own
Book.
C. F. de Jaenisch. Traité des Applications de l'Analyse
Mathématiques au Jeu des Échecs.
3 vols., no publisher, Saint-Pétersbourg. 1862-1863. Vol. 1: Livre I: Section III: De la marche
du cavalier, pp. 186-259 & Plate III.
Vol. 2: Livre II: Problème du Cavalier, pp. 1-296 & 31 plates
(some parts ??NYS). Vol. 3: Addition au
Livre II, pp. 239-243 (This Addition ??NYS).
This contains a vast amount of miscellaneous material and I have not yet
read it carefully. ??NYR
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 323, pp.
153-154 & 393: Rösselsprung-Aufgaben.
Three arrays of syllables and one must find a poetic riddle by following
a knight's tour. Arrays are 8 x 8,
8 x 8, 6 x 4.
C. Flye Sainte‑Marie. Bull. Soc. Math. de France (1876) 144‑150. ??NYS -- described by Jelliss. Shows there is no tour on a 4 x n
board and describes what a path must look like.
Mittenzwey. 1880.
Prob. 222-223, pp. 40 & 91;
1895?: 247-248, pp. 44 & 93;
1917: 247‑248, pp. 40-41 & 89. First is a knight's path.
Second is a board with word fragments and one has to make a poem, which
uses the same path as in the first problem.
Paul de Hijo [= Abbé
Jolivald]. Le Problème du Cavalier des
Échecs. Metz, 1882. ??NYS -- described by Jelliss and quoted by
Lucas. Jelliss notes the BL copy of de
Hijo was destroyed in the war, but he has since told me there are copies in The
Hague and Nijmegen. First determination
of the five 6 x 6 tours with 4-fold rotational symmetry, the 150
ways to cover the 8 x 8 with two circuits of length 32 giving a
pattern with 2‑fold rotational symmetry, the 378 ways giving reflectional
symmetry in a median, the 140 ways with four circuits giving 4-fold rotational
symmetry and the 301 ways giving symmetry in both medians (quoted in Lucas,
L'Arithmétique Amusante, pp. 238-241).
Lucas. Nouveaux jeux scientifiques ..., 1889, op. cit. in 4.B.3. (Described on p. 302, figure on p.
301.) 'La Fasioulette' is an 8 x 8
board with 64 links of length Ö5 to form knight's tours.
Knight's move puzzles. The Boy's Own Paper 11 (Nos. 557 &
558) (14 & 21 Sep 1889) 799 & 814. Four Shakespearean quotations concealed as
knight's tours on a 8 x 8 board.
Beginnings not indicated!
Hoffmann. 1893.
Chap. X, no. 6: The knight's tour, pp. 335-336 & 367-373
= Hoffmann‑Hordern, pp. 225-229.
Gives knight's paths due to Euler and Du Malabare, a knight's tour due
to Monneron, and four other unattributed tours. Gives Warnsdorff's rule, citing Walker's A New Treatise on Chess,
1832.
Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904, pp. 1080‑1093. Pp. 1084‑1086 gives many references to
19C work, including estimates of the number of tours and results on 'semi‑magic
tours'.
C. Planck. Chess Amateur (Dec 1908) 83. ??NYS -- described by Jelliss. Shows there
are 1728 paths on the 5 x 5 board.
Jelliss notes that this counts each path in both directions and there
are only 112 inequivalent tours.
Ahrens. 1910.
MUS I 325. Use of knight's tours
as a secret code.
Dudeney. AM.
1917. Prob. 339: The four
knight's tours, pp. 103 & 229.
Quadrisect the board into four congruent pieces such that there is a
knight's tour on the piece. Jelliss
asserts that the solution is unique and says this may be what Persian MS 260
(i.e. 211) intended. He notes that the
four tours can be joined to give a tour with four fold rotational symmetry.
W. H. Cozens. Cyclically symmetric knight's tours. MG 24 (No. 262) (Dec 1940) 315‑323. Finds symmetric tours on various odd‑shaped
boards.
H. J. R. Murray. The Knight's Tour. ??NYS. MS of 1942
described by G. P. Jelliss, G&PJ 2 (No. 17) (Oct 1999) 315. Observes that a knight can move from
the (0, 0) cell to the (2, 1) and
(1, 2) cella and that the angle
between these lines is the smaller angle of a
3, 4, 5 triangle. One can see this by extending the lines
to (8, 4) and (5, 10) and seeing these points form a 3, 4, 5
triangle with (0, 0).
W. H. Cozens. Note 2761:
On note 2592. MG 42 (No. 340)
(May 1958) 124‑125. Note 2592
tried to find the cyclically symmetric tours on the 6 x 6 board and found
4. Cozens notes two are reflections of
the other two and that three such tours were omitted. He found all these in his 1940 paper.
R. C. Read. Constructing open knight's tours
blindfold! Eureka 22 (Oct 1959)
5-9. Describes how to construct easily
a tour between given cells of opposite colours, correcting a method of Roget
described by Ball (MRE 11th ed, p. 181).
Says he can do it blindfold.
W. H. Cozens. Note 2884:
On note 2592. MG 44 (No. 348)
(May 1960) 117. Estimates there are 200,000
cyclically symmetric tours on the
10 x 10 board.
Roger F. Wheeler. Note 3059:
The KNIGHT's tour on 42 and other boards. MG 47 (No. 360) (May 1963) 136‑141. KNIGHT means a knight on a toroidal
board. He finds 2688 tours of 19 types
on the 42 toroid.
(Cf Tylor, 1982??)
J. J. Duby. Un algorithme graphique trouvant tous les
circuits Hamiltoniens d'un graphe.
Etude No. 8, IBM France, Paris, 22 Oct 1964. [In English with French title and summary.] Finds there are 9862 knight's tours on
the 6 x 6 board, where the tours all start at a fixed corner and then go to
a fixed one of the two cells reachable from the corner. He also finds 75,000 tours on the 8 x 8 board which have the same first 35 moves. He believes there may be over a million
tours.
Karl Fabel. Wanderungen von Schachfiguren. IN:
Eero Bonsdorff, Karl Fabel & Olavi Riihimaa; Schach und Zahl; Walter
Rau Verlag, Düsseldorf, 1966, pp. 40-50.
On p. 50, he says that there are
122,802,512 tours where the
knight does two joined half-board paths.
He also says there are upper bounds, determined by several authors, and
he gives 1.5 x 1026 as an example.
Gardner. SA (Oct 1967) = Magic Show, chap. 14. Surveys results of which boards have tours
or paths.
D. J. W. Stone. On the Knight's Tour Problem and Its Solution
by Graph‑Theoretic and Other Methods.
M.Sc. Thesis, Dept. of Computing Science, Univ. of Glasgow, Jan.
1969. Confirms Duby's 9862 tours on
the 6 x 6 board.
David Singmaster. Enumerating unlabelled Hamiltonian
circuits. International Series on
Numerical Mathematics, No. 29.
Birkhäuser, Basel, 1975, pp. 117‑130. Discusses the work of Duby and Stone and gives an estimate, which
Stone endorses, that there are 1023±3 tours on the 8 x 8 board.
C. M. B. Tylor. 2‑by‑2 tours. Chessics 14 (Jul‑Dec 1982) 14. Says there are 17 knight's tours on a 2 x 2
torus and gives them. Doesn't
mention Wheeler, 1963.
Robert Cannon & Stan
Dolan. The knight's tour. MG 70 (No. 452) (Jun 1986) 91‑100. A rectangular board is tourable if it has a
knight's path between any two cells of opposite colours. They prove that m x n is tourable if
and only if mn is even and
m ³ 6, n ³ 6. They also prove that m x
n has a knight's tour if and only
if mn is even and [(m ³ 5, n ³ 5)
or (m = 3, n ³ 10)] and that when
mn is even, m x n has a knight's path if and only if m ³ 3, n ³ 3, except for the 3 x 6 and 4 x 4
boards. (These later results are
well known -- see Gardner. The authors
only cite Ball's MRE.)
George Jelliss. Figured tours. MS 25:1 (1992/93) 16-20.
Exposition of paths and tours where certain stages of the path form an
interesting geometric figure. E.g.
Euler's first paper has a path on the 5
x 5 such that the points on one
diagonal are in arithmetic progression:
1, 7, 13, 19, 25.
Martin Loebbing & Ingo
Wegener. The number of knight's tours
equals 33,439,123,484,294 ‑--
Counting with binary decision diagrams.
Electronic Journal of Combinatorics 3 (1996) article R5. A somewhat vague description of a method for
counting knight's tours -- they speak of directed knight's tours, but it is not
clear if they have properly accounted for the symmetries of a tour or of the
board. Several people immediately
pointed out that the number is incorrect because it has to be divisible by
four. Two comments have appeared,
ibid. On 15 May 1996, the authors
admitted this and said they would redo the problem, but they have submitted no
further comment as of Jan 2001. On 18 Feb
1997, Brendan McKay announced that he had done the computation another way and
found 13,267,364,410,532.
In
view of the difference between this and my 1975 estimate of 1023±3 tours, it might be worth explaining my reasoning. In 1964, Duby found 75,000
tours with the same first 35 moves.
The average valence for a knight on an
8 x 8 board is 5.25, but one
cannot exit from a cell in the same direction as one entered, so we might
estimate the number of ways that the first 35 moves can be made as 4.2535 = 9.9 x 1021. Multiplying by 75,000 then gives 7.4 x 1026. I think I assumed that some of the first
moves had already been made, e.g. we only allow one move from the starting
cell, and factored by 8 for the symmetries of the square, to get 2.2 x 1025. I can't find my original calculations, and I
find the estimate 1025 in later papers, so I suppose I tried to
reduce the effect of the 4.2535 some more.
In retrospect, I had no knowledge of how many of these had already been
tried. If about half of all moves from
a cell had already been tried before any circuit was found, then the estimate
would be more like 2.2534 x 75,000 =
7.1 x 1016. If we
divide the given number of circuits by
75,000 and take the 34th root,
we get an average valence of 1.78 remaining, far less than I would have
guessed.
I
am grateful to Don Knuth for this reference.
Neither he nor I expected to ever see this number calculated!
5.F.2. OTHER HAMILTONIAN CIRCUITS
For
circuits on the n‑cube, see also 5.F.4 and 7.M.1,2,3.
For
circuits on the chessboard, see also 6.AK.
Le Nôtre. Le Labyrinte de Versailles, c1675. This was a hedge or garden maze, but the
objective was to visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at least one
fountain. Some fountains were not at
path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian
problem, except that one fountain was located at the end of a short dead
end. [Fisher, op. cit. in 5.E.1, pp.
49, 79, 130 & 144-145, with contemporary diagram on p. 144. He says there are 39 fountains, but the
diagram has 40.]
T. P. Kirkman. On the partitions of the R‑pyramid,
being the first class of R‑gonous X‑edra. Philos. Trans. Roy. Soc. 148 (1858) 145‑161.
W. R. Hamilton. The Icosian Game. 4pp instructions for the board game. J. Jaques and Son, London, 1859.
(Reproduced in BLW, pp. 32-35, with frontispiece photo of the board at
the Royal Irish Academy.)
For a long time, the only known
example of the game, produced by Jaques, was at the Royal Irish Academy in
Dublin. This example is inscribed on
the back as a present from Hamilton to his friend, J. T. Graves. It is complete, with pegs and instructions. None of the obvious museums have an
example. Diligent searching in the
antique trade failed to turn up an example in twenty years, but in Feb 1996,
James Dalgety found and acquired an example of the board -- sadly the pegs and
instructions were lacking. Dalgety
obtained another board in 1998, again without the pegs and instructions, but in
1999 he obtained another example, with the pegs.
Mittenzwey. 1880.
Prob. 281, pp. 50 & 100;
1895?: 310, pp. 53-54 & 102;
1917: 310, pp. 49 & 97.
The garden of a French palace has a maze with 31 points to see. Find a path past all of them with no repeated
edges and no crossings. The pattern is
clearly based on the Versailles maze of c1675 mentioned above, but I don't
recall the additional feature of no crossings occurring before.
T. P. Kirkman. Solution of problem 6610, proposed by
himself in verse. Math. Quest. Educ.
Times 35 (1881) 112‑116. On p.
115, he says Hamilton told him, upon occasion of Hamilton presenting him 'with
his handsomest copy of the puzzle', that Hamilton got the idea for the Icosian
Game from p. 160 of Kirkman's 1858 article,
Lucas. RM2, 1883, pp. 208‑210.
First? mention of the solid version.
The 2nd ed., 1893, has a footnote referring to Kirkman, 1858.
John Jaques & Son. The Traveller's Dodecahedron; or,
A Voyage Round the World. A New
Puzzle. "This amusing puzzle, exercising
considerable skill in its solution, forms a popular illustration of Sir William
Hamilton's Icosian Game. A wood
dodecahedron with the base pentagon stretched so that when it sits on the base,
all vertices are visible. With ivory?
pegs at the vertices, a handle that screws into the base, a string with rings
at the ends and one page of instructions, all in a box. No date.
The only known example was obtained by James Dalgety in 2002.
Pearson. 1907.
Part III, no. 60: The open door, pp. 60 & 130. Prisoner in one corner of an 8 x 8
array is allowed to exit from from the other corner provided he visits
every cell once. This requires him to
enter and leave a cell by the same door.
Ahrens. Mathematische Spiele. 2nd ed., Teubner, Leipzig, 1911. P. 44, note, says that a Dodekaederspiel is
available from Firma Paul Joschkowitz -- Magdeburg for .65
mark. This is not in the 1st ed.
of 1907 and the whole Chapter is dropped in the 3rd ed. of 1916 and the later
editions.
Anonymous. The problems drive. Eureka 12 (Oct 1949) 7-8 & 15. No. 3.
How many Hamiltonian circuits are there on a cube, starting from a given
point? Reflections and reversals count
as different tours. Answer is 12, but
this assumes also that rotations are different. See Singmaster, 1975, for careful definitions of how to
count. There are 96 labelled circuits,
of which 12 start at a given vertex.
But if one takes all the 48 symmetries of the cube as equivalences (six
of which fix the given vertex), there are just 2 circuits from a given starting
point. However, these are actually the
same circuit started at different points.
Presumably Kirkman and Hamilton knew of this.
C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed., Gollancz, London, 1971, pp.
24-25. Roman knobbed dodecahedra -- an
ancient solid version??
R. E. Ingram. Appendix 2: The Icosian Calculus. In:
The Mathematical Papers of Sir William Rowan Hamilton. Vol. III: Algebra. Ed. by H. Halberstam & R. E. Ingram. CUP, 1967, pp. 645‑647. [Halberstam told me that this Appendix is
due to Ingram.] Discusses the method
and asserts that the tetrahedron, cube and dodecahedron have only one
unlabelled circuit, the octahedron has two and the icosahedron has 17.
David Singmaster. Hamiltonian circuits on the regular
polyhedra. Notices Amer. Math. Soc. 20
(1973) A‑476, no. 73T‑A199.
Confirms Ingram's results and gives the number of labelled circuits.
David Singmaster. Op. cit. in 5.F.1. 1975. Carefully defines
labelled and unlabelled circuits.
Discusses results on regular polyhedra in 3 and higher dimensions.
David Singmaster. Hamiltonian circuits on the n‑dimensional octahedron. J. Combinatorial Theory (B) 18 (1975) 1‑4. Obtains an explicit formula for the number
of labelled circuits on the n‑dimensional
octahedron and shows it is @
(2n)!/e. Gives numbers for n £ 8. In unpublished work, it is shown that the number of unlabelled
circuits is asymptotic to the above divided by
n!2n×4n.
Angus Lavery. The Puzzle Box. G&P 2 (May 1994) 34-35.
Alternative solitaire, p. 34.
Asks for a knight's tour on the 33-hole solitaire board. Says he hasn't been able to do it and offers
a prize for a solution. In Solutions,
G&P 3 (Jun 1994) 44, he says it cannot be done and the proof will be given
in a future issue, but I never saw it.
5.F.3. KNIGHT'S TOURS IN HIGHER DIMENSIONS
A.‑T. Vandermonde. Remarques sur les problèmes de
situation. Hist. de l'Acad. des Sci.
avec les Mémoires (Paris) (1771 (1774)) Mémoires: pp. 566‑574 & Plates
I & II. ??NYS. First? mention of cubical problem. (Not
given in BLW excerpt.)
F. Maack. Mitt. über Raumschak. 1909, No. 2, p. 31. ??NYS -- cited by Gibbins, below. Knight's tour on 4 x 4 x 4 board.
Dudeney. AM.
1917. Prob. 340: The cubic knight's
tour, pp. 103 & 229. Says
Vandermonde asked for a tour on the faces of a
8 x 8 x 8 cube. He gives it as a problem with a solution.
N. M. Gibbins. Chess in three and four dimensions. MG 28 (No. 279) (1944) 46‑50. Gives knight's tour on 3 x 3 x 4
board -- an unpublished result due to E. Hubar‑Stockar of
Geneva. This is the smallest 3‑D
board with a tour. Gives Maack's tour
on 4 x 4 x 4 board.
Ian Stewart. Solid knight's tours. JRM 4:1 (Jan 1971) 1. Cites Dudeney. Gives a tour through the entire
8 x 8 x 8 cube by stacking 8
knight's paths.
T. W. Marlow. Closed knight tour of a 4 x 4 x 4
board. Chessics 29 & 30
(1987) 162. Inspired by Stewart.
5.F.4. OTHER CIRCUITS IN AND ON A CUBE
The number of Hamiltonian Circuits on
the n-dimensional cube is the same as
the number of Gray codes (see 7.M.3) and has been the subject of considerable
research. I will not try to cover this
in detail.
D. W. Crowe. The
n‑dimensional cube and the Tower of Hanoi. AMM 63:1 (Jan 1956) 29‑30.
E. N. Gilbert. Gray codes and paths on the n-cube.
Bell System Technical Journal 37 (1958) 815-826. Shows there are 9 inequivalent circuits on
the 4-cube and 1 on the n-cube for n =
1, 2, 3. The latter cases are
sufficiently easy that they may have been known before this.
Allen F. Dreyer. US Patent 3,222,072 -- Block Puzzle. Filed: 11 Jun 1965; patented: 7 Dec 1965. 4pp + 2pp diagrams. 27 cubes on an elastic. The holes are straight or diagonal so that
three consecutive cubes are either in a line or form a right angle. A solution is a Hamiltonian path through the
27 cells. Such puzzles were made in
Germany and I was given one about 1980 (see Singmaster and Haubrich &
Bordewijk below). Dreyer gives two
forms.
Gardner. The binary Gray code. SA (Aug 1972) c= Knotted, chap. 2.
Notes that the number of circuits on the n-cube, n > 4, is not known. SA (Apr 1973) reports that three (or four) groups had found the
number of circuits on the 4-cube -- this material is included in the Addendum
in Knotted, chap. 2, but none of the groups ever seem to have published their
results elsewhere. Unfortunately, none
of these found the number of inequivalent circuits since they failed to take
all the equivalences into account -- e.g. for
n = 1, 2, 3, 4, 5, their enumerations give: 2, 8, 96, 43008, 5 80189 28640 for the numbers of labelled circuits. Gardner's Addendum describes some further work
including some statistical work which estimates the number on the 6-cube is
about 2.4 x 1025.
David Singmaster. A cubical path puzzle. Written in 1980 and submitted to JRM, but
never published. For the 3 x 3 x 3
problem, the number, S, of straight through pieces (ignoring the
ends) satisfies 2 £ S £
11.
Mel A. Scott. Computer output, Jun 1986, 66pp. Determines there are 3599 circuits through
the 3 x 3 x 3 cube such that the resulting string of 27 cubes can be made into
a cube in just one way. But cf the next
article which gives a different number??
Jacques Haubrich & Nanco
Bordewijk. Cube chains. CFF 34 (Oct 1994) 12‑15. Erratum, CFF 35 (Dec 1994) 29. Says Dreyer is the first known reference to
the idea and that they were sold 'from about 1970' Reproduces the first page of diagrams from Dreyer's patent. Says his first version has a unique
solution, but the second has 38 solutions.
They have redone previous work and get new numbers. First, they consider all possible strings of
27 cubes with at most three in a line (i.e. with at most a single 'straight'
piece between two 'bend' pieces and they find there are 98,515
of these. Only 11,487
of these can be folded into a 3
x 3 x 3 cube. Of these, 3654 can be folded up in only one way. The chain with the most solutions had 142
different solutions. They refer to Mel
Scott's tables and indicate that the results correspond -- perhaps I miscounted
Scott's solutions??
5.G.1. GAS, WATER AND ELECTRICITY
Dudeney. Problem 146 -- Water, gas, and
electricity. Strand Mag. 46 (No. 271)
(Jul 1913) 110 & (No. 272) (Aug 1913) 221 (c= AM, prob. 251,
pp. 73 & 200‑201). Earlier
version is slightly more interesting, saying the problem 'that I have called
"Water, Gas, and Electricity" ... is as old as the hills'. Gives trick solution with pipe under one
house.
A. B. Nordmann. One Hundred More Parlour Tricks and
Problems. Wells, Gardner, Darton &
Co., London, nd [1927 -- BMC]. No. 96:
The "three houses" problem, pp. 89-90 & 114. "Were all the houses connected up with
all three supplies or not?" Answer
is no -- one connection cannot be made.
Loyd, Jr. SLAHP.
1928. The three houses and three
wells, pp. 6 & 87‑88. "A
puzzle ... which I first brought out in 1900 ..." The drawing is much less polished than
Dudeney's. Trick solution with a pipe
under one house, a bit differently laid out than Dudeney.
The Bile Beans Puzzle Book. 1933.
No. 46: Water, gas & electric light. Trick solution almost identical to Dudeney.
Philip Franklin. The four color problem. In:
Galois Lectures; Scripta Mathematica Library No. 5; Scripta Mathematica,
Yeshiva College, NY, 1941, pp. 49-85.
On p. 74, he refers to the graph as "the basis of a familiar
puzzle, to join each of three houses with each of three wells (or in a modern
version to a gas, water, and electricity plant)".
Leeming. 1946.
Chap. 6, prob. 4: Water, gas and electricity, pp. 71 & 185. Dudeney's trick solution.
H. ApSimon. Note 2312:
All modern conveniences. MG 36
(No. 318) (Dec 1952) 287‑288.
Given m houses and
n utilities, the maximum number
of non‑crossing connections is 2(m+n‑2) and this
occurs when all the resulting regions are 4‑sided. He extends to p‑partite graphs in general and a special case.
John Paul Adams. We Dare You to Solve This! Op. cit. in 5.C. 1957? Prob. 50: Another enduring
favorite appears below, pp. 30 & 49.
Electricity, gas, water.
Dudeney's trick solution.
Young World. c1960.
P. 4: Crossed lines.
Electricity, TV and public address lines. Trick solution with a line passing under a house.
T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751‑753. Shows you can join 4 utilities to 4 houses
on a torus without crossing.
5.H. COLOURED SQUARES AND CUBES, ETC.
5.H.1. INSTANT INSANITY = THE TANTALIZER
Note.
Often the diagrams do not show all sides of the pieces so I cannot tell
if one version is the same as another.
Frederick A. Schossow. US Patent 646,463 -- Puzzle. Applied: 19 May 1899; patented: 3 Apr 1900. 1p + 1p diagrams. Described in S&B, p. 38, which also says it is described in
O'Beirne, but I don't find it there??
Four cubes with suit patterns.
The net of each cube is shown.
The fourth cube has three clubs.
George Duncan Moffat. UK Patent 9810 -- Improvements in or
relating to Puzzle-apparatus. Applied:
28 May 1900; accepted: 30 Jun
1900. 2pp + 1p diagrams. For a six cube version with "letters R, K, B,
W, F and B-P, the initials of the names of General
Officers of the South African Field Force."
Joseph Meek. UK Patent 2775 -- Improved Puzzle Game. Applied: 5 Feb 1909; complete specification: 16 Jun 1909; accepted: 3 Feb 1910. 2pp + 1p diagrams. A four cube version with suit patterns. His discussion seems to describe the pieces drawn by Schossow.
Slocum. Compendium.
Shows: The Great Four Ace Puzzle
(Gamage's, 1913); Allies Flag Puzzle
(Gamage's, c1915); Katzenjammer Puzzle
(Johnson Smith, 1919).
Edwin F. Silkman. US Patent 2,024,541 -- Puzzle. Applied: 9 Sep 1932; patented: 17 Dec 1935. 2pp + 1 p diagrams. Four cubes marked with suits. The net of each cube is shown. The third cube has three hearts. This is just a relabelling of Schossow's
pattern, though two cubes have to be reflected which makes no difference to the
solution process.
E. M. Wyatt. The bewitching cubes. Puzzles in Wood. (Bruce Publishing, Co., Milwaukee, 1928) = Woodcraft Supply Corp., Woburn, Mass.,
1980, p. 13. A six cube, six way
version.
Abraham. 1933.
Prob. 303 -- The four cubes, p. 141 (100). 4 cube version "sold ... in 1932".
A. S. Filipiak. Four ace cube puzzle. 100 Puzzles, How do Make and How to Solve
Them. A. S. Barnes, NY, (1942) = Mathematical Puzzles, and Other Brain
Twisters; A. S. Barnes, NY, 1966;
Bell, NY, 1978; p. 108.
Leeming. 1946.
Chap. 10, prob. 9: The six cube puzzle, pp. 128‑129 &
212. Identical to Wyatt.
F. de Carteblanche [pseud. of
Cedric A. B. Smith]. The coloured cubes
problem. Eureka 9 (1947) 9‑11. General graphical solution method, now the
standard method.
T. H. O'Beirne. Note 2736:
Coloured cubes: A new "Tantalizer". MG 41 (No. 338) (Dec 1957) 292-293. Cites Carteblanche, but says the current
version is different. Gives a nicer
version.
T. H. O'Beirne. Note 2787:
Coloured cubes: a correction to Note 2736. MG 42 (No. 342) (Dec 1958) 284.
Finds more solutions than he had previously stated.
Norman T. Gridgeman. The 23 colored cubes. MM 44:5 (Nov 1971) 243-252. The
23 colored cubes are the
equivalence classes of ways of coloring the faces with 1
to 6 colors. He cites and
describes some later methods for attacking Instant Insanity problems.
Jozsef Bognár. UK Patent Application 2,076,663 A -- Spatial
Logical Puzzle. Filed 28 May 1981; published 9 Dec 1981. Cover page + 8pp + 3pp diagrams. Not clear if the patent was ever
granted. Describes Bognár's Planets,
which is a four piece instant insanity where the pieces are spherical and held
in a plastic tube. This was called
Bolygok in Hungarian and there is a reference to an earlier Hungarian patent. Also describes his version with eight pieces
held at the corners of a plastic cube.
Haubrich's
1995-1996 surveys, op. cit. in 5.H.4, include MacMahon puzzles as one class.
I
have just added the Carroll result that there are 30 six-coloured cubes, but
this must be older??
Frank H. Richards. US Patent 331,652 -- Domino. Applied: 13 Jun 1885; patented: 1 Dec 1885. 2pp + 2pp diagrams. Cited by Gardner in Magic Show, but with
date 1895. Reproduced in Haubrich,
About ..., 1996, op. cit. in 5.H.4. For
triangular matching games, specifically showing the MacMahon 5-coloured
triangles, but considering reflections as equivalences, so he has 35
pieces. [One of the colours is blank
and hence Gardner said it was a 4-colouring.]
Carroll-Wakeling. c1890?
Prob. 15: Painting cubes, pp. 18-19 & 67. This is one of the problems on undated sheets of paper that
Carroll sent to Bartholomew Price. How
many ways can one six-colour a cube?
Wakeling gives a solution, but this apparently is not on Carroll's MS.
Percy Alexander MacMahon &
Julian Robert John Jocelyn. UK
Patent 3927 A.D. 1892 -- Appliances to
be used in Playing a New Class of Games.
Applied: 29 Feb 1892; Complete
Specification Left: 28 Nov 1892;
Accepted: 28 Jan 1893. 5pp + 2pp
diagrams. Reproduced in Haubrich, About
..., 1996, op. cit. in 5.H.4. Describes
the 24 triangles with four types of edge and mentions other numbers of edge
types. Describes various games and
puzzles.
Percy Alexander MacMahon &
Julian Robert John Jocelyn. UK
Patent 8275 A.D. 1892 -- Appliances for
New Games of Puzzles. Applied: 2 May
1892; Complete Specification Left: 31
Jan 1893; Accepted: 4 Mar 1893. 2pp.
27 cubes with three colours, opposite faces having the same colour. Similar sets of 8, 27, 64, etc. cubes. Various matching games suggested. Using six colours and all six on each cube
gives 30 cubes -- the MacMahon Cubes.
Gives a complex matching problem of making two 2 x 2 x 2 cubes. Paul Garcia (email of 15 Nov 2002)
commented: "8275 describes 2
different sets of blocks, using either three colours or six colours. The three colour blocks form a set of 27
that can be assembled into a large cube with single coloured faces and internal
contact faces matching. For the six
colour cubes, the puzzle suggested is to pick out two associated cubes, and
find the sixteen cubes that can be assembled to make a copy of each. Not quite Mayblox, although using the same
colouring system."
James Dalgety. R. Journet & Company A Brief History of the Company & its
Puzzles. Published by the author, North
Barrow, Somerset, 1989. On p. 13, he
says Mayblox was patented in 1892. In
an email on 12 Nov 2002, he cited UK
Patent 8275.
Anon. Report:
"Mathematical Society, February 9". Nature 47 (No. 1217) (23 Feb 1893) 406. Report of MacMahon's talk:
The group of thirty cubes composed by six differently coloured squares.
See: Au Bon Marché, 1907, in 5.P.2, for a puzzle of hexagons with
matching edges.
Manson. 1911.
Likoh, pp. 171-172. MacMahon's
24 four-coloured isosceles right triangles, attributed to MacMahon.
"Toymaker". The Cubes of Mahomet Puzzle. Work, No. 1447 (9 Dec 1916) 168. 8 six-coloured cubes to be assembled into a
cube with singly-coloured faces and internal faces to have matching colours.
P. A. MacMahon. New Mathematical Pastimes. CUP, 1921.
The whole book deals with variations of the problem and calculates the
numbers of pieces of various types. In
particular, he describes the 24 4-coloured triangles, the 24
3-coloured squares, the MacMahon cubes, some right-triangular and
hexagonal sets and various subsets of these.
With n colours, there are n(n2+2)/3 triangles,
n(n+1)(n2‑n+2)/4
squares and n(n+1)(n4-n3+n2+2)/6 hexagons.
[If one allows reflectional equivalence, one gets n(n+1)(n+2)/6 triangles, n(n+1)(n2+n+2)/8 squares
and n(n+1)(n4-n3+4n2+2)/12 hexagons.
Problem -- is there an easy proof that the number of triangles is BC(n+2, 3)?] On p. 44, he says that Col. Julian R. Jocelyn told him some years
ago that one could duplicate any cube with 8 other cubes such that the internal
faces matched.
Slocum. Compendium.
Shows Mayblox made by R. Journet from Will Goldston's 1928 catalogue.
F. Winter. Das Spiel der 30 bunten Würfel MacMahon's Problem. Teubner, Leipzig, 1934, 128pp. ??NYR.
Clifford Montrose. Games to play by Yourself. Universal Publications, London, nd
[1930s?]. The coloured squares, pp.
78-80. Makes 16 squares with
four-coloured edges, using five colours, but there is no pattern to the
choice. Uses them to make a 4 x 4
array with matching edges, but seems to require the orientations to be
fixed.
M. R. Boothroyd &
J. H. Conway. Problems drive,
1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 6. There are twelve ways to colour the edges of a pentagon, when
rotations and reflections are considered as equivalences. Can you colour the edges of a dodecahedron
so each of these pentagonal colourings occurs once? [If one uses tiles, one has to have reversible tiles.] Solution says there are three distinct
solutions and describes them by describing contacts between 10 pentagons
forming a ring around the equator.
Richard K. Guy. Some mathematical recreations I &
II. Nabla [= Bull. Malayan Math.
Soc.] 7 (Oct & Dec 1960) 97-106 & 144-153. Pp. 101-104 discusses MacMahon triangles,
squares and hexagons.
T. H. O'Beirne. Puzzles and paradoxes 5: MacMahon's
three-colour set of squares. New
Scientist 9 (No. 220) (2 Feb 1961) 288-289.
Finds 18 of the 20 possible monochrome border patterns.
Gardner. SA (Mar 1961) = New MD, Chap. 16. MacMahon's 3-coloured squares and his
cubes. Addendum in New MD cites
Feldman, below.
Gary Feldman. Documentation of the MacMahon Squares
Problem. Stanford Artificial
Intelligence Project Memo No. 12, Stanford Computation Center, 16 Jan 1964. ??NYS
Finds 12,261 solutions for the 6 x 4 rectangle with
monochrome border -- but see Philpott, 1982, for 13,328 solutions!!
Gardner. SA (Oct 1968) = Magic Show, Chap. 16. MacMahon's four-coloured triangles and
numerous variants.
Wade E. Philpott. MacMahon's three-color squares. JRM 2:2 (1969) 67-78. Surveys the topic and repeats Feldman's
result.
N. T. Gridgeman, loc. cit. in
5.H.1, 1971, covers some ideas on the MacMahon cubes.
J. J. M. Verbakel. Digitale tegels (Digital tiles). Niet piekeren maar puzzelen (name of a
puzzle column). Trouw (a Dutch
newspaper) (1 Feb 1975). ??NYS --
described by Jacques Haubrich; Pantactic patterns and puzzles; CFF 34 (Oct
1994) 19-21. There are 16 ways to 2‑colour
the edges of a square if one does not allow them to rotate. Assemble these into a 4 x 4
square with matching edges.
There are 2,765,440 solutions in 172,840 classes of
16. One can add further constraints to
yield fewer solutions -- e.g. assume the
4 x 4 square is on a torus and
make all internal lines have a single colour.
Gardner. Puzzling over a problem‑solving
matrix, cubes of many colours and three‑dimensional dominoes. SA 239:3 (Sep 1978) 20‑30 &
242 c= Fractal, chap. 11. Good review of MacMahon (photo) and his
coloured cubes. Bibliography cites
recent work on Mayblox, etc.
Wade E. Philpott. Instructions for Multimatch. Kadon Enterprises, Pasadena, Maryland,
1982. Multimatch is just the 24
MacMahon 3-coloured squares. This
surveys the history, citing several articles ??NYS, up to the determination of
the 13,328 solutions for the 6 x
4 rectangle with monochrome border, by
Hilario Fernández Long (1977) and John W. Harris (1978).
Torsten Sillke. Three
3 x 3 matching puzzles. CFF 34 (Oct 1994) 22-23. He has wanted an interesting 9 element
subset of the MacMahon pieces and finds that of the 24 MacMahon 3-coloured
squares, just 9 of them contain all three colours. He considers both the corner and the edge versions. The editor notes that a 3 x 3
puzzle has 36 x 32/2 =
576 possible edge contacts and that the
number of these which match is a measure of the difficulty of the puzzle, with
most 3 x 3 puzzles having 60 to
80 matches. The corner version of Sillke's puzzle
has 78
matches and one solution. The
edge version has 189 matches and many solutions, hence Sillke
proposes various further conditions.
Here we have a set of pieces and one
has to join them so that some path is formed.
This is often due to a chain or a snake, etc. New section. Again,
Haubrich's 1995-1996 surveys, op. cit. in 5.H.4, include this as one class.
Hoffmann. 1893.
Chap. III, No. 18: The endless chain, pp. 99-100 & 131
= Hoffmann‑Hordern, pp. 91-92, with photo. 18 pieces, some with
parts of a chain, to make into an 8 x
8 array with the chain going
through 34 of the cells. All the
pieces are rectangles of width one.
Photo shows The Endless Chain, by The Reason Manufacturing Co.,
1880-1895. Hordern Collection, p. 62,
shows the same and La Chaine sans fin, 1880-1905.
Loyd. Cyclopedia. 1914. Sam Loyd's endless chain puzzle, pp. 280
& 377. Chain through all 64 cells
of a chessboard, cut into 13 pieces.
The chessboard dissection is of type:
13: 02213 131.
Hummerston. Fun, Mirth & Mystery. 1924.
The dissected serpent, p. 131.
Same pieces as Hoffmann, and almost the same pattern.
Collins. Book of Puzzles. 1927. The dissected snake
puzzle, pp. 126-127. 17 pieces forming
an 8 x 8 square. All the piece are
rectangular pieces of width one except for one L‑hexomino -- if this were
cut into straight tetromino and domino, the pieces would be identical to
Hoffmann. The pattern is identical to
Hummerston.
See Haubrich in 5.H.4.
These
all have coloured edges unless specified.
See S&B, p. 36, for examples.
Edwin L[ajette] Thurston. US Patent 487,797 -- Puzzle. Applied: 30 Sep 1890; patented: 13 Dec 1892. 3pp + 3pp diagrams. Reproduced in Haubrich, About ..., 1996, op.
cit. below. 4 x 4 puzzles with 6-coloured corners or edges,
but assuming no colour is repeated on a piece -- indeed he uses the 15 = BC(6,2) ways of choosing 4 out of 6 colours once only and then has a
sixteenth with the same colours as another, but in different order. Also a star-shaped puzzle of six
parallelograms.
Edwin L. Thurston. US Patent 487,798 -- Puzzle. Applied: 30 Sep 1890; patented: 13 Dec 1892. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. As far as I can see, this is the same as the 4 x 4
puzzle with 6‑coloured edges given above, but he seems to be
emphasising the 15 pieces.
Edwin L. Thurston. US Patent 490,689 -- Puzzle. Applied: 30 Sep 1890; patented: 31 Jun 1893. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. The patent is for 3 x 3 puzzles with 4‑coloured
corners or edges, but with pieces having no repeated colours and in a fixed
orientation. He selects some 8 of these
pieces for reasons not made clear and mentions moving them "after the manner of the old 13, 14, 15
puzzle." S&B, p. 36, describes
the Calumet Puzzle, Calumet Baking Powder Co., Chicago, which is a 3 x 3
head to tail puzzle, claimed to be covered by this patent.
Le Berger Malin. France, c1900. 3 x 3 head to tail
puzzle, but the edges are numbered and the matching edges must add to 10. ??NYS -- described by K. Takizawa, N.
Takashima & N. Yoshigahara; Vess Puzzle and Its Family -- A Compendium
of 3 by 3 Card Puzzles; published by the authors, Tokyo, 1983. Slocum has this in two different boxes and
dates it to c1900 -- I had c1915 previously.
Haubrich has one version, Produced by GB&O N.K. Atlas.
Angus K. Rankin. US Patent 1,006,878 -- Puzzle. Applied: 3 Feb 1911; patented: 24 Oct 1911. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. Described in S&B, p. 36. Grandpa's Wonder Puzzle. 3 x 3
square puzzle. Each piece has
corners coloured, using four colours, and the colours meeting at a corner must
differ. The patent doesn't show the
advertiser's name -- Grandpa's Wonder Soap -- but is otherwise identical to
S&B's photo.
Daily Mail World Record Net Sale
puzzle. 1920‑1921. Instructions and picture of the pieces. Letter from Whitehouse to me describing its
invention. 19 6-coloured hexagons without repeated colours. Daily Mail articles as follows. There may be others that I missed and
sometimes the page number is a bit unclear.
Note that 5 Dec was a Sunday.
9 Nov 1920, p. 5. "Daily Mail" puzzle.
To be issued on 7 Dec.
13 Nov
1920, p. 4. Hexagon mystery.
17 Nov
1920, p. 5. New mystery puzzle. Asserts the inventor does not know the
solution -- i.e. the solution has been locked up in a safe.
20 Nov
1920, p. 4. What is it?
23 Nov
1920, p. 5. Fascinating puzzle. The most fascinating puzzle since "Pigs
in Clover".
25 Nov
1920, p. 5. Can you do it?
29 Nov
1920, p. 5. £250 puzzle.
1 Dec 1920, p. 4. Mystery puzzle clues.
2 Dec 1920, p. 5. £250 puzzle race.
3 Dec 1920, p. 5. The puzzle.
4 Dec 1920, p. 4. The puzzle. Amplifies on
the inventor not knowing the solution -- after the idea was approved, a new
pattern was created by someone else and locked up.
6 Dec 1920, unnumbered back page. Photo with caption: £250 for solving this.
7 Dec 1920, p. 7. "Daily Mail" Puzzle. Released today. £100 for
getting the locked up solution. £100
for the first alternative solution and £50 for the next alternative solution. "It is believed that more than one
solution is possible."
8 Dec 1920, p. 5. "Daily Mail" puzzle.
9 Dec 1920, p. 5. Can you do it?
10 Dec
1920, p. 4. It can be done.
13 Dec
1920, p. 9. Most popular pastime. "More than 500,000 Daily Mail
Puzzles have been sold."
15 Dec
1920, p. 4. Puzzle king & the 19
hexagons. Dudeney says he does not
think it can be solved "except by trial."
16 Dec
1920, p. 4. Tantalising 19 hexagons.
16 Dec
1920, unnumbered back page. Banner at
top has: "The Daily Mail"
puzzle. Middle of page has a cartoon of
sailors trying to solve it.
17 Dec
1920, p. 5? The Xmas game.
18 Dec
1920, p. 7. Puzzle Xmas 'card'.
20 Dec
1920, p. 7. Hexagon fun.
22 Dec
1920, p. 3. 3,000,000 fascinated. It is assumed that about 5 people try each
example and so this indicates that nearly 600,000 have been sold.
23 Dec
1920, p. 3. Too many cooks.
23 Dec
1920, unnumbered back page.
Cartoon: The hexagonal dawn!
28 Dec
1920, p. 3? Puzzled millions. "On Christmas Eve the sales exceeded
600,000 ...."
29 Dec
1920, p. 3? "I will do
it."
30 Dec
1920, p. 8. Puzzle fun.
3 Jan 1921, p. 3. The Daily Mail Puzzle. C.
Lewis, aged 21, a postal clerk solved it within two hours of purchase and
submitted his solution on 7 Dec.
Hundreds of identical solutions were submitted, but no alternative
solutions have yet appeared. There are
two pairs of identical pieces: 1 &
12, 4 & 10.
3 Jan 1921, p. 10 = unnumbered back
page. Hexagon Puzzle Solved, with photo
of C. Lewis and diagram of solution.
10 Jan
1921, p. 4. Hexagon puzzle. Since no alternative hexagonal solutions
were received, the other £150 is awarded to those who submitted the most
ingenious other solution -- this was judged to be a butterfly shape, submitted
by 11 persons, who shared the £150.
Horace Hydes & Francis
Reginald Beaman Whitehouse. UK Patent
173,588 -- Improvements in Dominoes.
Applied: 29 Sep 1920; complete
application: 29 Jun 1921; accepted:
29 Dec 1921. Reproduced in
Haubrich, About ..., 1996, op. cit. below.
3pp + 1p diagrams. This is the
patent for the above puzzle, corresponding to provisional patent 27599/20
on the package. The illustration
shows a solved puzzle based on 'A
stitch in time saves nine'.
George Henry Haswell. US Patent 1,558,165 -- Puzzle. Applied: 3 Jul 1924; patented: 11 Sep 1925. Reproduced in Haubrich, About ..., 1996, op.
cit. below. 2pp + 1p diagrams. For edge-matching hexagons with further
internal markings which have to be aligned.
[E.g. one could draw a diagonal and require all diagonals to be vertical
-- this greatly simplifies the puzzle!]
If one numbers the vertices 1,
2, ..., 6, he gives an example formed
by drawing the diagonals 13, 15, 42,
46 which produces six triangles along
the edges and an internal rhombus.
C. Dudley Langford. Note 2829:
Dominoes numbered in the corners.
MG 43 (No. 344) (May 1959) 120‑122. Considers triangles, squares and hexagons with numbers at the
corners. There are the same number of
pieces as with numbers on the edges, but corner numbering gives many more kinds
of edges. E.g. with four numbers, there
are 24 triangles, but these have 16 edge patterns instead of 4. The editor (R. L. Goodstein) tells Langford
that he has made cubical dominoes "presumably with faces
numbered". Langford suggests cubes
with numbers at the corners. [I find 23
cubes with two corner numbers and 333 with three corner numbers. ??check]
Piet Hein. US Patent 4,005,868 -- Puzzle. Applied: 23 Jun 1975; patented: 1 Feb 1977. Front page + 8pp diagrams + 5pp text. Basically non-matching puzzles using marks
at the corners of faces of the regular polyhedra. He devises boards so the problems can be treated as planar.
Kiyoshi Takizawa; Naoaki Takashima & Nob.
Yoshigahara. Vess Puzzle and Its Family
-- A Compendium of 3 by
3 Card Puzzles. Published by the authors, Tokyo, Japan,
1983. Studies 32 types (in 48
versions) of 3 x 3 'head to tail' matching puzzles and 4
related types (in 4 versions).
All solutions are shown and most puzzles are illustrated with colour
photographs of one solution. (Haubrich
counts 51 versions -- check??)
Melford D. Clark. US Patent 4,410,180 -- Puzzle. Applied: 16 Nov 1981; patented: 18 Oct 1983. Reproduced in Haubrich, About ..., 1996, op.
cit. in 5.H.4. 2pp + 2pp diagrams. Corner matching squares, but with the pieces
marked 1, 2, ..., so that the pieces marked 1 form a
1 x 1 square, the pieces
marked 2 allow this to be extended to a
2 x 2 square, etc. There are
n2 - (n-1)2
pieces marked n.
Jacques Haubrich. Compendium of Card Matching Puzzles. Printed by the author, Aeneaslaan 21,
NL-5631 LA Eindhoven, Netherlands, 1995.
2 vol., 325pp. describing over
1050 puzzles. He classifies them by the nine most common
matching rules: Heads and Tails; Edge Matching (i.e. MacMahon); Path Matching; Corner Matching; Corner
Dismatching; Jig-Saw-Like; Continuous Path; Edge Dismatching;
Hybrid. He does not include
Jig-Saw-Like puzzles here. Using the
number of cards and their shape, then the matching rules, he has 136
types. 31 different numbers of cards occur: 4, 6-16, 18-21, 23-25, 28, 30, 36, 40, 45, 48, 56, 64, 70, 80,
85, 100. There is an index of 961
puzzle names. He says Hoffmann is the
earliest published example. He notes
that most path puzzles have a global criterion that the result have a single
circuit which slightly removes them from his matching criterion and he does not
treat them as thoroughly. He has
developed computer programs to solve each type of puzzle and has checked them
all.
Jacques Haubrich. About, Beyond and Behind Card Matching
Puzzles. [= Vol. 3 of above]. Ibid, Apr 1996, 87pp. This is a general discussion of the
different kinds of puzzles, how to solve them and their history, reproducing
ten patents and two obituaries.
5.I. LATIN SQUARES AND EULER SQUARES
This topic ties in with certain
tournament problems but I have not covered them. See also Hoffmann and Loughlin & Flood in 5.A.2 for examples
of two orthogonal 3 x 3 Latin squares.
The derangement problems in 5.K.2 give Latin rectangles.
Ahrens-1 & Ahrens-2. Opp. cit. in 7.N. 1917 & 1922. Ahrens-1
discusses and cites early examples of Latin squares, going back to medieval
Islam (c1200), where they were used on amulets. Ahrens-2 particularly discusses work of al‑Buni -- see
below.
(Ahmed [the h
should have an underdot] ibn ‘Alî ibn Jûsuf) el‑Bûni, (Abû'l‑‘Abbâs,
el‑Qoresî.) = Abu‑l‘Abbas
al‑Buni. (??= Muhyi'l‑Dîn
Abû’l-‘Abbâs al‑Bûnî -- can't
relocate my source of this form.) Sams
al‑ma‘ârif = Shams al‑ma‘ârif
al‑kubrâ = Šams
al-ma‘ārif. c1200. ??NYS.
Ahrens-1 describes this briefly and incorrectly. He expands and corrects this work in
Ahrens-2. See 7.N for more
details. Ahrens notes that a 4 x 4
magic square can be based on the pattern of two orthogonal Latin squares
of order 4, and Al-Buni's work indicates knowledge of such a pattern,
exemplified by the square
8, 11, 14,
1; 13, 2,
7, 12; 3, 16, 9,
6; 10, 5,
4, 15 considered (mod 4).
He also has Latin squares of order
4 using letters from a name of
God. He goes on to show 7
Latin squares of order 7, using the same 7 letters each time --
though four are corrupted. (Throughout,
the Latin squares also have 'Latin' diagonals, i.e. the diagonals contain all
the values.) These are arranged so each
has a different letter in the first place.
It is conjectured that these are associated with the days of the week or
the planets.
Tagliente. Libro de Abaco. (1515). 1541. F. 18v.
7 x 7 Latin square with
entries 1, 13, 2, 14, 3, 10,
4 cyclically shifted forward -- i.e.
the second row starts 13, 2, .... This is an elaborate plate which notes that
the sum of each file is 47 and has a motto: Sola Virtu la Fama Volla, but I
could find no text or other reason for its appearance!
Inscription on memorial to
Hannibal Bassett, d. 1708, in Meneage parish church, St. Mawgan, Cornwall. I first heard of this from Chris Abbess, who
reported it in some newsletter in c1993.
However, [Peter Haining; The Graveyard Wit; Frank Graham, Newcastle,
1973, p. 133] cites this as being at Cunwallow, near Helstone, Cornwall. [W. H. Howe; Everybody's Book of Epitaphs
Being for the Most Part What the Living Think of the Dead; Saxon & Co.,
London, nd [c1895] (facsimile by Pryor Publications, Whitstable, 1995); p. 173]
says it is in Gunwallow Churchyard.
Spelling and punctuation vary a bit.
The following gives a detailed account.
Alfred Hayman Cummings. The Churches and Antiquities of Cury &
Gunwalloe, in the Lizard District, including Local Traditions. E. Marlborough & Co., London & Truro,
1875, pp. 130-131. ??NX. "It has been said that there once
existed ... the curious epitaph --" and gives a considerable rearrangement
of the inscription below. He continues
"But this is in all probability a mistake, as repeated search has been
made for it, not only by the writer, but by a former Vicar of Gunwalloe, and it
could nowhere be found, while there is a plate with an inscription in
the church at Mawgan, the next parish, which might be very easily the one
referred to." He gives the
following inscription, saying it is to Hannibal Basset, d. 1708-9. Chris Weeks was kind enough to actually go
to the church of St. Winwaloe, Gunwalloe, where he found nothing, and to St.
Mawgan in Meneage, a few miles away.
Chris Weeks sent pictures of Gunwallowe -- the church is close to the
cliff edge and it looks like there could once have been more churchyard on the
other side of the church where the cliff has fallen away. In the church at St. Mawgan is the brass
plate with 'the Acrostic Brass Inscription', but it is not clearly associated
with a grave and I wonder if it may have been moved from Gunwallowe when a
grave was eroded by the sea. It is on
the left of the arch by the pulpit. I
reproduce Chris Weeks' copy of the text.
He has sent a photograph, but it was dark and the photo is not very
clear, but one can make out the Latin square part.
Hanniball
Baòòet here Inter'd doth lye
Who
dying lives to all Eternitye
hee
departed this life the 17th of Ian
1709/8
in the 22th year of his age ~
A
lover of learning
Shall wee all dye
Wee shall dye all
all dye shall wee
dye all wee shall
The òò are old style long esses.
The superscript th is actually over the numeral. The
9 is over the 8 in
the year and there is no stroke. This
is because it was before England adopted the Gregorian calendar and so the year
began on 25 Mar and was a year behind the continent between 1 Jan and 25
Mar. Correspondence of the time
commonly would show 1708/9 at this time, and I have used this form for
typographic convenience, but with the
9 over the 8 as
on the tomb.
A
word game book points out that this inscription is also palindromic!!
Richard Breen. Funny Endings. Penny Publishing, UK, 1999, p. 35. Gives the following form:
Shall we all die? / We shall die all. / All die shall we? /
Die all we shall and notes that
it is a word palindrome and says it comes from Gunwallam [sic], near Helstone.
Joseph Sauveur. Construction générale des quarrés
magiques. Mémoires de l'Académie Royale
des Sciences 1710(1711) 92‑138.
??NYS -- described in Cammann‑4, p. 297, (see 7.N for details of
Cammann) which says Sauveur invented Latin squares and describes some of his
work.
Ozanam. 1725.
1725: vol. IV, prob. 29, p. 434
& fig. 35, plate 10 (12). Two 4 x 4
orthogonal squares, using A, K,
Q, J of the 4 suits, but it looks
like:
J¨, A©, K§, Qª; Qª, K§, A¨, J©; A§, Jª, Q©, K¨; K©, Q¨, J§, Aª; but the
§ and ª look very similar. From later versions of the same diagram, it is clear that the
first row should have its § and ª reversed. Note the
diagonals also contain all four ranks and suits. (I have a reference for this to the 1723 edition.)
Minguet. 1733.
Pp. 146-148 (1864: 142-143; not noticed in other editions). Two
4 x 4 orthogonal squares,
using A, K, Q, J (= As, Rey, Caballo (knight), Sota (knave))
of the 4 suits, but the Spanish suits, in descending order, are: Espadas,
Bastos, Oros, Copas. The result is
described but not drawn, as:
RO,
AE, CC, SB; SC, CB, AO, RE; AB, RC, SE, CO; CE, SO, RB, AC;
which
would translate into the more usual cards as:
K¨, A§, Qª, J©; J§, Q©, A¨, Kª; A©, K§, Jª, Q¨; Qª, J¨, K©, A§.
However,
I'm not sure of the order of the Caballo and Sota; if they were reversed, which
would interchange Q and J in the latter pattern, then both Ozanam and Minguet
would have the property that each row is a cyclic shift or reversal of A, K, Q, J.
Alberti. 1747.
Art. 29, p. 203 (108) & fig. 36, plate IX, opp. p. 204
(108). Two 4 x 4 orthogonal squares,
figure simplified from the correct form of Ozanam, 1725.
L. Euler. Recherches sur une nouvelle espèce de
Quarrés Magiques. (Verhandelingen
uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen
(= Flessingue) 9 (1782) 85‑239.)
= Opera Omnia (1) 7 (1923) 291‑392. (= Comm. Arithm. 2 (1849) 302‑361.)
Manuel des Sorciers. 1825.
Pp. 78-79, art. 39. ??NX Correct form of Ozanam.
The Secret Out. 1859.
How to Arrange the Twelve Picture Cards and the four Aces of a Pack in
four Rows, so that there will be in Neither Row two Cards of the same Value nor
two of the same Suit, whether counted Horizontally or Perpendicularly, pp.
90-92. Two 4 x 4
orthogonal Latin squares, not the same as in Ozanam.
Bachet-Labosne. Problemes.
3rd ed., 1874. Supp. prob. XI,
1884: 200‑202. Two 4 x 4
orthogonal squares.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
Card Puzzles, No. XVI, pp. 17-18.
Similar to Ozanam.
Hoffmann. 1893.
Chap. X, no. 14: Another card puzzle, pp. 342 & 378-379
= Hoffmann‑Hordern, pp. 234 & 236. Two orthogonal Latin squares, but the diagonals do not contain
all the suits and ranks.
Aª, J©, Q¨, K§; J¨, A§, Kª, Q©; Q§, K¨, A©, Jª; K©, Qª, J§, A¨.
G. Tarry. Le probleme de 36 officiers. Comptes Rendus de l'Association Française pour
l'Avancement de Science Naturel 1 (1900) 122‑123 &
2 (1901) 170‑203. ??NYS
Dudeney. Problem 521. Weekly Dispatch (1 Nov, 15 Nov, 1903) both p. 10.
H. A. Thurston. Latin squares. Eureka 9 (Apr 1947) 19-21.
Survey of current knowledge.
T. G. Room. Note 2569:
A new type of magic square. MG
39 (No. 330) (Dec 1955) 307. Introduces
'Room Squares'. Take the 2n(2n‑1)/2 combinations from 2n symbols and insert them in a 2n‑1 x 2n‑1 grid so that each row and column contains
all 2n
symbols. There are n
entries and n‑1 blanks in each row and column. There is an easy solution for n = 1.
n = 2 and n = 3
are impossible. Gives a solution
for n = 4. This is a design for a round‑robin tournament with the
additional constraint of 2n‑1 sites such that each player plays once at
each site.
Parker shows there are two
orthogonal Latin squares of order 10 in 1959.
R. C. Bose &
S. S. Shrikande. On the falsity
of Euler's conjecture about the nonexistence of two orthogonal Latin squares of
order 4t+2. Proc. Nat. Acad. Sci. (USA) 45: 5 (1959) 734‑737.
Gardner. SA (Nov 1959) c= New MD, chap. 14.
Describes Bose & Shrikande's work.
SA cover shows a 10 x 10 counterexample in colour. Kara Lynn and David Klarner actually made a quilt
of this, thereby producing a counterpane counterexample! They told me that the hardest part of the
task was finding ten sufficiently contrasting colours of material.
H. Howard Frisinger. Note:
The solution of a famous two-centuries-old problem: the Leonhard Euler-Latin square
conjecture. HM 8 (1981) 56-60. Good survey of the history.
Jacques Bouteloup. Carrés Magiques, Carrés Latins et
Eulériens. Éditions du Choix, Bréançon,
1991. Nice systematic survey of this
field, analysing many classic methods.
An Eulerian square is essentially two orthogonal Latin squares.
See MUS I 210-284. S&B 37 shows examples. See also 5.Z. See also 6.T for examples where no three are in a row.
Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904.
Pp. 1082‑1084 discusses history and results for the n
queens problem, with many references.
Paul J. Campbell. Gauss and the eight queens problem. HM 4:4 (Nov 1977) 397‑404. Detailed history. Demonstrates that Gauss did not obtain a complete solution and
traces how this misconception originated and spread.
"Schachfreund" (Max
Bezzel). Berliner Schachzeitung 3 (Sep
1848) 363. ??NYS
Solutions. Ibid. 4 (Jan 1849) 40. ??NYS
(Ahrens says this only gives two solutions. A. C. White says two or three. Jaenisch says a total of 5
solutions were published here and in 1854.)
Franz Nauck. Eine in das Gebiet der Mathematik fallende
Aufgabe von Herrn Dr. Nauck in Schleusingen.
Illustrirte Zeitung (Leipzig) 14 (No. 361) (1 Jun 1850) 352. Reposes problem. [The papers do not give a first name or initial. The only Nauck in the first six volumes of
Poggendorff is Ernst Friedrich (1819-1875), a geologist. Ahrens gives no initial. Campbell gives Franz.]
Franz Nauck. Briefwechseln mit Allen für Alle. Illustrirte Zeitung (Leipzig) 15 (No. 377)
(21 Sep 1850) 182. Complete
solution.
Editorial comments:
Briefwechsel. Illustrirte Zeitung
(Leipzig) 15 (No. 378) (28 Sep 1850) 207.
Thanks 6 correspondents for the complete solution and says Nauck reports
that a blind person has also found all
92 solutions.
Gauss read the Illustrirte
Zeitung and worked on the problem, corresponding with his friend Schumacher
starting on 1 Sep 1850. Campbell
discusses the content of the letters, which were published in: C. A. F. Peters, ed; Briefwechsel zwischen
C. F. Gauss und H. C. Schumacher; vol. 6, Altona, 1865, ??NYS. John Brillhart writes that there is some
material in Gauss' Werke, vol. XII: Varia kleine Notizen verschiednen Inhalts
... 5, pp. 19-28, ??NYS -- not
cited by Campbell.
F. J. E. Lionnet. Question 251. Nouvelles Annales de Mathématiques 11 (1852) 114‑115. Reposes problem and gives an abstract
version.
Giusto Bellavitis. Terza rivista di alcuni articoli dei Comptes
Rendus dell'Accademia delle Scienze di Francia e di alcuni questioni des
Nouvelles Annales des mathématiques.
Atti dell'I. R. Istituto Veneto di Scienze, Lettere ed Arti (3) 6 [=
vol. 19] (1860/61) 376-392 & 411‑436 (as part of Adunanza del Giorno
17 Marzo 1861 on pp. 347‑436).
The material of interest is: Q. 251.
Disposizione sullo scacchiere di otto regine, on pp. 434‑435. Gives the
12 essentially different
solutions. Lucas (1895) says Bellavitis
was the first to find all solutions, but see above. However this may be the first appearance of the 12
essentially different solutions.
C. F. de Jaenisch. Op. cit. in 5.F.1. 1862. Vol. 1, pp.
122-135. Gives the 12 basic solutions and
shows they produce 92. Notes that in
every solution, 4 queens are on white squares and 4 are on black.
A. C. Cretaine. Études sur le Problème de la Marche du
Cavalier au Jeu des Échecs et Solution du Problème des Huit Dames. A. Cretaine, Paris, 1865. ??NYS -- cited by Lucas (1895). Shows it is possible to solve the eight
queens problem after placing one queen arbitrarily.
G. Bellavitis. Algebra N. 72 Lionnet. Atti dell'Istituto Veneto (3) 15 (1869/70)
844‑845.
Siegmund Günther. Zur mathematische Theorie des
Schachbretts. Grunert's Archiv der
Mathematik und Physik 56 (1874) 281-292.
??NYS. Sketches history of the
problem -- see Campbell. He gives a
theoretical, but not very practical, approach via determinants which he carries
out for 4 x 4 and 5 x 5.
J. W. L. Glaisher. On the problem of the eight queens. Philosophical Magazine (4) 48 (1874)
457-467. Gives a sketch of Günther's
history which creates several errors, in particular attributing the solution to
Gauss -- see Campbell, who suggests Glaisher could not read German well. (However, in 1921 & 1923, Glaisher
published two long articles involving the history of 15-16C German mathematics,
showing great familiarity with the language.)
Simplifies and extends Günther's approach and does 6 x 6,
7 x 7, 8 x 8 boards.
Lucas. RM2, 1883. Note V: Additions du Tome premier. Pp. 238-240. Gives the solutions on the
9 x 9 board, due to P. H.
Schoute, in a series of articles titled Wiskundige Verpoozingen in Eigen
Haard. Gives the solutions on the 10 x 10
board, found by M. Delannoy.
S&B, p. 37, show an 1886
puzzle version of the six queens problem.
A. Pein. Aufstellung von n Königinnen auf einem
Schachbrett von n2 Feldern.
Leipzig. ??NYS -- cited by Ball,
MRE, 4th ed., 1905 as giving the 92 inequivalent solutions on the 10 x 10.
Ball. MRE, 1st ed., 1892. The
eight queens problem, pp. 85-88. Cites
Günther and Glaisher and repeats the historical errors. Sketches Günther's approach, but only cites
Glaisher's extension of it. He gives
the numbers of solutions and of inequivalent solutions up through 10 x 10
-- see Dudeney below for these numbers, but the two values in ( )
are not given by Dudeney. He
states results for the 9 x 9 and
10 x 10, citing Lucas. Says that a
6 x 6 version "is sold in
the streets of London for a penny".
Hoffmann. 1893.
Chap. VI, pp. 272‑273 & 286 = Hoffmann-Hordern,
pp. 187-189, with photo.
No.
24: No two in a row. Eight queens. Photo on p. 188 shows Jeu des Sentinelles,
by Watilliaux, dated 1874-1895.
No.
25: The "Simple" Puzzle. Nine
queens. Says a version was sold by
Messrs. Feltham, with a notched board but the pieces were allowed to move over
the gaps, so it was really a 9 x 9 board.
Chap. X, No. 18: The Treasure at Medinet, pp. 343‑344
& 381 = Hoffmann-Hordern, pp. 237-239. This is a solution of the eight queens problem, cut into four
quadrants and jumbled. The goal is to
reconstruct the solution. Photo on p.
239 shows Jeu des Manifestants, with box.
Hordern
Collection, p. 94, and S&B, p. 37, show a version of this with same box,
but which divides the board into eight
2 x 4 rectangles.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co.,
NY), nd [1895]. P. 1: The famous
Italian pin puzzle. 6 queens puzzle. No solution.
Lucas. L'Arithmétique Amusante.
1895. Note IV: Section I: Les huit dames, pp. 210-220. Asserts Bellavitis was the first to find all
solutions. Discusses symmetries and
shows the 12 basic solutions. Correctly
describes Jaenisch as obscure. Gives an
easy solution of Cretaine's problem which can be remembered as a trick. Shows there are six solutions which can be
superimposed with no overlap, i.e. six solutions using disjoint sets of cells.
C. D. Locock, conductor. Chess Column. Knowledge 19 (Jan 1896)
23-24; (Feb 1896) 47‑48; (May 1896) 119; (Jul 1896) 167-168. This
series begins by saying most players know there is a solution, "but,
possibly, some may be surprised to learn that there are ninety-two ways of performing
the feat, ...." He then enumerates
them. Second article studies various properties
of the solutions, particularly looking for examples where one solution shifts
to produce another one. Third article
notes some readers' comments. Fourth
article is a long communication from W. J. Ashdown about the number of distinct
solutions, which he gets as 24 rather than the usual 12.
T. B. Sprague. Proc. Edinburgh Math. Soc. 17 (1898-9)
43-68. ??NYS -- cited by Ball, MRE, 4th
ed., 1905, as giving the 341 inequivalent solutions on the 11 x 11.
Benson. 1904.
Pins and dots puzzle, p. 253. 6
queens problem, one solution.
Ball. MRE, 4th ed., 1905. The
eight queens problem, pp. 114-120.
Corrects some history by citing MUS, 1st ed., 1901. Gives one instance of Glaisher's method --
going from 4 x 4 to 5
x 5 and its results going up to 8 x 8.
Says the 92 inequivalent solutions on the 10 x 10 were given
by Pein and the 341 inequivalent solutions on the 11 x 11 were given by
Sprague. The 5th ed., pp. 113-119 calls
it "One of the classical problems connected with a chess-board" and
adds examples of solutions up to 21 x
21 due to Mr. Derington.
Pearson. 1907.
Part III, no. 59: Stray dots, pp. 59 & 130. Same as Hoffmann's Treasure at Medinet.
Burren Loughlin &
L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909.
The eight provinces, pp. 14-15 & 65. Same as Hoffmann's Treasure at Medinet.
A. C. White. Sam Loyd and His Chess Problems. 1913.
Op. cit. in 1. P. 101 says Loyd
discovered that all solutions have a piece at
d1 or equivalent.
Williams. Home Entertainments. 1914.
A draughtboard puzzle, p. 115.
"Arrange eight men on a draughtboard in such a way that no two are
upon the same line in any direction." This is not well stated!!
Gives one solution:
52468317 and says "Work out other solutions for
yourself."
Dudeney. AM.
1917. The guarded chessboard,
pp. 95‑96. Gives the number of
ways of placing n queens and the number of inequivalent ways. The values in ( ) are given by Ball,
but not by Dudeney.
n 4 5 6 7 8 9 10 11 12 13
ways 2 10 4 40 92 (352) (724) ‑ - -
inequivalent ways 1 2 1 6 12 46 92 341 (1766) (1346)
Ball. MRE, 9th ed., 1920. The
eight queens problem, pp. 113-119.
Omits references to Pein and Sprague and adds the number of inequivalent
solutions for the 12 x 12 and
13 x 13.
Blyth. Match-Stick Magic.
1921. No pairs allowed, p.
74. 6 queens problem.
Hummerston. Fun, Mirth & Mystery. 1924.
No two in a line, p. 48.
Chessboard. Place 'so that no
two are upon the same line in any direction along straight or diagonal
lines?' Gives one solution: 47531682,
'but there are hundreds of other ways'.
You can let someone place the first piece.
Rohrbough. Puzzle Craft. 1932. Houdini Puzzle, p.
17. 6 x 6 case.
Rohrbough. Brain Resters and Testers. c1935.
Houdini Puzzle, p. 25. 6 x
6 problem. "-- From New York World some years ago, credited to
Harry Houdini." I have never seen
this attribution elsewhere.
Pál Révész. Mathematik auf dem Schachbrett. In:
Endre Hódi, ed. Mathematisches Mosaik.
(As: Matematikai Érdekességek;
Gondolat, Budapest, 1969.) Translated
by Günther Eisenreich. Urania‑Verlag,
Leipzig, 1977. Pp. 20‑27. On p. 24, he says that all solutions have 4
queens on white and 4 on black. He says
that one can place at most 5 non‑attacking queens on one colour.
Doubleday - 2. 1971.
Too easy?, pp. 97-98. The two
solutions on the 4 x 4 board are disjoint.
Dean S. Clark & Oved
Shisha. Proof without words: Inductive
construction of an infinite chessboard with maximal placement of nonattacking
queens. MM 61:2 (1988) 98. Consider a
5 x 5 board with queens in cells (1,1), (2,4), (3,2), (4,5), (5,3). 5 such boards can be similarly placed within
a 25 x 25 board viewed as a 5 x
5 array of 5 x 5 boards and this has
no queens attacking. Repeating the
inflationary process gives a solution on the board of edge 53, then the board of edge 54, ....
They cite their paper:
Invulnerable queens on an infinite chessboard; Annals of the NY Acad. of
Sci.: Third Intern. Conf. on Comb. Math.; to appear. ??NYS.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B.
1991. Squares before your eyes,
pp. 21 & 106. Asks for solutions of
the eight queens problem with no piece on either main diagonal. Two of the 12 basic solutions have this, but
one of these is the symmetric case, so there are 12 solutions of this problem.
Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by the
author. See the discussion in 6.F. He finds the following numbers of solutions
for placing n queens, n = 1, 2, ...,
18.
1, 0,
0, 2, 10, 4, 40,
92, 352, 724,
2680, 14200, 73712,
3 65596, 22 79184, 147 72512,
958 15104, 6660 90624.
5.I.2. COLOURING CHESSBOARD WITH NO REPEATS IN A LINE
New
section. I know there is a general
result that an n x n board can be n‑coloured if
n satisfies some condition
like n º 1 or 5
(mod 6), but I don't recall any other
old examples of the problem.
Dudeney. Problem 50: A problem in mosaics. Tit‑Bits 32 (11 Sep 1897) 439 &
33 (2 Oct 1897) 3.
An 8 x 8 board with two adjacent corners omitted can
be 8‑coloured with no two in a row, column or diagonal. = Anon. & Dudeney; A chat with the
Puzzle King; The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599
& 3:1 (Apr 1900) 89.
Dudeney. AM.
1917. Prob. 302: A problem in
mosaics, pp. 90 & 215-216. The
solution to the previous problem is given and then it is asked to relay the
tiles so that the omitted squares are the
(3,3) and (3,6)
cells.
Hummerston. Fun, Mirth & Mystery. 1924.
Q.E.D. -- The office boy problem, Puzzle no. 30, pp. 82 & 176. Wants to mark the cells of a 4 x 4
board with no two the same in any 'straight line ..., either
horizontally, vertically, or diagonally.'
His answer is:
ABCD, CDEA,
EABC, BCDA, which has no two the same on any short
diagonal. The problem uses coins of
values: A, B, C, D, E =
12, 30, 120, 24, 6 and the
object is to maximize the total value of the arrangement. In fact, there are only two ways to 5-colour
the board and they are mirror images.
Four colours are used three times and one is used four times -- setting
the value 120 on the latter cells gives the maximum value of 696.
NOTE.
Perfect means no two squares are the same size. Compound means there is a squared
subrectangle. Simple means not
compound.
Dudeney. Puzzling Times at Solvamhall Castle: Lady
Isabel's casket. London Mag. 7
(No. 42) (Jan 1902) 584 & 8 (No. 43) (Feb 1902) 56. = CP, prob. 40, pp. 67 & 191‑193. Square into
12 unequal squares and a
rectangle.
Max Dehn. Über die Zerlegung von Rechtecken in
Rechtecke. Math. Annalen 57 (1903) 314‑332. Long and technical. No examples. Shows sides must be parallel and commensurable.
Loyd. The patch quilt puzzle.
Cyclopedia, 1914, pp. 39 & 344.
= MPSL1, prob. 76, pp. 73 & 147‑148. c= SLAHP: Building a patchquilt, pp. 30
& 92. 13 x 13 into
11 squares, not simple nor
perfect. (Gardner, in 536, says this
appeared in Loyd's "Our Puzzle Magazine", issue 1 (1907), ??NYS.)
Loyd. The darktown patch quilt party.
Cyclopedia, 1914, pp. 65 & 347.
12 x 12 into 11
squares, not simple nor perfect, in two ways.
P. J. Federico. Squaring rectangles and squares -- A
historical review with annotated bibliography.
In: Graph Theory and Related
Topics; ed. by J. A. Bondy & U. S. R. Murty; Academic Press, NY, 1979, pp.
173‑196. Pp. 189‑190 give
the background to Moroń's work.
Moroń later found the first example of Sprague but did not publish
it.
Z. Moroń. O rozkładach prostokątów na
kwadraty (In Polish) (On the dissection of a rectangle into squares). Przegląd Matematyczno‑Fizyczny
(Warsaw) 3 (1925) 152‑153.
Decomposes rectangles into 9 and 10 unequal squares. (Translation provided by A. Mąkowski,
1p. Translation also available from M.
Goldberg, ??NYS.)
M. Kraitchik. La Mathématique des Jeux, 1930, op. cit. in
4.A.2, p. 272. Gives Loyd's "Patch
quilt puzzle" solution and Lusin's opinion that there is no perfect solution.
A. Schoenflies. Einführung in der analytische Geometrie der
Ebene und des Raumes. 2nd ed., revised
and extended by M. Dehn, Springer, Berlin, 1931. Appendix VI: Ungelöste Probleme der Analytischen Geometrie,
pp. 402‑411. Same results as
in Dehn's 1903 paper.
Michio Abe. On the problem to cover simply and without
gap the inside of a square with a finite number of squares which are all
different from one another (in Japanese).
Proc. Phys.‑Math. Soc. Japan 4 (1931) 359‑366. ??NYS
Michio Abe. Same title (in English). Ibid. (3) 14 (1932) 385‑387. Gives
191 x 195 rectangle into 11
squares. Shows there are squared
rectangles arbitrarily close to squares.
Alfred Stöhr. Über Zerlegung von Rechtecken in
inkongruente Quadrate. Schr. Math.
Inst. und Inst. angew. Math. Univ. Berlin 4:5 (1939), Teubner, Leipzig, pp. 119‑140. ??NYR.
(This was his dissertation at the Univ. of Berlin.)
S. Chowla. The division of a rectangle into unequal
squares. Math. Student 7 (1939) 69‑70. Reconstructs Moroń's 9
square decomposition.
Minutes of the 203rd Meeting of
the Trinity Mathematical Society (Cambridge) (13 Mar 1939). Minute Books, vol. III, pp. 244‑246. Minutes of A. Stone's lecture: "Squaring the Square". Announces Brooks's example with 39
elements, side 4639, but containing a perfect subrectangle.
Minutes of the 204th Meeting of
the Trinity Mathematical Society (Cambridge) (24 Apr 1939). Minute Books, vol. III, p. 248. Announcement by C. A. B. Smith that Tutte
had found a perfect squared square with no perfect subrectangle.
R. Sprague. Recreation in Mathematics. Op. cit. in 4.A.1. 1963. The expanded
foreword of the English edition adds comments on Dudeney's "Lady Isabel's
Casket", which led to the following paper.
R. Sprague. Beispiel einer Zerlegung des Quadrats in
lauter verschiedene Quadrate. Math.
Zeitschr. 45 (1939) 607‑608.
First perfect squared square --
55 elements, side 4205.
R. Sprague. Zur Abschätzung der Mindestzahl
inkongruenter Quadrate, die ein gegebenes Rechteck ausfüllen. Math. Zeitschrift 46 (1940) 460‑471. Tutte's 1979 commentary says this shows
every rectangle with commensurable sides can be dissected into unequal squares.
A. H. Stone, proposer; M. Goldberg & W. T. Tutte, solvers. Problem E401. AMM 47:1 (Jan 1940) 48
& AMM 47:8 (Oct 1940) 570‑572. Perfect squared square -- 28
elements, side 1015.
R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte. The dissection of rectangles into squares. Duke Math. J. 7 (1940) 312‑340. = Selected Papers of W. T. Tutte; Charles
Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 10-38, with
commentary by Tutte on pp. 1-9. Tutte's
1979 commentary says Smith was perplexed by the solution of Dudeney's
"Lady Isabel's Casket" -- see also his 1958 article.
A. H. Stone, proposer; Michael Goldberg, solver. Problem E476. AMM 48 (1941) 405 ??NYS
& 49 (1942) 198-199. An isosceles right triangle can be dissected
into 6
similar figures, all of different sizes. Editorial notes say that Douglas and Starke found a different
solution and that one can replace
6 by any larger number, but it
is not known if 6 is the least such. Stone asks if there is any solution where the smaller triangles
have no common sides.
M. Kraitchik. Mathematical Recreations, op. cit. in 4.A.2,
1943. P. 198. Shows the compound perfect squared square with 26
elements and side 608 from Brooks, et al.
C. J. Bouwkamp. On the construction of simple perfect
squared squares. Konink. Neder. Akad.
van Wetensch. Proc. 50 (1947) 72-78 = Indag. Math. 9 (1947) 57-63. This criticised the method of Brooks, Smith,
Stone & Tutte, but was later retracted.
Brooks, Smith, Stone &
Tutte. A simple perfect square. Konink. Neder. Akad. van Wetensch. Proc. 50
(1947) 1300‑1301. = Selected
Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre,
Manitoba, 1979; pp. 99-100, with commentary by Tutte on p. 98. Bouwkamp had published several notes and was
unable to make the authors' 1940 method work.
Here they clarify the situation and give an example. One writer said they give details of Sprague's
first example, but the example is not described as being the same as in
Sprague.
W. T. Tutte. The dissection of equilateral triangles into
equilateral triangles. Proc. Camb. Phil
Soc. 44 (1948) 464‑482. =
Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre,
Manitoba, 1979; pp. 106-125, with commentary by Tutte on pp. 101-105.
T. H. Willcocks, proposer and
solver. Problem 7795. Fairy Chess Review 7:1 (Aug 1948) 97 &
106 (misnumberings for 5 & 14). Refers
to prob. 7523 -- ??NYS. Finds compound
perfect squares of orders 27, 27, 28
and 24.
T. H. Willcocks. A note on some perfect squares. Canadian J. Math. 3 (1951) 304‑308. Describes the result in Fairy Chess Review
prob. 7795.
T. H. Willcocks. Fairy Chess Review (Feb & Jun
1951). Prob. 8972. ??NYS -- cited and described by G. P.
Jelliss; Prob. 44 -- A double squaring, G&PJ 2 (No. 17) (Oct 1999) 318-319. Squares of edges 3, 5, 9, 11, 14, 19, 20, 24, 31, 33, 36, 39, 42 can be formed into a 75 x 112
rectangle in two different ways.
{These are reproduced, without attribution, as Fig. 21, p. 33 of Joseph
S. Madachy; Madachy's Mathematical Recreations; Dover, 1979 (this is a
corrected reprint of Mathematics on Vacation, 1966, ??NYS). The 1979 ed. has an errata slip inserted for
p. 33 as the description of Fig. 21 was omitted in the text, but the erratum
doesn't cite a source for the result.}
The G&PJ problem then poses a new problem from Willcocks involving 21
squares to be made into a rectangle in two different ways -- it is not
clear if these have to be the same shape.
M. Goldberg. The squaring of developable surfaces. SM 18 (1952) 17‑24. Squares cylinder, Möbius strip, cone.
W. T. Tutte. Squaring the square. Guest column for SA (Nov 1958). c= Gardner's 2nd Book, pp. 186‑209. The latter
= Selected Papers of W. T.
Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp.
244-266, with a note by Tutte on p. 244, but the references have been
omitted. Historical account -- cites
Dudeney as the original inspiration of Smith.
R. L. Hutchings &
J. D. Blake. Problems drive
1962. Eureka 25 (Oct 1962) 20-21 &
34-35. Prob. G. Assemble squares of sides 2, 5, 7, 9, 16, 25, 28, 33, 36 into a rectangle. The rectangle is 69 x
61 and is not either of Moroń's
examples.
W. T. Tutte. The quest of the perfect square. AMM 72:2, part II (Feb 1965) 29-35. = Selected Papers of W. T. Tutte;
Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 432-438,
with brief commentary by Tutte on p. 431.
General survey, updating his 1958 survey.
Blanche Descartes [pseud. of
Cedric A. B. Smith]. Division of a
square into rectangles. Eureka 34
(1971) 31-35. Surveys some history and
Stone's dissection of an isosceles right triangle into 6
others of different sizes (see above).
Tutte has a dissection of an equilateral triangle into 15
equilateral triangles -- but some of the pieces must have the same area
so we consider up and down pointing triangles as + and - areas and then all the
areas are different. Author then
considers dissecting a square into incongruent but equiareal rectangles. He finds it can be done in n
pieces for any n ³ 7.
A. J. W. Duijvestijn. Simple perfect squared square of lowest
order. J. Combinatorial Thy. B 25
(1978) 240‑243. Finds a perfect
square of minimal order 21.
A. J. W. Duijvestin, P. J.
Federico & P. Leeuw. Compound
perfect squares. AMM 89 (1982) 15‑32. Shows Willcocks' example has the smallest
order for a compound perfect square and is the only example of its order, 24.
This is the problem of cutting a
square into smaller squares.
Loyd. Cyclopedia, 1914, pp. 248 & 372, 307 & 380. Cut 3 x 3
into 6 squares: 2 x 2 and
5 1 x 1.
Dudeney. AM.
1917. Prob. 173: Mrs Perkins's
quilt, pp. 47 & 180. Same as Loyd's
"Patch quilt puzzle" in 5.J.
Dudeney. PCP.
1932. Prob. 117: Square of
Squares, pp. 53 & 148‑149.
= 536, prob. 343, pp. 120 & 324‑325. c= "Mrs Perkins's quilt".
N. J. Fine & I. Niven,
proposers; F. Herzog, solver. Problem E724 -- Admissible Numbers. AMM 53 (1946) 271 & 54 (1947) 41‑42. Cubical version.
J. H. Conway. Mrs Perkins's quilt. Proc. Camb. Phil. Soc. 60 (1964) 363‑368.
G. B. Trustrum. Mrs Perkins's quilt. Ibid. 61 (1965) 7‑11.
Ripley's Puzzles and Games. 1966.
Pp. 16-17, item 7. "Can you
divide a square into 6 perfect squares?"
Answer as in Loyd.
Nick Lord. Note 72.11:
Subdividing hypercubes. MG 72
(No. 459) (Mar 1988) 47‑48. Gives
an upper bound for impossible numbers in
d dimensions.
David Tall. To prove or not to prove. Mathematics Review 1:3 (Jan 1991)
29-32. Tall regularly uses the question
as an exercise in problem solving.
About ten years earlier, a 14 year old girl pointed out that the problem
doesn't clearly rule out rejoining pieces.
E.g. by cutting along the diagonals and rejoining, one can make two
squares.
S. Chowla. Problem
1779. Math. Student 7 (1939) 80. (Solution given in Brooks, et al., Duke
Math. J., op. cit. in 5.J, section 10.4, but they give no reference to a
solution in Math. Student.)
5.J.3. TILING A SQUARE OF SIDE 70 WITH SQUARES OF SIDES 1, 2, ..., 24
J. R. Bitner. Use of Macros in Backtrack Programming. M.Sc. Thesis, ref. UIUCDCS‑R‑74‑687,
Univ. of Illinois, Urbana‑Champaign, 1974, ??NYS. Shows such a tiling is impossible.
Let D(n) = the number of derangements of n
things, i.e. permutations
leaving no point fixed.
Eberhard Knobloch. Euler and the history of a problem in
probability theory.
Gaņita-Bhāratī [NOTE:
ņ denotes an n
with an underdot] (Bull. Ind. Soc. Hist. Math.) 6 (1984) 1‑12. Discusses the history, noting that many 19C
authors were unaware of Euler's work.
There is some ambiguity in his descriptions due to early confusion
of n
as the number of cards and
n as the number of the card on
which a match first occurs. Describes
numerous others who worked on the problem up to about 1900: De Moivre, Waring, Lambert, Laplace, Cantor,
etc.
Pierre Rémond de Montmort. Essai d'analyse sur les jeux de
hazards. (1708); Seconde edition revue & augmentee de
plusieurs lettres, (Quillau, Paris,
1713 (reprinted by Chelsea, NY, 1980));
2nd issue, Jombert & Quillau, 1714.
Problèmes divers sur le jeu du trieze, pp. 54‑64. In the original game, one has a deck of 52
cards and counts 1, 2, ...,
13 as one turns over the cards. If a card of rank i occurs at the i-th count, then the player wins. In general, one simplifies by assuming there
are n
distinct cards numbered 1, ...,
n and one counts 1, ..., n.
One can ask for the probability of winning at some time and of winning
at the k-th draw. In 1708, Montmort already gives tables of
the number of permutations of n cards such that one wins on the k-th draw, for n = 1, ..., 6.
He gives various recurrences and the series expression for the
probability and (more or less) finds its limit. In the 2nd ed., he gives a proof of the series expression, due to
Nicholas Bernoulli, and John Bernoulli says he has found it also. Nicholas' solution covers the general case
with repeated cards. [See: F. N. David; Games, Gods and Gambling; Griffin,
London, 1962, pp. 144‑146 & 157.]
(Comtet and David say it is in the 1708 ed. I have seen it on pp. 54-64 of an edition which is uncertain, but
probably 1708, ??NX. Knobloch cites
1713, pp. 130-143, but adds that Montmort gave the results without proofs in
the 1708 ed. and includes several letters from and to John I and Nicholas I
Bernoulli in the 1713 ed., pp. 290-324, and mentions the problem in his Preface
-- ??NYS.)
Abraham de Moivre. The Doctrine of Chances: or, A Method of
Calculating the Probability of Events in Play.
W. Pearson for the Author, London, 1718. Prob. XXV, pp. 59-63.
(= 2nd ed, H. Woodfall for the Author, London, 1738. Prob. XXXIV, pp. 95-98.) States and demonstrates the formula for
finding the probability of p items to be correct and q
items to be incorrect out of
n items. One of his examples is the probability of
six items being deranged being 53/144.
L. Euler. Calcul de la probabilité dans le jeu de
rencontre. Mémoires de l'Académie des
Sciences de Berlin (7) (1751(1753)) 255‑270. = Opera Omnia (1) 7 (1923) 11‑25. Obtains the series for the probability and
notes it approaches 1/e.
L. Euler. Fragmenta ex Adversariis Mathematicis
Deprompta. MS of 1750‑1755. Pp. 287‑288: Problema de
permutationibus. First published in
Opera Omnia (1) 7 (1923) 542‑545.
Obtains alternating series for
D(n).
Ozanam-Montucla. 1778.
Prob. 5, 1778: 125-126; 1803:
123-124; 1814: 108-109; 1840: omitted. Describes Jeu du Treize, where a person
takes a whole deck and turns up the cards, counting 1, 2, ..., 13 as he
goes. He wins if a card of rank i
appears at the i‑th
count. Montucla's description is brief
and indicates there are several variations of the game. Hutton gives a lengthier description of one
version. Cites Montmort for the
probability of winning as .632..
L. Euler. Solutio quaestionis curiosae ex doctrina
combinationum. (Mem. Acad. Sci. St.
Pétersbourg 3 (1809/10(1811)) 57‑64.)
= Opera Omnia (1) 7 (1923) 435‑440. (This was presented to the Acad. on 18 Oct 1779.) Shows
D(n) = (n‑1) [D(n‑1) + D(n‑2)] and
D(n) = nD(n‑1) + (‑1)n.
Ball. MRE. 1st ed., 1892. Pp. 106-107: The mousetrap and Treize. In the first, one puts out n
cards in a circle and counts out.
If the count k occurs on the k-th card, the card is removed and one starts again. Says Cayley and Steen have studied
this. It looks a bit like a derangement
question.
Bill Severn. Packs of Fun. 101 Unusual Things to Do with Playing Cards and To Know about
Them. David McKay, NY, 1967. P. 24: Games for One: Up and down. Using a deck of 52 cards, count through 1, 2, ..., 13 four times. You lose if a
card of rank i appears when you count i, i.e. you win if the cards are a
generalized derangement. Though a
natural extension of the problem, I can't recall seeing it treated, perhaps
because it seems to get very messy.
However, a quick investigation reveals that the probability of such a
generalized derangement should approach
e-4.
Brian R. Stonebridge. Derangements of a multiset. Bull. Inst. Math. Appl. 28:3 (Mar 1992)
47-49. Gets a reasonable extension to
multisets, i.e. sets with repeated elements.
5.K.1. DERANGED BOXES OF A, B AND A & B
Three boxes contain A
or B or A & B, but they have been shifted about so each is
in one of the other boxes. You can look
at one item from one box to determine what is in all of them. This is just added and is certainly older
than the examples below.
Simon Dresner. Science World Book of Brain Teasers. 1962.
Op. cit. in 5.B.1. Prob. 84:
Marble garble, pp. 40 & 110. Black
and white marbles.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 26: Mexican
jumping beans, pp. 40-41 & 96. Red
and black beans in matchboxes. The
problem continues with a Bertrand box paradox -- see 8.H.1.
Doubleday - 3. 1972.
Open the box, pp. 147-148. Black
and white marbles.
5.K.2. OTHER LOGIC PUZZLES BASED ON DERANGEMENTS
These
typically involve a butcher, a baker and a brewer whose surnames are Butcher,
Baker and Brewer, but no one has the profession of his name. I generally only state the beginning of the
problem.
New
section -- there must be older examples.
Gardner, in an article: My ten
favorite brainteasers in Games (collected in Games Big Book of Games, 1984, pp.
130-131) says this is one of his favorite problems. ??locate
I
now see these lead to Latin rectangles, cf Section 5.I.
R. Turner, proposer: The sons of
the dons; Eureka 2 (May 1939) 9-10. K.
Tweedie, solver: On the problem of the sons of the dons. Eureka 4 (May 1940) 21-23. Six dons, in analysis, geometry, algebra,
dynamics, physics and astronomy, each have a son who studies one of these
subjects, but none studies the subject of his father. Several further restrictions, e.g., there are no two students who
each study the subject of the other's father.
M. Adams. Puzzle Book. 1939. Prob. B.91: Easter
bonnet, pp. 80 & 107. Women named
Green, Black, Brown and White with 4 colours of hats and 4 colours of dresses,
but name, hat and dress are always distinct.
J. B. Parker. Round the table. Eureka 5 (Jan 1941)
20-21 & 6 (May 1941) 11. Seven
men, whose names are colours, with ties, socks and cars, being coloured with
three of the names of other men and all colours used for each item, sitting at
a table with eight places.
Anonymous. The umbrella problem. Eureka
9 (Apr 1947) 22 & 10 (Mar 1948) 25. Six men 'of negligible honesty' each go away with another's
umbrella.
Jonathan Always. Puzzles to Puzzle You. Tandem, London, 1965. No. 30: Something about ties, pp. 16 &
74-75. Black, Green and Brown are
wearing ties, but none has the colour of his name, remarked the green tie
wearer to Mr. Black.
David Singmaster. The deranged secretary. If a secretary puts n
letters all in wrong envelopes, how many envelopes must one open before
one knows what is each of the unopened envelopes?
Problem
proposal and solution 71.B. MG 71 (No.
455) (Mar 1987) 65 & 71 (No. 457) (Oct 1987) 238-239.
Open
question. The Weekend Telegraph (11 Jun
1988) XV & (18 Jun 1988) XV.
This is a solitaire (= patience) game
developed by Cayley, based on Treize.
Take a deck of cards, numbered
1, 2, ..., n, and shuffle them. Count through them. If a card does not match its count, put it
on bottom and continue. If it matches,
set it aside and start counting again from 1. One wins if all cards are set aside. In this case, pick up the deck and start a
new game.
T. W. O. Richards,
proposer; Richard I. Hess, solver. Prob. 1828.
CM 19 (1993) 78 & 20 (1994) 77-78. Asks whether there is any arrangement which allows three or more
consecutive wins. No theoretical
solution. Searching finds one solution
for n = 6 and n = 8 and
8 solutions for n = 9.
How
many ways can n couples be seated, alternating sexes, with
no couples adjacent?
A. Cayley. On a problem of arrangements. Proc. Roy. Soc. Edin. 9 (1878) 338‑342. Problem raised by Tait. Uses inclusion/exclusion to get a closed
sum.
T. Muir. On Professor Tait's problem of
arrangements. Ibid., 382‑387. Uses determinants to get a simple n‑term recurrence.
A. Cayley. Note on Mr. Muir's solution of a problem of
arrangement. Ibid., 388‑391. Uses generating function to simplify to a
usable form.
T. Muir. Additional note on a problem of
arrangement. Ibid., 11 (1882) 187‑190. Obtains Laisant's 2nd order and 4th order
recurrences.
É. Lucas. Théorie des Nombres. Gauthier‑Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. Section 123, example II, p. 215 &
Note III, pp. 491‑495.
Lucas appears not to have known of the work of Cayley and Muir. He describes Laisant's results. The 2nd order, non‑homogeneous
recurrence, on pp. 494‑495, is attributed to Moreau.
C. Laisant. Sur deux problèmes de permutations. Bull. Soc. Math. de France 19 (1890‑91)
105‑108. General approach to
problems of restricted occupancy. His
work yields a 2nd order non-homogeneous recurrence and homogeneous 3rd and 4th
order recurrences. He cites Lucas, but
says Moreau's work is unpublished.
H. M. Taylor. A problem on arrangements. Messenger of Math. 32 (1903) 60‑63. Gets almost to Muir & Laisant's 4th
order recurrence.
J. Touchard. Sur un problème de permutations. C. R. Acad. Sci. Paris 198 (1934) 631‑633. Solution in terms of a complicated
integral. States the explicit
summation.
I. Kaplansky. Solution of the "problème des
ménages". Bull. Amer. Math. Soc.
49 (1943) 784‑785. Obtains the
now usual explicit summation.
I. Kaplansky & J.
Riordan. The problème de ménages. SM 12 (1946) 113‑124. Gives the history and a uniform approach.
J. Touchard. Permutations discordant with two given
permutations. SM 19 (1953) 109‑119. Says he prepared a 65pp MS developing the
results announced in 1934 and rediscovered in Kaplansky and in Kaplansky &
Riordan. Proves Kaplansky's lemma on
selections by finding the generating functions which involve Chebyshev
polynomials. Obtains the explicit
summation, as done by Kaplansky.
Extends to more general problems.
M. Wyman & L. Moser. On the 'problème des ménages'. Canadian J. Math. 10 (1958) 468‑480. Analytic study. Updates the history -- 26 references. Gives table of values for
n = 0 (1) 65.
Jacques Dutka. On the 'Problème des ménages'. Math. Intell. 8:3 (1986) 18‑25 &
33. Thorough survey & history -- 25
references.
Kenneth P. Bogart & Peter G.
Doyle. Non‑sexist solution of the
ménage problem. AMM 93 (1986) 514‑518. 14 references.
5.M. SIX PEOPLE AT A PARTY -- RAMSEY THEORY
In a group of six people, there is a
triple who all know each other or there is a triple who are all strangers. I.e., the Ramsey number R(3,3) = 6.
I will not go into the more complex aspects of this -- see Graham &
Spencer for a survey.
P. Erdös & G. Szekeres. A combinatorial problem in geometry. Compositio Math. 2 (1935) 463‑470. [= Paul Erdös; The Art of Counting; Ed. by
Joel Spencer, MIT Press, 1973, pp. 5‑12.] They prove that if n ³
BC(a+b-2, a-1), then any two‑colouring
of Ka contains a monochromatic Ka or Kb.
William Lowell Putnam Examination,
1953, part I, problem 2. In: L. E. Bush; The William Lowell Putnam
Mathematical Competition; AMM 60 (1953) 539-542. Reprinted in: A. M.
Gleason, R. E. Greenwood & L. M. Kelly; The William Lowell Putnam
Mathematical Competition Problems and Solutions -- 1938‑1964; MAA, 1980;
pp. 38 & 365‑366. The classic
six people at a party problem.
R. E. Greenwood & A. M.
Gleason. Combinatorial relations and
chromatic graphs. Canadian J. Math. 7
(1955) 1-7. Considers n = n(a,b,...) such that a two colouring of
Kn contains a Ka of the first colour or a
Kb of the second
colour or .... Thus n(3,3) = 6.
They find the bound and many other results of Erdös & Szekeres.
C. W. Bostwick, proposer; John Rainwater & J. D. Baum,
solvers. Problem E1321 -- A gathering
of six people. AMM 65 (1958) 446 &
66 (1959) 141‑142.
Gamow & Stern. 1958.
Diagonal strings. Pp. 93‑95.
G. J. Simmons. The Game of Sim. JRM 2 (1969) 66.
M. Gardner. SA (Jan 1973) c= Knotted, chap. 9. Exposits Sim. Reports Simmons' result that it is second person (determined
after his 1969 article above). The
Addendum in Knotted reports that several people have shown that Sim on five
points is a draw. Numerous references.
Ronald L. Graham & Joel H.
Spencer. Ramsey theory. SA 263:1 (Jul 1990) 80‑85. Popular survey of Ramsey theory beginning
from Ramsey and Erdös & Szekeres.
5.N. JEEP OR EXPLORER'S PROBLEM
See
Ball for some general discussion and notation.
Alcuin. 9C.
Prob. 52: Propositio de homine patrefamilias. Wants to get 90 measures over a distance of 30 leagues. He is trying to get the most to the other
side, so this is different than the 20C versions. Solution is confusing, but Folkerts rectifies a misprint and this
makes it less confusing. Alcuin's
camels only eat when loaded!! (Or else
they perish when their carrying is done??)
The camel take a load to a point 20 leagues away and leaves 10 there,
then returns. This results in getting
20 to the destination.
The
optimum solution is for the camel to make two return trips and a single trip
to 10
leucas, so he will have consumed
30 measures and he has 60
measures to carry on. He now
makes one return and a single trip of another
15 leucas, so he will
have consumed another 30 measures, leaving 30 to carry on the
last 5
leucas, so he reaches home with
25 measures.
Pacioli. De Viribus.
c1500. Probs. 49‑52. Agostini only describes Prob. 49 in some
detail.
Ff.
94r - 95v. XLIX. (Capitolo) de doi
aportare pome ch' piu navanza (Of two ways to transport as many apples as
possible). = Peirani 134‑135. One has 90 apples to transport 30 miles from
Borgo [San Sepolcro] to Perosia [Perugia], but one eats one apple per mile and
one can carry at most 30 apples. He
carries 30 apples 20 miles and leaves 10 there and returns, without eating on
the return trip! (So this = Alcuin.) Pacioli continues and gives the optimum solution!
F.
95v. L. C(apitolo). de .3. navi per
.30. gabelle 90. mesure (Of three ships
holding 90 measures, passing 30 customs points). Each ship has to pay one measure at each customs point. Mathematically the same as the
previous.
F.
96r. LI. C(apitolo). de portar .100.
perle .10. miglia lontano 10. per volta
et ogni miglio lascia 1a (To carry 100 pearls 10 miles, 10 at a time,
leaving one every mile). = Peirani
136-137. Takes them 2 miles in ten
trips, giving 80 there. Then takes them
to the destination in 8 trips, getting 16 to the destination.
Ff.
96v - 97r. LII. C(apitolo). el medesimo
con piu avanzo per altro modo (The same with more carried by another
method). Continues the previous problem
and takes them 5 miles in ten trips, giving 50 there. Then takes them to the destination in 5 trips, getting 25 to the
destination.
[This is optimal for a single stop --
if one makes the stop at distance
a, then one gets a(10-a)
to the destination. One can make
more stops, but this is restricted by the fact that pearls cannot be
divided. Assuming that the amount of
pearls accumulated at each depot is a multiple of ten, one can get 28 to the
destination by using depots at 2 and
7 or
5 and 7. One can get 27 to the destination
with depots at 4 and 9 or 5
and 9. These are all the ways one can
put in two depots with integral multiples of 10 at each depot and none of these
can be extended to three such depots.
If the material being transported was a continuous material like grain,
then I think the optimal method is to first move 1 mile to get 90 there, then
move another 10/9 to get 80 there, then another 10/8
to get 70 there, ..., continuing until we get 40 at 8.4563...,
and then make four trips to the destination. This gets 33.8254 to the destination. Is this the best method??]
Cardan. Practica Arithmetice. 1539.
Chap. 66, section 57, ff. EE.vi.v - EE.vii.v (pp. 152‑153). Complicated problem involving carrying food
and material up the Tower of Babel!
Tower is assumed 36 miles high and seems to require 15625 porters.
Mittenzwey. 1880.
Prob. 135, pp. 28-29; 1895?:
153, p. 32; 1917: 153, pp. 29. If eight porters can carry eight full loads
from A
to B in an hour, how long will it take four porters? The obvious answer is two hours, but he
observes that the porters have to return from
B to A and it will take three
hours. [Probably a little less as they
should return in less time than they go.]
Pearson. 1907.
Part II, pp. 139 & 216. Two
explorers who can carry food for 12 days.
(No depots, i.e. form A of Ball, below.)
Loyd. A dash for the South Pole.
Ladies' Home Journal (15 Dec 1910).
??NYS -- source?? -- WS??
Ball. MRE, 5th ed., 1911.
Exploration problems, pp. 23‑24.
He distinguishes two forms of the problem, with n
explorers who can carry food for
d days.
A. Without depots, they can get one man nd/(n+1)
days into the desert and back.
B. With depots permitted, they can get a
man
d/2 (1/1 + 1/2 + ... + 1/n) into the desert and back. This is the more common form.
Dudeney. Problem 744: Exploring the desert. Strand Mag. (1925). ??NX. (??= MP 49)
Dudeney. MP.
1926. Prob. 49: Exploring the
desert, pp. 21 & 111 (= 536, prob. 76, pp. 22 & 240). A version of Ball's form A,
with n = 9, d = 10,
but replacing days by stages of length 40 miles.
Abraham. 1933.
Prob. 34 -- The explorers, pp. 13 & 25 (9‑10 & 112). 4 explorers, each carrying food for 5
days. Mentions general case. This is Ball's form A.
Haldeman-Julius. 1937.
No. 10: The four explorers, pp. 4 & 21. Ball's form A, with n =
4, d = 5.
Olaf Helmer. Problem in logistics: The Jeep problem. Project Rand Report RA‑15015
(1 Dec 1946) 7pp.
N. J. Fine. The jeep problem. AMM 54 (1947) 24‑31.
C. G. Phipps. The jeep problem: a more general
solution. AMM 54 (1947) 458‑462.
G. G. Alway. Note 2707:
Crossing the desert. MG 41 (No.
337) (1957) 209. If a jeep can carry
enough fuel to get halfway across, how much fuel is needed to get across? For a desert of width 2,
this leads to the series 1 + 1/3
+ 1/5 + 1/7 + .... See Lehmann and Pyle
below.
G. C. S[hephard, ed.] The problems drive. Eureka 11 (Jan 1949) 10-11 & 30. No. 2.
Four explorers, starting from a supply base. Each can carry food for 100 miles and goes 25 miles per day. Two men do the returning to base and
bringing out more supplies. the third
man does ferrying to the fourth man. How
far can the fourth man get into the desert and return? Answer is 100 miles. Ball's form B would give 104 1/6.
Gamow & Stern. 1958.
Refueling. Pp. 114‑115.
Pyle, I. C. The explorer's problem. Eureka 21 (Oct 1958) 5-7. Considers a lorry whose load of fuel takes
it a distance which we assume as the unit.
What is the widest desert one can cross? And how do you do it?
This is similar to Alway, above.
He starts at the far side and sees you have to have a load at distance 1
from the far side, then two loads at distance
1 + 1/3 from the far side, then three loads at 1 + 1/3 + 1/5, .... This diverges, so
any width can be crossed. Does examples
with given widths of 2, 3 and 4 units.
Editor notes that he is not convinced the method is optimal.
Martin Gardner, SA (May &
June 1959) c= 2nd Book, chap. 14, prob. 1. (The book gives extensive references which were not in SA.)
R. L. Goodstein. Letter: Explorer's problem. Eureka 22 (Oct 1959) 23. Says Alway shows that Pyle's method is
optimal. Editor notes Gardner's article
and that Eureka was cited in the solutions in Jun.
David Gale. The jeep once more or jeeper by the
dozen & Correction to "The Jeep once more or jeeper by the
dozen". AMM 77:5 (May 1970)
493-501 & 78:6 (Jun-Jul 1971) 644-645.
Gives an elaborate approach via a formula of Banach for path lengths in
one dimension. This formally proves
that the various methods used are actually optimal and that a continuous string
of depots cannot help, etc. Notes that
the cost for a round trip is only slightly more than for a one-way trip -- but
the Correction points out that this is wrong and indeed the round trip is
nearly four times as expensive as a one-way trip. Considers sending several jeeps.
Says he hasn't been able to do the round trip problem when there is fuel
on both sides of the desert. Comments
on use of dynamic programming, noting that R. E. Bellman [Dynamic Programming;
Princeton Univ. Press, 1955, p. 103, ex. 54-55] gives the problem as exercises
without solution and that he cannot see how to do it!
Birtwistle. Math. Puzzles & Perplexities. 1971.
The
expedition, pp. 124-125, 183 & 194.
Ball's form A, first with n =
5, d = 6, then in general.
Second
expedition, pp. 125-126. Ball's form B,
done in general.
Third
expedition, pp. 126, 183-184 & 194.
Three men want to cross a 180 mile wide desert. They can travel 20 miles per day and can
carry food for six days, which can be stored at depots. Minimize the total distance travelled. Solution seems erroneous to me.
A. K. Austin. Jeep trips and card stacks. MTg 58 (1972) 24‑25. There are
n flags located at
distances a1, a1 + a2, a1 + a2 + a3, ....
Jeep has to begin at the origin, go to the first flag, return to the
origin, go to the second flag, return, ....
He can unload and load fuel at the flags. Can he do this with
F fuel? Author shows this is equivalent to
successfully stacking cards over a cliff with successive overhangs being a1, a2, a3, ....
Doubleday - 3. 1972.
Traveller's Tale, pp. 63-64. d =
8 and we want one man to get across the
desert of width 12. How many porters, who return to base, are
needed? The solution implies that no
depots are used. Reasoning as in Ball's
case A, we see that n men can support one man crossing a desert of
width 2nd/(n+1). If depots are permitted, this is essentially
the jeep problem and n men can support a man getting across a
desert of width d [1 + 1/3 + 1/5 + ...
+ 1/(2n-1)]
Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 13, pp. 27 & 129. d = 4
and we want to get a man across a desert of width 6.
Similar to Doubleday - 3.
Pierre Berloquin. [Le Jardin du Sphinx. Dunod, Paris, 1981.] Translated by Charles Scribner Jr as: The Garden of the Sphinx. Scribner's, NY, 1985.
Prob.
1: Water in the desert, pp. 3 & 85.
Prob.
40: Less water in the desert, pp. 26 & 111.
Prob.
80: Beyond thirst, pp. 48 & 140.
Prob.
141: The barrier of thirst, pp. 79 & 181.
Prob.
150: No holds barred, pp. 82 & 150.
In
all of these, d = 5 and we want to get a man across a desert of
width 4, and sometimes back, which is slightly different than the problem
of getting to the maximum distance and back.
Prob.
1 is Ball's form A, with
n = 4 men, using 20
days' water.
Prob.
40 is Ball's form B, but using only whole day trips, using 14
days' water.
Prob.
80 is Ball's form B, optimized for width 4, using
11½ days' water.
Prob.
141 uses depots and bearers who don't return, as in Alcuin?? You can get one man, who is the only one to
return, a distance d (1/2 + 1/3 + ... +
1/(n+1)) into the desert this way. He gives the optimum form for width 4,
using 9½ days' water.
Prob.
150 is like prob. 141, except that no one returns! You can get him d (1 + 1/2
+ ... + 1/n) into the desert this
way. The optimum here uses 4
days' water.
D. R. Westbrook. Note 74.7:
The desert fox, a variation of the jeep problem. MG 74 (No. 467) (1990) 49‑50. A more complex version, posed by A. K.
Dewdney in SA (Jan 1987), is solved here.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B.
1991. Round-trip, pp. 96-97
& 140. Plane wants to circle the
earth, but can only carry fuel to go half-way.
Other planes can accompany and transfer fuel, but must return to base.
Dylan Gow. Flyaway.
MS 25:3 (1992/3) 84-86.
Considers the standard problem without return as in Alway, Pyle and
Lehmann -- but finds a non-optimal solution.
Wolfram Hinderer. Optimal crossing of a desert. MS 26:4 (1993/4) 100-102. Finds optimal solutions for Gow's problem
and for the case with return -- i.e. Ball's
B. Also considers use of extra
jeeps that do not return, i.e. Berloquin's 141 & 150. Notes that extra jeeps that must return to
base do not change the distance that one jeep can reach. [But it changes the time required.]
Harold Boas. Letter:
Crossing deserts. MS 26:4
(1993/4) 122. Notes the problem has a
long history and cites Fine, Phipps, Gale (and correction), Alway.
David Singmaster. Letter:
Crossing deserts. MS 27:3
(1994/5) 63. Points out that the
history is far older and sketches the history given above.
Günter Rote & Guochuan
Zhang. Optimal Logistics for
Expeditions: The Jeep Problem with Complete Refilling. Karl-Franzens-Universität Graz &
Technische Universität Graz. Bericht 71
(24 Jun 1996). This deals with a
variant. "We have n
cans of fuel on the edge of a desert and a jeep with an empty tank whose
capacity is just one can. The jeep can
carry one can in addition to the fuel in its tank. Moreover, when a can is opened, the fuel must immediately be
filled into the jeep's tank. The goal
is to find the farthest point in the desert which the jeep can reach by
consuming the n cans of fuel. Derick Wood [1984] treated this problem similarly to the
classical problem and gave the first solution.
Ute and Wilfried Brauer [1989] presented a new strategy and got a better
solution than Wood's. They also
conjectured that their solution was optimal for infinitely many values of n.
We give an algorithm which produces a better solution than Brauers' for
all n > 6, and we use a linear
programming formulation to derive an upper bound which shows that our solution
is optimal." 14 references,
several not given above.
5.O. TAIT'S COUNTER PUZZLE: BBBBWWWW TO WBWBWBWB
See
S&B 125.
The
rules are that one can move two counters as an ordered pair, e.g. from BBBBWWWW
to BBB..WWWBW, but not to
BBB..WWWWB -- except in Lucas
(1895) and AM prob. 237, where such reversal must be done. Also, moving to BBB..WWW.BW is sometimes
explicitly prohibited, but it is not always clear just where one can move
to. It is also not always specified
where the blank spaces are at the beginning and end positions.
Gardner,
1961, requires that the two counters must be
BW or WB.
Barbeau, 1995, notes that moving
to BWBWBWBW is a different problem, requiring an extra move. I had not noticed this difference before --
indeed I previously had it the wrong way round in the heading of this
section. I must check to see if this
occurs earlier. See Achugbue &
Chin, 1979-80, for this version.
Genjun Nakane (= Hōjiku
Nakane). Kanja‑otogi‑soshi
(Book of amusing problems for the entertainment of thinkers). 1743.
??NYS. (See: T. Hayashi; Tait's problem with counters in
the Japanese mathematics; Bibl. Mathem. (3) 6 (1905) 323, for this and other
Japanese references of 1844 and 1879, ??NYS.)
P. G. Tait. Listing's Topologie. Philosophical Mag. (Ser. 5) 17 (No. 103)
(Jan 1884) 30‑46 & plate opp. p. 80.
Section 12, pp. 39‑40. He
says he recently saw it being played on a train.
George Hope Verney (= Lloyd‑Verney). Chess eccentricities. Longmans, 1885. P. 193: The pawn puzzle.
??NX With 4 & 4.
Lucas. Amusements par les jetons.
La Nature 15 (1887, 2nd sem.) 10-11.
??NYS -- cited by Ahrens, title obtained from Harkin. Probably c= the material in RM3, below.
Ball. MRE, 1st ed., 1892, pp. 48‑49.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
Card Puzzles No. XIV: The eight-card puzzle, pp. 14-15. Uses cards:
BRBRBRBR and asks to bring the
colours together, explicitly requiring the moved cards to be placed in contact
with the unmoved cards.
Hoffmann. 1893.
Chap. VI, pp. 270‑271 & 284‑285 = Hoffmann-Hordern,
pp. 184-186, with photo.
No.
19: The "Four and Four" puzzle.
Photo on p. 184 shows a version named Monkey Puzzle advertising Brooke's
Soap to go from BBBBBWWWW.. to
..WBWBWBWB .
No.
20: The "Five and Five" puzzle.
No.
21: The "Six and Six" puzzle.
Lucas. RM3. 1893. Amusements par les jetons, pp. 145‑151. He gives Delannoy's general solution
for n
of each colour in n moves.
Remarks that one can reverse the moved pair.
Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co.,
NY), nd [1895]. P. 11: The Egyptian
disc puzzle. 4 & 4. "Two discs adjoining each other to be
moved at a time; no gaps to be left in the line." -- this seems to prevent one from making any
moves at all!! No solution.
Lucas. L'Arithmétique Amusante.
1895. Pp. 84-108.
Prob.
XXI - XXIV and Méthode générale, pp. 84-97.
Gives solution for 4, 5, 6,
7 and the general solution for n & n
in n moves due to Delannoy.
Rouges
et noires, avec interversion, prob. XXV - XXVIII and Méthode générale, pp. 97‑108. Interversion means that the two pieces being
moved are reversed or turned over, e.g. from
BBBBWWWW to BBB..WWWWB,
but not to BBB..WWWBW. Gives solutions for 4, 5, 6, 7, 8 pairs and in general in n moves, but he ends with a gap, e.g. ....BB..BB
and it takes an extra move to close up the gap.
Ball. MRE, 3rd ed., 1896, pp. 65‑66. Cites Delannoy's solution as being in La Nature (Jun 1887)
10. ??NYS.
Ahrens. MUS I.
1910. Pp. 14-15 &
19-25. Cites Tait and gives Delannoy's
general solution, from Lucas.
Ball. MRE, 5th ed., 1911, pp. 75-77.
Adds a citation to Hayashi, but incorrectly gives the date as 1896.
Loyd. Cyclopedia. 1914. After dinner tricks, pp. 41 & 344. 4 & 4.
Williams. Home Entertainments. 1914.
The eight counters puzzle, pp. 116-117.
Standard version, but with black and white reversed, in four moves. Says the moved counters must be placed in
line with and touching the others.
Dudeney. AM.
1917.
Prob.
236: The hat puzzle, pp. 67 & 196-197.
BWBWBWBWBW.. to have the Bs
and Ws together and two blanks at an end. Uses 5 moves to get to ..WWWWWBBBBB.
Prob.
237: Boys and girls, pp. 67-68 & 197.
..BWBWBWBW to have the Bs
and Ws together with two blanks at an end, but pairs must be reversed as
they are moved. Solution in 5 moves to WWWWBBBB...
= Putnam, no. 2. Cf Lucas, 1895.
Blyth. Match-Stick Magic.
1921. Transferring in twos, pp.
80-81. WBWBWBWB.. to
..BBBBWWWW in four moves.
King. Best 100. 1927. No. 66, pp. 27 & 55. = Foulsham's, no. 9, pp. 9 & 13. BWBWBWBW..
to ..WWWWBBBB, specifically prescribed.
Rohrbough. Brain Resters and Testers. c1935.
Alternate in Four Moves, p. 4.
..BBBBWWWW to WBWBWBWB.. , but he doesn't specify the
blanks, showing all stages as closed up to 8 spaces, except the first two
stages have a gap in the middle.
McKay. At Home Tonight. 1940.
Prob.
43: Arranging counters, pp. 73 & 87-88.
RBRBRB.... to ....BBBRRR
in three moves. Sketches general
solution.
Prob.
45: Triplets, pp. 74 & 88.
YRBYRBYRB.. to BBBYYYRRR..
in 5 moves.
McKay. Party Night. 1940. Heads and tails again, p. 151. RBRBR..
to ..BBRRR in three moves. RBRBRB.. to ..BBBRRR in four moves. RBRBRBRB.. to ..BBBBRRRR
in four moves. Notes that the
first move takes coins 2 & 3 to the end and thereafter one is always
filling the spaces just vacated.
Gardner. SA (Jun & Jul 1961) =
New MD, chap. 19, no. 1: Collating the coins. BWBWB to BBBWW,
moving pairs of BW or
WB only, but the final position
may be shifted. Gardner thanks H. S.
Percival for the idea. Solution in 4
moves, using gaps and with the solution shifted by six spaces to the
right. Thanks to Heinrich Hemme for
this reference.
Joseph S. Madachy. Mathematics on Vacation. (Scribners, NY, 1966, ??NYS); c= Madachy's Mathematical Recreations. Dover, 1979. Prob. 3: Nine-coin move, pp. 115 & 128-129 (where the
solution is headed Eight-coin move).
This uses three types of coin, which I will denote by B, R, W.
BRWBRWBRW ® WWWRRRBBB by moving two adjacent unlike coins
at a time and not placing the two coins away from the rest. Eight move solution leaves the coins in the
same places, but uses two extra cells at each end. From the discussion of Bergerson's problem, see below, it is
clear that the earlier book omitted the word unlike and had a nine move
solution, which has been replaced by Bergerson's eight move solution.
Yeong‑Wen Hwang. An interlacing transformation problem. AMM 67 (1967) 974‑976. Shows the problem with 2n
pieces, n > 2, can be solved in n moves and this is
minimal.
Doubleday - 1. 1969.
Prob. 70: Oranges and lemons, pp. 86 & 170. = Doubleday - 4, pp. 95‑96. BWBWBWBWBW.. considered as a cycle.
There are two solutions in five moves:
to ..WWWWWBBBBB, which never
uses the cycle; and to: BBWWWWWW..BBB.
Howard W. Bergerson,
proposer; Editorial discussion; D. Dobrev, further solver; R. H. Jones, further solver. JRM
2:2 (Apr 1969) 97; 3:1 (Jan
1970) 47-48; 3:4 (Oct 1970)
233-234; 6:2 (Spring 1973) 158. Gives Madachy's 1966 problem and says there
is a shorter solution. The editor points
out that Madachy's book and Bergerson have omitted unlike. Bergerson has an eight move solution of the intended
problem, using two extra cells at each end, and Leigh James gives a six move
solution of the stated problem, also using two extra cells at each end. Dobrev gives solutions in six and five
steps, using only two extra cells at the right. Jones notes that the problem does not state that the coins have
to be adjacent and produces a four move solution of the stated problem, going
from ....BRWBRWBRW.... to
WWW..R..R..R..BBB.
Jan M. Gombert. Coin strings. MM 42:5 (Nov 1969) 244-247.
Notes that BWBWB...... ®
......BBBWW can be done in four
moves. In general, BWB...BWB,
with n Ws and n+1
Bs alternating can be
transformed to BB...BWW...W in n2 moves and this is minimal. This requires shifting the whole string n(n+1)
to the right and a move can go to places separated from the rest of the
pieces. By symmetry, ......BWBWB ® WWBBB...... in the same number of moves.
Doubleday - 2. 1971.
Two by two, pp. 107-108.
..BWBWBWBW to WWWWBBBB...
He doesn't specify where the extra spaces are, but says the first two
must move to the end of the row, then two more into the space, and so on. The solution always has two moving into an internal
space after the first move.
Wayne A. Wickelgren. How to Solve Problems. Freeman, 1974. Checker-rearrangement problem, pp. 144‑146. BWBWB
to BBBWW by moving two adjacent checkers, of
different colours, at a time. Solves in
four moves, but the pattern moves six places to the left.
Putnam. Puzzle Fun.
1978.
No. 1:
Nickles [sic] & dimes, pp. 1 & 25.
Usual version with 8 coins.
Solution has blanks at the opposite end to where they began.
No. 2:
Nickles [sic] & dimes variation, pp. 1 & 25. Same, except the order of each pair must be reversed as it
moves. Solution in five moves with
blanks at opposite end to where they started.
= AM 237. Cf Lucas, 1895.
James O. Achugbue &
Francis Y. Chin. Some new
results on a shuffling problem. JRM 12:2
(1979-80) 125-129. They demonstrate
that any pattern of n & n occupying
2n consecutive cells can be
transformed into any other pattern in the same cells, using only two extra
cells at the right, except for the case
n = 3 where 10
cells are used. They then find
an optimal solution for BB...BW...WW ®
BWBW...BW in n+1 moves using two extra
cells. They seem to leave open the
question of whether the number of moves could be shortened by using more cells.
Walter Gibson. Big Book of Magic for All Ages. Kaye & Ward, Kingswood, Surrey,
1982.
Six
cents at a time, p. 117. Uses pennies
and nickels. .....PNPNP to
NNPPP..... in four moves.
Tricky
turnover, p. 137. HTHTHT to
HHHTTT in two moves. This requires turning over one of the two
coins on each move.
Ed Barbeau. After Math.
Wall & Emerson, Toronto, 1995.
Pp. 117, 119 & 123-126. He
asks to move BBBWWW to
WBWBWB and to BWBWBW
and notes that the latter takes an extra move. He sketches the general solutions.
5.P. GENERAL MOVING PIECE PUZZLES
See
also under 5.A.
See Hordern, op. cit. in 5.A, pp. 167‑177,
for a survey of these puzzles. The
Chifu‑Chemulpo (or Russo‑Jap Railway) Puzzle of 1903 is actually
not of this type since all the pieces can move by themselves -- Hordern, pp.
124‑125 & plate VIII.
See
S&B 124‑125.
A
'spur' is a dead‑end line. A
'side‑line' is a line or siding joined to another at both ends.
Mittenzwey. 1880.
Prob. 219-221, pp. 39-40 & 91;
1895?: 244-246, pp. 43-44 & 93;
1917: 244-246, pp. 40 & 89.
First two have a canal too narrow to permit boats to pass, with a
'bight', or widening, big enough to hold one boat while another passes. First problem has two boats meeting one
boat; second problem has two boats meeting two boats. The third problem has a single track railway with a side-line big
enough to hold an engine and 16 wagons on the side-line or on the main line
between the switches. Two trains
consisting of an engine and 20 wagons meet.
Lucas. RM2, 1883, pp. 131‑133.
Passing with a spur and with a side‑line.
Alexander Henry Reed. UK Patent 15,051 -- Improvements in
Puzzles. Complete specification: 8 Dec
1885. 4pp + 1p diagrams. Reverse a train using a small turntable on
the line. This has forms with one line
and with two crossing lines. One object
is to spell 'Humpty Dumptie'. He also
has a circular line with three turntables (equivalent to the recent Top-Spin
Puzzle of F. Lammertinck).
Pryse Protheroe. US Patent 332,211 -- Puzzle. Applied: 18 Sep 1885; patented: 8 Dec 1885. 3pp + 1p diagrams. Described in Hordern, p. 167.
Identical to the Reed patent above!
Both Reed and Protheroe are described as residents of suburban
London. The Reed patent says it was
communicated from abroad by an Israel J. Merritt Jr of New York and it doesn't
assert that Reed is the inventor, so perhaps Reed and Merritt were agents for
Protheroe.
Jeffrey & Son (Syracuse,
NY). Great Railroad Puzzle. Postcard puzzle produced in 1888. ??NYS.
Described in Hordern, pp. 175‑176. Passing with a turntable that holds two wagons.
Arthur G. Farwell. US Patent 437,186 -- Toy or Puzzle. Applied: 20 May 1889; patented: 30 Sep 1890. 1p + 1p diagrams. Described in Hordern, pp.
167‑169. Great Northern Puzzle. This requires interchanging two cars on the
legs of a 'delta' switch which is too short to allow the engine through, but
will let the cars through. Hordern
lists 6 later patents on the same basic idea.
Ball. MRE, 1st ed., 1892, pp. 43‑44. Great Northern Puzzle "which I bought some eight or nine
years ago." (Hordern, p. 167,
erroneously attributes this quote to Ahrens.)
Loyd. Problem 28: A railway puzzle.
Tit‑Bits 32 (10 Apr
& 1 May 1897) 23 &
79. Engine and 3 cars need to
pass 4 cars by means of a 'delta' switch whose branches and tail hold only one
car. Solution with 28 reversals.
Loyd. Problem 31: The turn‑table puzzle. Tit‑Bits 32 (1 &
22 May 1897) 79 & 135.
Reverse an engine and 9 cars with an 8 track turntable whose lines hold
3 cars. The turntable is a double
curved connection which connects, e.g. track 1 to tracks 4 or 6.
E. Fourrey. Récréations Arithmétiques. Op. cit. in 4.A.1. 1899. Art. 239: Problèmes
de Chemin de fer, pp. 184-189.
I. Three parallel tracks with two switched
crossing tracks. Train of 21 wagons on
the first track must leave wagons 9 & 12 on third track.
II.
Delta shape with a turntable at the point of the delta, which can only hold the
wagons and not the engine, so this is isomorphic to Farwell.
III. This
is a more complex railway problem involving timetables on a circular line.
J. W. B. Shunting!
c1900. ??NYS. Described in Hordern, pp. 176‑177
& plate XII. Reversing a train with
a turntable that holds three wagons.
Orril L. Hubbard. US Patent 753,266 -- Puzzle. Applied: 21 Apr 1902; patented: 1 Mar 1904. 3pp + 1p diagrams. Great Railroad Puzzle, described in Hordern, pp. 175‑176. Improved version of the Jeffrey & Son
puzzle of 1888. Engine & 2 cars to
pass engine & 3 cars, using a turntable that holds two cars, preserving
order of each train.
Harry Lionel Hook &
George Frederick White. UK
Patent 26,645 -- An Improved Puzzle or Game.
Applied: 3 Dec 1902; accepted:
11 Jun 1903. 2pp + 1p diagrams. This is very cryptic, but appears to be a
kind of sliding piece Puzzle using turntables.
Mr. X [cf 4.A.1]. His Pages.
The Royal Magazine 10:1 (May 1903) 50-51 & 10:2 (Jun 1903)
140-141 & 10:4 (Aug 1903)
336-337. A railway puzzle. One north-south line with a spur heading
north which is holding 7 trucks, but cannot hold the engine as well, so the
engine is on the main line heading south.
An engine pulling seven trucks arrives from the north and wants to get
past. First solution uses 17 stages;
second uses 12 stages.
Mr. X [cf 4.A.1]. His Pages.
The Royal Magazine 10:5 (Sep 1903) 426-427 & 10:6
(Oct 1903) 530‑531. A
shunting problem. Same as Fourrey - II,
hence isomorphic to Farwell. Solution
in 17 stages.
Celluloid Starch Puzzle. c1905.
Described in Hordern, pp. 169‑170. Cars on the three parts of a 'delta' switch with an engine
approaching. Reverse the engine,
leaving all cars on their original places.
More complexly, suppose the tail of the 'delta' only holds one car or
the engine.
Livingston B. Pennell. US Patent 783,589 -- Game Apparatus. Applied: 20 Mar 1902; patented: 28 Feb 1905. 3pp + 1p diagrams. Described in Hordern, p. 173.
Passing with a side line -- engine & 3 cars to pass engine & 3
cars using a siding which already contains 3 cars, without couplings, so these
three can only be pushed. Also the
engines can move at most three cars at a time.
William Rich & Harry
Pritchard. UK Patent 7647 -- Railway
Game and Puzzle. Applied: 11 Apr
1905; complete specification: 11 Oct
1905; accepted: 14 Dec 1905. 2pp + 1p diagrams. Main line with two short and two long spurs.
Ball. MRE, 4th ed., 1905, pp. 61-63, adds a problem with a side-line,
"on sale in the streets in 1905“.
The 5th ed., 1911, pp. 69-71 & 82, adds the name "Chifu-Chemulpo
Puzzle" and that the minimum number of moves is 26, in more than one
way. P. 82 gives solutions of both
problems.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Great Northern
Puzzle. He says the "Railway
puzzle" was very popular "about twenty years ago".
Ahrens. MUS I.
1910. Pp. 3-4. Great Northern. Says it is apparently modern and cites Fourrey for other
examples.
Anon. Prob. 6. Hobbies 32 (No. 814) (20 May 1911) 145 &
(No. 817) (10 Jun 1911) 208.
Great Northern Puzzle. Solution
asks if readers know any other railway puzzles.
Loyd. The switch problem
& Primitive railroading
problem. Cyclopedia, 1914, pp. 167
& 361; 89 & 350 (= MPSL2, prob.
24, pp. 18‑19; MPSL1, prob. 95,
pp. 92 & 155). Passing with a
'delta' switch & passing with a spur. The first is like Tit-Bits Problem 28, but
the engine and 3 cars have to pass 5 cars.
Solution in 32 moves. See
Hordern, pp. 170‑171.
Hummerston. Fun, Mirth & Mystery. 1924.
The Chinese railways, pp. 103 & 188. Imagine a line of positions:
ABCEHGJLMN with single
positions D, I, F, K attached to
positions C, H, G, L. You have eight engines at ABCD
and KLMN and the object is to exchange them,
preserving the order. He does it in 18
moves, where a move can be of any length.
King. Best 100. 1927. No. 14, pp. 12 & 41. Side‑line with a bridge over it too
low for the engine. Must interchange
two wagons on the side‑line which are on opposite sides of the bridge.
B. M. Fairbanks. Railroad switching problems. IN:
S. Loyd Jr., ed.; Tricks and Puzzles; op. cit. in 5.D.1 under
Chapin; 1927. P. 85 & Answers p.
7. Three realistic problems with
several spurs and sidelines.
Loyd Jr. SLAHP.
1928. Switching cars, pp. 54
& 106. Great Northern puzzle. See Hordern, pp. 168‑169.
Doubleday - 2. 1971.
Traffic jam, pp. 85-86. Version
with cars in a narrow lane and a lay-by.
Two cars going each way. Though
the lay-by is three cars wide and just over a car long, he restricts its use so
that it acts like it is two cars wide.
Jacques
Haubrich has kindly enlightened me that 'taquin' simply means 'teaser'. So these items should be re-categorised.
Lucas. RM3. 1893. 3ème Récréation -- Le jeu du caméléon et le
jeu des jonctions de points, pp. 89‑103.
Pp. 91‑97 -- Le taquin de neuf cases avec un seul port. I thought that taquin was the French generic
term for such puzzles, but I find no other usage than that below, except in
referring to the 15 Puzzle -- see references to taquin in 5.A.
Au Bon Marché (the Paris
department store). Catalogue of 1907,
p. 13. Reproduced in Mary Hillier;
Automata and Mechanical Toys; An Illustrated History; Jupiter Books, London,
1976, p. 179. This shows Le Taquin Japonais Jeu de Patience
Casse-tete. This comprises 16
hexagonal pieces, looking like a corner view of a die, so each has three
rhombic parts containing a pattern of pips.
They are to be placed as the corners of four interlocked hexagons with
the numbers on adjacent rhombi matching.
5.Q. NUMBER OF REGIONS DETERMINED BY N LINES OR PLANES
Mittenzwey. 1880.
Prob. 200, pp. 37 & 89;
1895?: 225, pp. 41 & 91;
1917: 225, pp. 38 & 88. Family
of 4 adults and 4 children. With three
cuts, divide a cake so the adults and the children get equal pieces. He makes two perpendicular diametrical cuts
and then a circular cut around the middle.
He seems to mean the adults get equal pieces and the children get equal
pieces, not necessarily the same. But
if the circular cut is at Ö2/2 of the radius, then the areas are all
equal. Not clear where this should go
-- also entered in 5.T.
Jakob Steiner. Einige Gesetze über die Theilung der Ebene
und des Raumes. (J. reine u. angew.
Math. 1 (1826) 349‑364) = Gesam.
Werke, 1881, vol. 1, pp. 77‑94.
Says the plane problem has been raised before, even in a Pestalozzi school
book, but believes he is first to consider 3‑space. Considers division by lines and circles
(planes and spheres) and allows parallel families, but no three coincident.
Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306
& Three puzzles; Knowledge 9 (Sep 1886) 336-337. "3.
A man marks 6 straight lines on a field in such a way as to enclose 10
spaces. How does he manage
this?" Solution begins: "III.
To inclose ten spaces by six ropes fastened to nine pegs." Take
(0,0), (1,0), ..., (n,0), (0,n), ..., (0,1), as 2n+1 points, using n+2 ropes from (0,0) to (n,0) and to (0,n) and from
(i,0) to (0,n+1-i) to enclose n(n+1)/2
areas.
Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886)
9 & (Dec 1886) 39-40.
Describes several ways of solving previous problem and asks for a
symmetric version.
G.
Chrystal. Algebra -- An Elementary
Text-Book. Vol. 2, A. & C. Black,
Edinburgh, 1889. [Note -- the 1889
version of vol. 1 is a 2nd ed.] Chap.
23, Exercises IV, p. 34. Several
similar problems and the following.
No. 7
-- find number of interior and of exterior intersections of the diagonals of a
convex n-gon.
No. 8
-- n
points in general position in space, draw planes through every three and
find number of lines and of points of intersection.
L. Schläfli. Theorie der vielfachen Kontinuität. Neue Denkschriften der allgemeinen
schweizerischen Gesellschaft für die Naturwissenschaften 38:IV, Zürich, 1901,
239 pp. = Ges. Math. Abh., Birkhäuser,
Basel, 1950‑1956, vol. 1, pp. 167‑392. (Pp. 388‑392 are a Nachwort by J. J. Burckhardt.) Material of interest is Art. 16: Über die
Zahl der Teile, ..., pp. 209‑212.
Obtains formula for k hyperplanes in n space.
Loyd, Dudeney, Pearson &
Loyd Jr. give various puzzles based on this topic.
Howard D. Grossman. Plane- and space-dissection. SM 11 (1945) 189-190. Notes Schläfli's result and observes that
the number of regions determined by
k+1 hyperspheres in n
space is twice the number of regions determined by k
hyperplanes and gives a two to one correspondence for the case n = 2.
Leo Moser, solver. MM 26 (Mar 1953) 226. ??NYS.
Given in: Charles W. Trigg;
Mathematical Quickies; (McGraw‑Hill, NY, 1967); corrected ed., Dover, 1985.
Quickie 32: Triangles in a circle, pp. 11 & 90‑91. N
points on a circle with all diagonals drawn. Assume no three diagonals are concurrent. How many triangles are formed whose vertices
are internal intersections?
Timothy Murphy. The dissection of a circle by chords. MG 56 (No. 396) (May 1972) 113‑115 +
Correction (No. 397) (Oct 1972) 235‑236. N points on a circle, in
a plane or on a sphere; or N
lines in a plane or on a sphere, all simply done, using Euler's formula.
Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Slicing cakes, pp. 33 & 61.
Cut a circular cake into 12 equal pieces with 4 cuts. [From this, we see that N
full cuts can yield either
2N or 4(N-1) equal pieces. Further, if we make k
circular cuts producing k+1 regions of equal area and then make N-k
diametric cuts equally spaced, we get
2(k+1)(N-k) pieces of the same
size.]
Looking at this problem, I see
that one can obtain any number of pieces from
N+1 up through the maximum.
5.Q.1. NUMBER OF INTERSECTIONS DETERMINED BY N LINES
Chrystal. Text Book of Algebra. 2nd ed., vol. 2, 1889, p. 34, ex. 7. See above.
Loyd Jr. SLAHP.
1928. When drummers meet, pp. 74
& 115. Six straight railroads can
meet in 15 points.
Paul Erdös, proposer; Norbert Kaufman & R. H. Koch and Arthur
Rosenthal, solvers. Problem E750. AMM 53 (1946) 591 & 54 (1947) 344. The first solution is given in Trigg, op.
cit. in 5.Q, Quickie 191: Intersections of diagonals, pp. 53 & 166‑167. In a convex
n‑gon, how many intersections of diagonals are there? This counts a triple intersection as three
ordinary (i.e. double) intersections or assumes no three diagonals are
concurrent. Editorial notes add some
extra results and cite Chrystal.
See
also 5.O. Some of these are puzzles, but
some are games and are described in the standard works on games -- see the
beginning of 4.B.
See
MUS I 182-210.
Ahrens, MUS I 182‑183,
gives legend associating this with American Indians. Bergholt, below, and Beasley, below, find this legend in the 1799
Encyclopédie Méthodique: Dictionnaire des Jeux Mathématique (??*), ??NYS. Ahrens also cites some early 19C material
which has not been located. Bergholt
says some maintain the game comes from China.
Thomas Hyde. Historia Nerdiludii, hoc est dicere,
Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From
the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a
5 x 5 board with each
side having 12 men, but the description is extremely
brief. It seems to have two players,
but this may simply refer to the two types of piece. I'm not clear whether it's played like solitaire (with the jumped
pieces being removed) or like frogs & toads. I would be grateful if someone could read the Latin
carefully. The name of the puzzle is
clearly Arabic and Hyde cites an Arabic source, Hanzoanitas (not further
identified on the pages I have) -- I would be grateful to anyone who can track
down and translate Arabic sources.
G. W. Leibniz. Le Jeu du Solitaire. Unpublished MS LH XXXV 3 A 10 f. 1-2,
of c1678. Transcribed in: S. de Mora-Charles; Quelques jeux de hazard
selon Leibniz; HM 19 (1992) 125-157.
Text is on pp. 152-154. 37
hole board. Says the Germans call it
'Die Melancholy' and that it is now the mode at the French court.
Claude‑Auguste Berey. Engraving:
Madame la Princesse de Soubize jouant au Jeu de Solitaire. 1697(?).
Beasley (below) discovered and added this while his book was in
proof. It shows the 37‑hole
French board. Reproduced in: Pieter van Delft & Jack Botermans;
Creative Puzzles of the World; op. cit. in 5.E.2.a, p. 170.
G. W. Leibniz. Jeu des Productions. Unpublished MS LH XXXV 8,30 f. 4, of
1698. Transcribed in: de Mora-Charles, loc. cit. above. Text is on pp. 154-155. 37 hole board. Considers the game in reverse.
Trouvain. Engraving:
Dame de Qualité Jouant au Solitaire.
1698(?).
Claude‑Auguste Berey. Engraving:
Nouveau Jeu de Solitaire.
Undated, but Berey was active c1690‑c1730. Reproduced in: R. C. Bell; The Board Game Book; Marshall Cavendish, London,
1979, pp. 54‑55 and in: Jasia
Reichardt, ed.; Play Orbit [catalogue of an exhibition at the ICA, London, and
elsewhere in 1969-1970]; Studio International, 1969, p. 38. Beasley's additional notes point out that
this engraving is well known, but he had not realised its date until the
earlier Berey engraving was discovered.
This engraving includes the legend associating the game with the American
indians -- "son origine vient de l'amerique ou les Peuples vont seuls à la
chasse, et au retour plantent leurs flèches en des trous de leur cases, ce qui
donna idée a un françois de composer ce jeu ...." Reichardt says the original is in the Bibliothèque
Nationale.
The three engravings above are
reproduced in: Henri d'Allemagne; Musée
rétrospectif de la classe 100, Jeux, à l'exposition universelle international
de 1900 à Paris, Tome II, pp. 152‑158. D'Allemagne says the originals are in the Bibliothèque Nationale,
Paris. He (and de Mora-Charles) also
cites Rémond de Montmort, 2nd ed., 1713 -- see below.
G. W. Leibniz. Annotatio de quibusdam Ludis; inprimis de
Ludo quodam Sinico, differentiaque Scachici et Latrunculorum & novo genere
Ludi Navalis. Misc. Berolinensia (= Misc.
Soc. Reg., Berlin) 1 (1710) 24. Last
para. on p. 24 relates to solitaire.
(English translation on p. xii of Beasley, below.)
Pierre Rémond de Montmort. Essai d'analyse sur les jeux de
hazards. (1708); Seconde edition revue & augmentee de
plusieurs lettres, (Quillau, Paris,
1713 (reprinted by Chelsea, NY, 1980));
2nd issue, Jombert & Quillau, 1714.
Avertissement (to the 2nd ed.), xli-xl.
"J'ai trouvé dans le premier volume de l'Academie Royale de Berlin,
...; il propose ensuite des Problèmes sur un jeu qui a été à la mode en France
il y a douze ou quinze ans, qui se nomme Le Solitaire."
Edward Hordern's collection has
a wooden 37 hole board on the back of which is inscribed "Invented by Lord
Derwentwater when Imprisoned in the Tower". The writing is old, at least 19C, possibly earlier. However the Encyclopedia Britannica article
on Derwentwater and the DNB article on Radcliffe, James, shows that the
relevant Lord was most likely to have been James Radcliffe (1689-1716), the 3rd
Earl from 1705, who joined the Stuart rising in 1715, was captured at Preston,
was imprisoned in the Tower and was beheaded on 24 Feb 1716, so the
implied date of invention is 1715 or 1716.
The third Earl became a figure of romance and many stories and books
appeared about him, so the invention of solitaire could well have been
attributed to him.
Though
the title was attainted and hence legally extinct, it was claimed by
relatives. Both James's brother Charles
(1693‑1746), the claimed 5th Earl from 1731, and Charles's son James Bartholomew
(1725-1786), the claimed 6th Earl from 1746, spent time in prison for their
Stuart sympathies. Charles escaped from
Newgate Prison after the 1715 rising, but both were captured on their way to
the 1745 rising and taken to the Tower where Charles was beheaded. If either of these is the Lord Derwentwater
referred to, then the date must be 1745 or 1746. A guide book to Northumberland, where the family lived at
Dyvelston (or Dilston) Castle, near Hexham, asserts the last Derwentwater was
executed in 1745, while the [Blue Guide] says the last was executed for
his part in the 1715 uprising.
In
any case, the claim seems unlikely.
G. W. Leibniz. Letter to de Montmort (17 Jan 1716). In:
C. J. Gerhardt, ed.; Die Philosophischen Schriften von Gottfried
Wilhelm Leibniz; (Berlin, 1887) = Olms,
Hildesheim, 1960; Vol. 3, pp. 667‑669.
Relevant passage is on pp. 668‑669. (Poinsot, op. cit. in 5.E, p. 17, quotes this as letter VIII
in Leibn. Opera philologica.)
J. C. Wiegleb. Unterricht in der natürlichen Magie. Nicolai, Berlin & Stettin, 1779. Anhang von dreyen Solitärspielen, pp. 413‑416,
??NYS -- cited by Beasley. First known
diagram of the 33‑hole board.
Catel. Kunst-Cabinet. 1790. Das Grillenspiel (Solitaire), p. 50 &
fig. 167 on plate VI. 33 hole
board. (Das Schaaf- und Wolfspiel, p.
52 & fig. 169 on plate VI, is a game on the 33-hole board.)
Bestelmeier. 1801.
Item 511: Ein Solitair, oder Nonnenspiel. 33 hole board.
Strutt. Op. cit. in 4.B.1. The Solitary Game. (1801:
Book IV, p. 238. ??NYS -- cited by
Beasley -- may be actually 1791??)
1833: Book IV, chap. II, art. XV, p. 319. c= Strutt-Cox, p. 259.
Beasley says this is the first attribution to a prisoner in the
Bastille. The description is vague:
"fifty or sixty" holes and "a certain number of pegs". Strutt-Cox adds a note that "The game
of Solitaire, reimported from France, ..., came again into Fashion in England
in the late" 1850s and early 1860s.
Ada Lovelace. Letter of 16 Feb 1840 to Charles
Babbage. BM MSS 37191, f. 331. ??NYS -- reproduced in Teri Perl; Math
Equals; Addison-Wesley, Menlo Park, California, 1978, pp. 109-110. Discusses the 37 hole board and wonders if
there is a mathematical formula for it.
M. Reiss. Beiträge zur Theorie des Solitär‑Spiels. J. reine angew. Math. 54 (1857) 376‑379.
St. v. Kosiński & Louis
Wolfsberg. German Patent 42919 --
Geduldspiel. Patented:
25 Sep 1877. 1p + 1p
diagrams. 33 hole version.
The Sociable. 1858.
The game of solitaire, pp. 282-284.
37 hole board. "It is
supposed to have been invented in America, by a Frenchman, to beguile the
wearisomeness attendant upon forest life, and for the amusement of the Indians,
who pass much of their time alone at the chase, ...."
Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and Sons,
London, 1883, HB. Section 135:
Solitaire, p. 49. Mentions a 37
hole board but shows a 33 hole board.
This material presumably goes back some time before this edition. It later shows Fox and Geese on the 33 hole
board.
Hoffmann. 1893.
Chap. X, no. 11: Solitaire problems, pp. 339-340 & 376-377
= Hoffmann‑Hordern, pp. 232-233, with photo on p. 235. Three problems. Photo on p. 235 shows a 33-hole board in a square frame,
1820-1840, and a 37-hole board with a holding handle, 1840-1890.
Ernest Bergholt. Complete Handbook to the Game of Solitaire
on the English Board of Thirty-three Holes.
Routledge, London, nd [Preface dated Nov 1920] -- facsimile produced by
Naoaki Takashima, 1993. This is the
best general survey of the game prior to Beasley.
King. Best 100. 1927. No. 68, pp. 28 & 55. = Foulsham's no. 24, pp. 9 & 13. 3 x 3
array of men in the middle of a
5 x 5 board. Men can jump diagonally as well as
orthogonally. Object is to leave one
man in the centre.
Rohrbough. Puzzle Craft. 1932. Note on
Solitaire & French Solitaire, pp. 14-15
(= pp. 6-7 of 1940s?). 33
hole board, despite being called French.
B. M. Stewart. Solitaire on a checkerboard. AMM 48 (1941) 228-233. This surveys the history and then considers
the game on the 32 cell board comprising the squares of one colour on a
chessboard. He tilts this by 45o to get a board with 7 rows, having 2, 4, 6, 8, 6, 4, 2 cells in each row. He
shows that each beginning-ending problem which is permitted by the parity rules
is actually solvable, but he gives examples to show this need not happen on
other boards.
Gardner. SA (Jun 1962). Much amended as: Unexpected, chap. 11, citing results of Beasley,
Conway, et al. Cites Leibniz and
mentions Bastille story.
J. D. Beasley. Some notes on solitaire. Eureka 25 (Oct 1962) 13-18. No history of the game.
Jeanine Cabrera & René
Houot. Traité Pratique du
Solitaire. Librairie Saint‑Germain,
Paris, 1977. On p. 2, they give the
story that it was invented by a prisoner in the Bastille, late 18C, and they
even give the name of the reputed inventor:
"Comte"(?) Pellisson.
They say that a Paul Pellisson‑Fontanier was in the Bastille in
1661‑1666 and was a man of some note, but history records no connection
between him and the game.
The Diagram Group. Baffle Puzzles -- 3: Practical Puzzles. Sphere, 1983. No. 12. On the 33-hole
board place 16 markers:
1 in row 2; 3 in row 3; 5 in row 4;
7 in row 5; making a triangle
centred on the mid-line. Can you remove
all the men, except for one in the central square? Gives a solution in 15 jumps.
J. D. Beasley. The Ins and Outs of Peg Solitaire. OUP, 1985.
History, pp. 3‑7; Selected
Bibliography, pp. 253‑261.
PLUS Additional notes, from the
author, 1p, Aug 1985. 57 references and
5 patents, including everything known before 1850.
Franco Agostini & Nicola
Alberto De Carlo. Intelligence
Games. (As: Giochi della Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. This gives the legend of the nobleman in the
Bastille. Then says that "it would
appear that a very similar game" is mentioned by Ovid "and again, it
was widely played in ancient China -- hence its still frequent alternative name,
"Chinese checkers."" I
have included this as an excellent example of how unreferenced statements are
made in popular literature. I have
never seen either of these latter statements made elsewhere. The connection with Ovid is pretty tenuous
-- he mentions a game involving three in a row and otherwise is pretty cryptic
and I haven't seen anyone else claiming Ovid is referring to a solitaire game
-- cf 4.B.5. The connection with
Chinese checkers is so far off that I wonder if there is a translation problem
-- i.e. does the Italian name refer to some game other than what is known as
Chinese checkers in English??
Nob Yoshigahara. Puzzlart.
Tokyo, 1992. Coin solitaire, pp.
5 & 90. Four problems on a 4 x 4
board.
Marc Wellens, et al. Speelgoed Museum Vlaanderen -- Musée du
Jouet Flandre -- Spielzeug Museum Flandern -- Flanders Toy Museum. Speelgoedmuseum Mechelen, Belgium, 1996, p.
90 (in English), asserts 'It was
invented by the French nobleman Palissen, who had been imprisoned in the
Bastille by Louis XIV' in the early 18C.
The triangular version of the game has
only recently been investigated. The
triangular board is generally numbered as below.
1
2 3
4 5
6
7 8 9 10
11 12 13 14
15
Herbert M. Smith. US Patent 462,170 -- Puzzle. Filed: 13 Mar 1891; issued: 27 Oct 1891. 2pp + 1p diagrams. A board based on a triangular lattice.
Rohrbough. Puzzle Craft. 1932. Triangle Puzzle, p.
5 (= p. 6 in 1940s?). Remove peg 13 and
leave last peg in hole 13.
Maxey Brooke. (Fun for the Money, Scribner's, 1963); reprinted as: Coin Games and Puzzles; Dover, 1973. All the following are on the 15 hole board.
Prob.
1: Triangular jump, pp. 10-11 & 75.
Remove one man and jump to leave one man on the board. Says Wesley Edwards asserts there are just
six solutions. He removes the middle
man of an edge and leaves the last man there.
Prob.
2: Triangular jump, Ltd., pp. 12-13 & 75.
Removes some of the possible jumps.
Prob.
3: Headless triangle, pp. 14 & 75.
Remove a corner man and leave last man there.
M. Gardner. SA (Feb 1966) c= Carnival, 1975, chap. 2.
Says a 15 hole version has been on sale as Ke Puzzle Game by S. S. Adams
for some years. Addendum cites Brooke
and Hentzel and says much unpublished work has been done.
Irvin Roy Hentzel. Triangular puzzle peg. JRM 6:4 (1973) 280-283. Gives basic theory for the triangular
version. Cites Gardner.
[Henry] Joseph & Lenore
Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books (Charter Communications), NY,
1975. Pennies for your thoughts, pp.
179-182. Remove a coin and solve. Hint says to remove the coin at 13
and that you should be able to have the last coin at 13.
The solution has this property.
Alan G. Henney & Dagmar R.
Henney. Computer oriented
solutions. CM 4:8 (1978) 212‑216. Considers the 'Canadian I. Q. Problem',
which is the 15 hole board, but they also permit such jumps as 1 to 13,
removing 5. They find solutions from each initial
removal by random trial and error on a computer.
Putnam. Puzzle Fun.
1978. No. 15: Jumping coins, pp.
5 & 28. 15 hole version, remove peg 1
and leave last man there.
Benjamin L. Schwarz & Hayo
Ahlburg. Triangular peg solitaire -- A
new result. JRM 16:2 (1983-84)
97-101. General study of the 15 hole
board showing that starting and ending with
5 is impossible.
J. D. Beasley. The Ins and Outs of Peg Solitaire. Op. cit. above, 1985. Pp. 229-232 discusses the triangular
version, citing Smith, Gardner and Hentzel, saying that little has been
published on it.
Irvin Roy Hentzel & Robert
Roy Hentzel. Triangular puzzle
peg. JRM 18:4 (1985-86) 253‑256. Develops theory.
John Duncan & Donald
Hayes. Triangular solitaire. JRM 23:1 (1991) 26-37. Extended analysis. Studies army advancement problem.
William A. Miller. Triangular peg solitaire on a
microcomputer. JRM 23:2 (1991)
109-115 & 24:1 (1992) 11.
Summarises and extends previous work.
On the 10 hole triangular board, the classic problem has essentially a
unique solution -- the removed man must be an edge man (e.g. 2) and the last
man must be on the adjacent edge and a neighbour of the starting hole (i.e. 3
if one starts with 2). On the 15 hole
board, the removed man can be anywhere and there are many solutions in each
case.
Remove
man from hole: 1 2 4 5
Number
of solutions: 29760 14880 85258 1550
Considers
the 'tree' formed by the first four rows and hole 13.
New section. See also King and Stewart in 5.R.1 for some forms based on a
square board.
A
B C D E
F G
H
I J
Putnam. Puzzle Fun.
1978. No. 53: Checker star, pp.
10 & 34. Use the 10 points of a
pentagram, as above, and leave one of the inner points empty. Reduce to one man. [Parity shows the one man must be at an outer point and any outer
point can be achieved. If one leaves an
outer point empty, then the last man must be on an inner point and any of these
can be achieved.]
Hummerston. Fun, Mirth & Mystery. 1924.
Perplexity,
pp. 22-23. Using the octagram board
shown in 5.A, place 15 markers on it, leaving cell 16 empty. It is possible to remove all but one man. [I can't see how to apply parity to this
board.]
Solplex,
p. 25. In playing his Perplexity,
specify where you will leave the last man?
Leap
frog, Puzzle no. 22, pp. 64 & 175.
Take a 4 x 3 board with the long edge extended by one
more cell at the upper left and lower right.
Put white counters on the 4 x
3 area, put a black counter in one of
the extra cells and leave the other extra cell empty. Remove all but the black man.
Counting multiple jumps of the same man as a single move, he does it in
eight moves, getting the black man back to its starting point.
5.R.2. FROGS AND TOADS: BBB_WWW TO WWW_BBB
In
the simplest version, one has n black men at the left and n
white men at the right of a strip of
2n+1 cells, e.g. BBB_WWW.
One can slide a piece forward (i.e. blacks go left and whites go right)
into an adjacent place or one can jump forward over one man of the other colour
into an empty place. The object is to
reverse the colours, i.e. to get
WWW_BBB. S&B 121 & 125,
shows versions.
One
finds that the solution never has a man moving backward nor a man jumping
another man of the same colour. Some
authors have considered relaxing these restrictions, particularly if one has
more blank spaces, when these unusual moves permit shorter solutions. Perhaps the most general form of the
one-dimensional problem would be the following. Suppose we have m men at the left of the board, n men
at the right and b blank spaces in the middle. The usual case has b = 1, but when b > 1,
the kinds of move permitted do change the number of moves in a minimal
solution. First, considering slides,
can a piece slide backward? Can a piece
slide more than one space? If so, is
there a maximum distance, s, that it is allowed to slide? (The usual problem has s = 1.)
Of course s £
b. Second, considering jumps, can a
piece jump backward? Can a piece jump
over a piece (or pieces) of its own colour and/or a blank space (or spaces)
and/or a mixture of these? If so, is
there a maximum number of pieces,
p, that it can jump over? (The usual problem has p = 1.)
It is not hard to construct simple examples with s > 1
such that shorter solutions exist when unusual moves are permitted. Are there situations where one can show that
backward moves are not needed?
The
game is sometimes played on a 2-dimensional board, where one colour can move
down or right and the other can move up or left. See: Hyde ??; Lucas (1883); Ball; Hoffmann and 5.R.3.
Chinese checkers is a later variation of this same idea. On these more complex boards, one is usually
allowed to make multiple jumps and the object is usually to minimize the number
of moves to accomplish the interchange of pieces.
There
is a trick version to convert full and empty glasses: FFFEEE to FEFEFE
in one move, which is done by pouring.
I've just noted this in a 1992 book and I'll look for earlier examples.
Thomas Hyde. Historia Nerdiludii, hoc est dicere,
Trunculorum; .... (= Vol. 2 of De Ludis
Orientalibus, see 7.B for vol. 1.) From
the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo dicto Ufuba wa Hulana, p. 233. This has a
5 x 5 board with each
side having 12 men, but the description is extremely brief. It seems to have two players, but this may
simply refer to the two types of piece.
I'm not clear whether it's played like solitaire (with the jumped pieces
being removed) or like frogs & toads.
I would be grateful if someone could read the Latin carefully. The name of the puzzle is clearly Arabic and
Hyde cites an Arabic source, Hanzoanitas (not further identified on the pages I
have) -- I would be grateful to anyone who can track down and translate Arabic
sources.
American Agriculturist (Jun
1867). Spanish Puzzle. ??NYR -- copy sent by Will Shortz.
Anonymous. Every Little Boy's Book A Complete Cyclopædia of in and outdoor games
with and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and fifty
illustrations. Routledge, London,
nd. HPL gives c1850, but the material
is clearly derived from Every Boy's Book, whose first edition was 1856. But the text considered here is not in the
1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor), nor in the 8th
ed of 1868 (published for Christmas 1867), which was the first seriously
revised edition, with Edmund Routledge as editor, nor in the 13th ed. of
1878. So this material is hard to date,
though in 4.A.1, I've guessed this book may be c1868.
P.
12: Frogs and toads. "A new and
fascinating game of skill for two players; played on a leather board with
twelve reptiles; the toads crawling, and the frogs hopping, according to
certain laws laid down in the rules.
The game occupies but a few minutes, but in playing it there is scarcely
any limit to the skill that can be exhibited, thus forming a lasting amusement. (Published by Jaques, Hatton Garden.)" This does not sound like our puzzle, but
perhaps it is related. Unfortunately
Jaques' records were destroyed in WW2, so it is unlikely they can shed any
light on what the game was. Does anyone
know what it was?
Hanky Panky. 1872.
Checker puzzle, p. 124.
Three and three, with solution.
Mittenzwey. 1880.
Prob. 239, pp. 44 & 94;
1895?: 267-268, pp. 48 & 96;
1917: 267-268, pp. 44 & 91‑92. Problem with 3 & 3 brown and white
horses in stalls. 1895? adds a version
with 4 & 4.
Bazemore Bros. (Chattanooga,
Tennessee). The Great "13"
Puzzle! Copyright No. 1033 ‑ O ‑ 1883. Hammond & Jones Printers.
Advertising puzzle consisting of two
3 and 3 versions arranged in
an X
pattern.
Lucas. RM2. 1883.
Pp.
141‑143. Finds number of moves
for n and n.
Pp.
144‑145. Considers game on 5 x 5,
7 x 7, ..., boards and gives number of moves.
Edward Hordern's collection has
an example called Sphinxes and
Pyramids from the 1880s.
Sophus Tromholt. Streichholzspiele. (1889; 5th ed,
1892.) Revised from 14th ed. of 1909 by
R. Thiele; Zentralantiquariat der DDR, Leipzig, 1986. Prob. 11, 41, 81 are the game for 4 & 4, 2 & 2, 3 & 3.
Ball. MRE, 1st ed., 1892, pp. 49‑51. 3 & 3 case, citing
Lucas, with generalization to n &
n; 7 x 7 board, citing Lucas, with generalization to 2n+1 x 2n+1.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
German counter puzzle, p. 112. 3
& 3 case.
Hoffmann. 1893.
Chap. VI, pp. 269‑270 & 282‑284 = Hoffmann-Hordern,
pp. 182-185, with photo.
No.
17: The "Right and Left" puzzle.
Three and three. Photo on p. 184
shows: a cartoon from Punch (18 Dec 1880): The Irish Frog Puzzle -- with a Deal
of Croaking; and an example of a handsome carved board with square pieces with
black and white frogs on the tops, registered 1880. Hordern Collection, p. 77, shows the latter board and two
further versions: Combat Sino-Japonais (1894‑1895) and Anglais &
Boers (1899-1902).
No.
18. Extends to a 7 x 5
board.
Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??)
751. 3 and 3.
Lucas. L'Arithmétique Amusante.
1895. Prob. XXXV: Le bal des
crapauds et des grenouilles, pp. 117-124.
Does 2 and 2, 3 and 3,
4 and 4 and the general case of n and n,
showing it can be done in
n(n+2) moves -- n2 jumps and
2n steps. The general solution is attributed to M. Van
den Berg. M. Schoute notes that each
move should make as little change as possible from the previous with respect to
the two aspects of changing type of piece and changing type of move.
Clark. Mental Nuts. 1904, no.
72; 1916, no. 62. A good study. 3 and 3.
Burren Loughlin &
L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909.
Doola's Game, pp. 42-43 & 61-62.
3 and 3.
Anon. Prob. 47: The monkey's dilemma.
Hobbies 30 (No. 762) (21 May
1910) 168 & 182 & (No. 765) (11 Jun 1910) 228. Basically
3 & 3, but there are eight
posts for crossing a river, with the monkeys on 1,2,3 and 6,7,8.
The monkeys can jump onto the bank and we want the monkeys to all get to
the bank they are headed for, so this is not the same as BBB..WWW
to WWW..BBB. The solution doesn't spell out all the
steps, so it's not clear what the minimum number of moves is -- could we have a
monkey jumping another of the same colour?
Ahrens. MUS I.
1910. Pp. 17-19. Basically repeats some of Lucas's work from
1883 & 1895.
Williams. Home Entertainments. 1914.
The cross-over puzzle, pp. 119-120.
3 and 3 with red and white
counters. Doesn't say how many moves
are required.
Dudeney. AM.
1917. Prob. 216: The educated
frogs, pp. 59-60 & 194.
_WWWBBB to BBBWWW_
with frogs able to jump either way over one or two men of either
colour. Solution in 10 jumps.
Ball. MRE, 9th ed., 1920, pp. 77-79, considers the m & n
case, giving the number of steps in the solution.
Blyth. Match-Stick Magic.
1921. Matchstick circle
transfer, pp. 81‑82. 3 and 3 in 15 moves.
Hummerston. Fun, Mirth & Mystery. 1924.
The frolicsome frogs, Puzzle no. 2, pp. 17 & 172. Two
3 & 3 problems with the
boards crossing at the centre cell. He
notes that the easiest solution is to solve the boards one at a a time. He says: "It is not good play to jump a
counter over another of the same colour."
Lynn Rohrbough, ed. Socializers. Handy Series, Kit G, Cooperative Recreation Service, Delaware,
Ohio, 1925. Six Frogs, p. 5. Dudeney's 1917 problem done in 11 moves.
Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 235 describes
this as The Sphinx Puzzle, "very popular around the turn of the century,
particularly in the United States and France" and they show an example of
the period labelled The Sphinx and Pyramid Puzzle -- An Egyptian Novelty.
Haldeman-Julius. 1937.
No. 162: Checker problem, pp. 18 & 29. 3 & 3.
See Harbin in 5.R.4 for a 1963
example.
Doubleday - 1. 1969.
Prob.
77: Square dance, pp. 93 & 171. =
Doubleday - 5, pp. 103-104. Start
with _WWWBBB. He says they must change places, with a piece able to move into
the vacant space by sliding (either way) or by jumping one or two pieces of any
colour. Asks for a solution in 10
moves. His solution gets to BBBWWW_,
which does not seem to be 'changing places' to me.
Prob.
79: All change, pp. 95 & 171. =
Doubleday - 5, pp. 105-106. BB_WW
Start with the pattern at the right and change
the whites and BB_WW
blacks in 10 moves, where a piece can slide one
place into an
adjacent vacant square or jump one or two
pieces into a vacant square. However,
the solution simply does each row separately.
Katharina Zechlin. (Dekorative Spiele zum Selbermachen; Verlag
Frech, WWWWW
Stuttgart-Botnang,
1973.) Translated as: Making Games in Wood Games BWWWW
you
can build yourself. Sterling, 1975, pp.
24-27: The chess knight game. BBOWW
5
x 5 board with 12 knights of each of
two colours, arranged as at the right. BBBBW
The
object is to reverse them by knight's moves.
Says it can be done within BBBBB
50
moves and 'is almost impossible to do it in less than 45'.
Wickelgren. How to Solve Problems. Op. cit. in 5.O. 1974. Discrimination
reversal problem, pp. 78‑81.
_WWWBBB to BBBWWW
with the extra place not specified in the goal, with pieces allowed to move
into the vacant space by sliding or by hopping over one or two pieces. Gets to
BBBWW_W in 9 moves. [I find it takes 10 moves to get to BBBWWW_ .]
Joe Celko. Jumping frogs and the Dutch national
flag. Abacus 4:1 (Fall 1986)
71-73. Same as Wickelgren. Celko attributes this to Dudeney. Gives a solution to BBBWWW_
in 10 moves and asks for results for higher numbers.
Johnston Anderson. Seeing induction at work. MG 75 (No. 474) (Dec 1991) 406-414. Example 2: Frogs, pp. 408-411. Careful proof that BB...BB_WW...WW to WW...WW_BB...BB, with n counters of each colour, requires n2 + 2n moves.
5.R.3. FORE AND AFT -- 3 BY 3 SQUARES MEETING AT A CORNER
This is Frogs and Toads on part of
the 5 x 5 board consisting of two 3
x 3 subarrays at diagonally opposite
corners. They overlap in the central
square. One square has 8 black men and
the other has 8 white men, with the centre left vacant.
Ball. MRE, 1st ed., 1892, pp. 51‑52. 51 move solution. In the
third ed., 1896, pp. 69‑70, he says he believes he was the first to
publish the puzzle but "that it has been since widely distributed in
connexion with an advertisement and probably now is well known". He gives a 48 move solution.
Hoffmann. 1893.
Chap. VI, no. 26: The "English Sixteen" puzzle, pp. 273‑274
& 287 = Hoffmann-Hordern, pp. 188-189, with photo. Mentions that it is produced by Messrs
Heywood, as below. Solution in 52
moves, which he believes is minimal.
Hordern notes that the minimum is 46.
Photo on p. 188 of the Heywood version, see next entry.
John Heywood, Manchester,
produced a version called 'The English Sixteen Puzzle', undated, but by 1893 as
Hoffmann cites it. Photo in
Hoffmann-Hordern, p. 188, dated 1880‑1895.
Charles A. Emerson. US Patent 522,250 -- Puzzle. Applied: 3 Nov 1893; patented: 3 July 1894. 2pp + 1p diagrams. The Fore and Aft Puzzle.
Says it can be done in 48, 49,
50, 51 or 52 moves.
Dudeney. Problem 66: The sixteen puzzle. Tit‑Bits 33 (1 Jan &
5 Feb 1898) 257 & 355.
"It was produced, I believe, in America, many years ago, and has
since been issued over here in the form of an advertisement by a prominent
commercial house." Solution in 46
moves. He says published solutions
assert the minimum number of moves is
53, 52 or 50. The 46 move
solution is given in Ball, MRE, 5th ed., 1911, 79‑80.
Ball. MRE, 5th ed., 1911, pp. 79-80.
Drops his historical claims and includes a 46 move solution due to
Dudeney.
Loyd. Fore and aft puzzle.
Cyclopedia, 1914, pp. 108 & 353 (solution misprinted, but claimed to
be 47 moves in contrast to 52 move solutions 'in the puzzle books'.) (c= MPSL1, prob. 4, pp. 3‑4 &
121 (only referring to Dudeney's 46 move solution)).
Loyd Jr. SLAHP.
1928. A joke on granddad, pp. 29
& 93. Says 'our granddaddies, who
used to play this puzzle game 75 years ago, when it was universally popular. The old‑time books explain how the
solution is accomplished in 52 moves, "the shortest possible
method."' He then asks for and
gives a 46 move solution.
M. Adams. Puzzles That Everyone Can Do. 1931.
Prob. 24, pp. 17 & 132: "General post". Gives a solution which takes 46 moves, but
gives no discussion of it.
Rohrbough. Puzzle Craft. 1932. Migration (or Fore
and Aft), p. 12 (= p. 15 of 1940s?).
Says it was popular 75 years ago and it has recently been shown that it
can be done in 46 moves, then gives a solution which stops at 42 moves!
M. Gardner. SA (Sep 1959) = 2nd Book, pp. 210‑219. Discusses the puzzle. On pp. 218‑219, he gives
Dudeney's 46 move solution and says 48 different solutions and several proofs
that 46 is minimal were sent to him.
Uwe Schult. Das Seemanns‑Spiel: Mathematisch
erledigt. Reported in Das Mathematische
Kabinett column, Bild der Wissenschaft 19:11 (Nov 1982) 181-184. (A version is given in Neues aus dem
Mathematischen Kabinett, ed. by Thiagar Devendran, Hugendubel, Munich, 1985,
pp. 102‑103.) There are 218,790
possible patterns of the pieces.
Reversing black and white takes
46 moves and there are 1026
different halfway positions that can occur in a 46
move solution. There are two
patterns which require 47 moves, namely, after reversing black and
white, put one of the far corner pieces in the centre.
Nob Yoshigahara, postcard to me
on 18 Aug 1994, announces he has found the worst solution -- in 58 moves.
5.R.4. REVERSING FROGS AND TOADS: _12...n TO _n...21 , ETC.
A
piece can slide into the empty cell or jump another piece into the empty cell.
Dudeney. AM.
1917.
Prob.
214: The six frogs, pp. 59 & 193.
Case of n = 6, solved in
21 moves, which he says is
minimal. In general, the minimal solution
takes n(n+1)/2 moves, including n steps, when n is
even and (n2+3n-8)/2
moves, including 2n-4 steps, when
n is odd. "This complete general solution is
published here for the first time."
Prob.
215: The grasshopper puzzle, pp. 59 & 193-194. Problem for a circular arrangement. Example has n = 12. Says he invented it in 1900. Solvable in
44 moves. General solution is complex -- he says that
for n > 4, it can be done in (n2+4n-16)/4 moves when
n is even and in (n2+6n-31)/4 moves when
n is odd.
Rohrbough. Puzzle Craft. 1932. The Reversible
Frogs, p. 22 (= The Jumping Frogs, pp. 20‑21 of 1940s?). n = 8, citing Dudeney, AM.
Robert Harbin. Party Lines. Op. cit. in 5.B.1.
1963. Hopover, p. 89. First gives
3 and 3 Frogs and Toads, then
asks for complete reversal from
123_456 to 654_321.
[Henry] Joseph and Lenore
Scott. Master Mind Pencil Puzzles. Tempo Books (Grosset & Dunlap), NY, 1973
(& 1978?? -- both dates are given -- I'm presuming the 1978 is a 2nd ptg or
a reissue under a different imprint??).
Reverse the numbers, pp. 117‑118. Give the problem for n =
6 and a solution in 21 moves. For
n even, the method gives a
solution in n(n+1)/2, it is not shown that this is optimal, nor is
a general method given for odd n.
[Henry] Joseph & Lenore
Scott. Master Mind Brain Teasers. 1973.
Op. cit. in 5.E. 13-hour clock,
pp. 43-44. Case n = 12 considered in a circle can be done in 44 moves.
Joe Celko. Jumping frogs and the Dutch national
flag. Abacus 4:1 (Fall 1986)
71-73. Cites Dudeney and gives the
results.
Jim Howson. The Computer Weekly Book of Puzzlers. Computer Weekly Publications, Sutton,
Surrey, 1988, unpaginated. [The
material comes from his column which started in 1966, so an item may go back to
then.] Prob. 54 -- same as the Scotts
in Master Mind Pencil Puzzles.
There
are a number of similar games on different boards -- too many to describe
completely here, so I will generally just cite extensive descriptions. See any of the main books on games mentioned
at the beginning of 4.B, such as Bell or Falkener. The key feature is that one side has more, but weaker, pieces. These are sometimes called hunt games. The standard Fox and Geese is played on a 33
hole Solitaire board, with diagonal moves allowed. I have recently acquired but not yet read Murray's History of
Board Games other than Chess which should have lots of material.
Gretti's Saga, late 12C. Mention of Fox and Geese. Also in Edward IV's accounts. ??NYS -- cited by Botermans et al, below.
Shackerley (or Schackerley or
Shakerley) Marmion. A Fine Companion (a
play). 1633. IN: The Dramatic Works of Shackerley Marmion; William Paterson,
Edinburgh & H. Sotheran & Co., London, 1875.
II, v, pp. 140-141. "...,
let him sit in the shop ..., and play at fox and geese with the foreman,
....." Earliest English occurrence
of fox-and-geese. Quoted by OED and
cited by Fiske, below.
Richard Lovelace. To His Honoured Friend On His Game of
Chesse-Play or To Dr. F. B. on his Book of Chesse. 1656?, published in his Posthume Poems,
1659. Lines 1-4. My edition of Lovelace notes that F. B. was
Francis Beale, author of 'Royall Game of Chesse Play,' 1656. Lovelace died in 1658.
Sir,
now unravell'd is the Golden Fleece,
Men
that could onely fool at Fox and Geese,
Are
new-made Polititians by the Book,
And
can both judge and conquer with a look.
Henry Brooke. Fool of Quality. [A novel.]
1766-1768. Vol. I, p. 367. ??NYS -- quoted by Fiske, below. "Can you play at no kind of game,
Master Harry?" "A little at
fox‑and‑geese, madam."
Catel. Kunst-Cabinet. 1790.
Das
Fuchs- und Hühnerspiel, pp. 51-52 & fig. 168 on plate VI. 11 chickens against one fox on a 4 x 4
board with all diagonals drawn, giving
16 + 9 playing points.
Das
Schaaf- und Wolfspiel, p. 52 & fig. 169 on plate VI, is the same game on
the 33-hole solitaire board with 11 sheep and one wolf, no diagonals
Bestelmeier. 1801.
Item
83: Das Schaaf- und Wolfspiel. Same
diagram and game as Catel, p. 52.
Item
833: Ein Belagerungspiel. 33 hole board
with a fortress on one arm, with diagonals drawn.
Strutt. Op. cit. in 4.B.1. Fox and Geese. 1833: Book
IV, chap. II, art. XIV, pp. 318‑319.
= Strutt-Cox, p. 258 & plate opp. p. 246. Fig. 107 (= plate opp. p. 246) shows the 33
hole board with its diagonals drawn.
Gomme. Op. cit. in 4.B.1. I 141‑142
refers to Strutt and Micklethwaite.
Illustrated Boy's Own
Treasury. 1860. Fox and Geese, pp. 406‑407. 33 hole Solitaire board with diagonals
drawn.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 320, p. 152:
Fuchs und Gänse. Shows 33 hole
solitaire board with diagonals drawn.
Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S. National Museum
for 1893, pp. 489‑537. Pp.
874-877 describes: the Japanese game
of Juroku Musashi (Sixteen Soldiers) with 16 men versus a general;
the Chinese game of Shap luk kon
tséung kwan (The sixteen pursue the
commander); another Chinese game
of Yeung luk sz' kon tséung kwan with 27 men against a commander (described
by Hyde -- ?? I didn't see this); the
Malayan game of Dam Hariman (Tiger Game), identical to the Hindu game of
Mogol Putt'han (= Mogul Pathan
(Mogul against Pathan)), similar to a
Peruvian game of Solitario and the Mexican game of Coyote; the Siamese game of Sua ghin gnua (Tiger and Oxen) and the
similar Burmese game of Lay gwet
kyah, with three big tigers versus 11
or 12 little tigers; the Samoan game of Moo;
the Hawaiian game of Konane; a
similar Madagascarian game; the Hindu
game of Pulijudam (Tiger Game) with three tigers versus 15
lambs.
Fiske. Op. cit. in 4.B.1.
1905. Fox-and-Geese, pp. 146-156
& 359, discusses the history of the game, especially as to whether it is
identical to the old Norse game of Hnefatafl.
On p. 359, he says that John of Salisbury (c1150) used 'vulpes' as
the name of a game, but there is no indication of what it was. He says "the fox-and-geese board, in
comparatively modern times, has begun to be used for games more or less
different in their nature, especially for one called in England solitaire and
in France "English solitaire", and for another, known in Spain and
Italy as asalto (assalto), in French as assaut, in Danish as
belejringsspel." He then surveys
the various sources that he treated under Mérelles -- see 4.B.1 and 4.B.5 for
details. He is not sure that Brunet is
really describing the game in the Alfonso MS (op. cit. in 4.B.5 and
below). He cites an 1855 Italian usage
as Jeu de Renard or Giuoco della Volpe.
In Come Posso Divertirmi? (Milan, 1901, pp. 231-233), it is said that
the game is usually played with 17 geese rather than 13 -- Fiske notes that
this assertion is of "some historical value, if it be true." Moulidars calls it Marelle Quintuple, quotes
Maison des Jeux Académiques (Paris, 1668) for a story that it was invented by
the Lydians and gives the game with 13 or 17 geese. Asalto has 2 men against 24.
Fiske quotes Shackley Marmion, above, for the oldest English occurrence
of fox-and-geese and then Henry Brooke, above.
Fiske follows with German, Swedish and Icelandic (with 13 geese)
references.
H. Parker. Ancient Ceylon. Op. cit. in 4.B.1, 1909.
Pp. 580-583 & 585 describe four forms of The Leopards Game, with one
tiger against seven leopards, three leopards against 15 dogs, two leopards
against 24 cattle and one leopard against six cattle on a 12 x 12
board. The first two are played
on a triangular board.
Robert Kanigel. The Man Who Knew Infinity. A Life of the Genius Ramanujan. (Scribner's, NY, 1991); Abacus (Little, Brown & Co. (UK)),
London, 1992. Pp. 18 & 377: Ramanujan and his mother used to play the
game with three tigers and fifteen goats on a kind of triangular board.
The Spanish Treatise on
Chess-Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of
Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototypic
Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913. See 4.B.5 for more details of this work. See below.
Botermans et al. The World of Games. Op. cit. in 4.B.5. 1989. P. 147 says De
Cercar La Liebre (Catch the Hare) occurs in the Alfonso MS and is the earliest
example of a hunt game in European literature, but undoubtedly derived from an
Arabic game of the Alquerque type -- I didn't see this when I briefly looked at
the facsimile -- ??NYS. They say Murray
has noted that hunt games are popular in Asia, but not in Africa, leading to
the conjecture that they originated in Asia.
They describe it on a 5 x 5 array of points with verticals and
horizontals and some diagonals drawn, with one hare against 12 hunters.
Botermans
et al. continue on pp. 148-155 to describe the following.
Shap
Luk Kon Tseung Kwan (Sixteen Pursue the General) played on a 5 x 5
board like Catch the Hare with an extra triangle on one side and
capturing by interception.
Yeung
Luk Sz'Kon Tseung Kwan, seen in Nanking by Hyde and described by him in 1694,
somewhat similar to the above, but with 26 rebels against a general. (??NYS)
Fox
and Geese, mentioned in Gretti's Saga of late 12C and in Edward IV's
accounts. They give a version called
Lupo e Pecore from a 16C Venetian book, using a Solitaire board extended by
three points on each arm, giving 45 points.
They give a 1664 engraving showing Le Jeu du Roi which they say is a
rather complex form of fox and geese, but looks like a four-handed game on a
cross-shaped board with 7 x 5 arms on a
7 x 7 central square and 4
groups of 7 x 4 men.
Leopard
games, from Southeast Asia, with a kind of triangular board. Len Choa, from Thailand, has a tiger against
six leopards. Hat Diviyan Keliya, from
Sri Lanka, has a tiger against seven leopards.
Tiger
games, also from Southeast Asia, are similar to leopard games, but use an
extended Alquerque board (as in Catch the Hare). Rimau (Tiger), from Malaysia, has 24 men versus a tiger and
Rimau-Rimau (Tigers) is a version with two tigers versus 22 men.
Murray. 1913.
P. 347
cites a 1901 Indian book for 2 lions against 32 goats on a chessboard.
P. 371
cites a Soyat (North Asia) example (19C?) of Bouge‑Shodra (Boar's Chess)
with 2 boars against 24 calves on a chessboard.
Pp.
569 & 616‑617 cite the Alfonso MS of 1283 for 'De cercar la liebre',
played on a 5 x 5 board with
10, 11 or 12 men against a hare.
P. 585
shows Cott. 6 (c1275) of 8 pawns against a king on a chessboard.
Pp.
587 & 590 give Cott. 11 = K6: Le Guy de Alfins with king and 4 bishops
against a king on a chessboard.
Pp.
589-590 shows K4 = CB249: Le Guy de Dames and No. 5 = K5: Le Guy de
Damoyselles, which have 16 pawns against a king on a chessboard.
P. 617
discusses Fox and Geese, with 13, 15 or 17 geese against a fox on the solitaire
board. Edward IV, c1470, bought
"two foxis and 46 hounds".
Murray says more elaborate forms exist and refers to Hyde and Fiske (see
4.B.1 and 5.F.1 for more on these), ??NYS.
Pp.
675 & 692 show CB258: Partitum regis Francorum with king and four pawns
against king on the chessboard. It says
the first side wins.
P. 758
describes a 16C Venetian board (then) at South Kensington (V&A??) with the
Solitaire board for Fox and Geese and an enlarged board for Fox and Geese.
P. 857
mentions Fox and Geese in Iceland.
Family Friend 2 (1850) 59. Fox and geese. 4 geese against 1 fox on a chess board.
The Sociable. 1858.
Fox and geese, p. 281. 17 geese
against a fox on the solitaire board.
Four men versus a king on the draughts board, saying the first side wins
even allowing the king to be placed anywhere against the men who start on one
side.
Stewart Culin. Korean Games, op. cit. in 4.B.5, 1895. Pp. 76-77 describes some games of this type,
in particular a Japanese game called Yasasukari Musashi with 16 soldiers versus
a general on a 5 x 5 board, taken from a 1714 (or 1712) Japanese
book: Wa Kan san sai dzu e
"Japanese, Chinese, Three Powers picture collection", published in
Osaka.
Anonymous. Enquire Within upon Everything. 66th ed., 862nd thousand, Houlston and Sons,
London, 1883, HB. Section 2593: Fox and
Geese, p. 364. 33 hole Solitaire board
with 17 geese against a fox. 4 geese
against a fox on the chessboard. Says
the geese should win in both cases.
Slocum. Compendium.
Shows Solitaire and Solitaire & Tactic Board from Gamage's 1913
catalogue. Like Bestelmeier's 833, but
without diagonals.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Games involving unequal
forces, pp. 43-52. Discusses the
following.
The
Maharajah and the Sepoys. 1 against 16
on a chessboard.
Fox
and Geese. Cites an Icelandic work of
c1300 (probably Gretti's Saga?). 1
against 13 or 17 on a Solitaire board.
Lambs
and Tigers, from India. 3 against 15.
Cows
and Leopards, from SE Asia. 2 against
24.
Vultures
and Crows, also called Kaooa, from India.
1 against 7 on a pentagram board.
The
New Military Game of German Tactics, c1870.
2 against 24 on a Solitaire Board with a fortress, as in Bestelmeier.
Yuri I. Averbakh. Board games and real events. IN: Alexander J. de Voogt, ed.; New
Approaches to Board Games Research: Asian Origins and Future Perspectives;
International Institute for Asian Studies, Leiden, 1995; pp. 17-23. Notes that Murray believes hunt games
evolved from war games, but he feels the opposite is true. He describes a Nepalese game of Baghachal
with four tigers versus 20 goats -- this is Murray's 5.6.22. He corrects some of Murray's assertions
about Boar Chess and describes other Tuvinian hunt games: Bull's Chess and
Calves' Chess, probably borrowed from the Mongols. The latter has a three-in-a-row pattern and he wonders if there
is some connection with morris or noughts and crosses (which he says is
"played everywhere"). He
mentions Cercar la Liebre from the Alfonso MS.
Fox and Geese type games are mentioned in the Icelandic sagas as 'the
fox game'. He describes several forms.
One
has an octagram and seven men. One has to
place a man on a vacant point and then slide him to an adjacent vacant point,
then do the same with the next man, ...,
so as to cover seven of the points.
The diagram is just an 8-cycle and is the same as the knight's
connections on the 3 x 3 board, so the octagram puzzle is equivalent
to the 7 knights problem mentioned in 5.F.1.
Further, the 4 knights problem of 5.F.1 has the same 8-cycle, with men
at alternate points of it.
Versions
with different numbers of points.
5 points:
Rohrbough.
7 points: Mittenzwey;
Meyer.
9 points: Dudeney.
10 points: Bell & Cornelius; Hoffmann; Cohen;
Williams; Toymaker; Rohrbough; Putnam.
13 points: Berkeley & Rowland.
Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pentalpha, p.
15. Says that a pentagram board occurs
at Kurna, Egypt, c-1400 and that the solitaire game of Pentalpha is played in
Crete. This has 9 men to be placed on
the vertices and the intersections of the pentagram. Each man must be placed on a vacant point, then slid ahead two
positions along one straight line. The
intermediate point may be occupied, but the ending point must be
unoccupied. Unfortunately we don't know
if the Egyptian board was used for this game.
Pacioli. De Viribus.
c1500.
Ff.
112r - 113v. .C(apitolo). LXVIII. D(e).
cita ch' a .8. porti ch' cosa convi(e)ne arepararli (Chap. 68. Of a city with 8 gates which admits of
division ??). = Peirani 158-160. Octagram puzzle with a complex story about a
city with 8 gates and 7 disputing factions to be placed at the gates.
F.
IVv. = Peirani 8. The Index gives the above as Problem
83. Problem 82: De .8. donne ch' sonno
aun ballo et de .7. giovini quali con loro sa con pagnano (Of 8 ladies who are
at a ball and of 7 youths who accompany them).
Schwenter. 1636.
Part 2, exercise 36, pp. 149-150.
Octagram.
Witgeest. Het Natuurlyk Tover-Boek. 1686.
Prob. 4, pp. 224-225. Octagram,
taken from Schwenter.
Les Amusemens. 1749.
P. xxxiii.
Catel. Kunst-Cabinet. 1790. Das Achteck, pp. 12-13 & fig. 36 on
plate II. The rules are not clearly
described.
Bestelmeier. 1801.
Item 290: Das Achteckspiel. Text
copies part of Catel.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more
details. Ff. 131-133 are an analysis of
the heptagram puzzle.
Rational Recreations. 1824.
Feat 34, pp. 161-164. Octagram.
Endless Amusement II. 1826?
Prob. 28, pp. 203-204. = New
Sphinx, c1840, pp. 137-138.
Nuts to Crack IV (1835), no. 194
-- part of a long section called Tricks upon Travellers.
Family Friend (Dec 1858)
359. Practical puzzles -- 1. I don't have the answer.
The Boy's Own Magazine 3 (1857)
159 & 192. Puzzle of the points.
Illustrated Boy's Own
Treasury. 1860. Practical Puzzles No. 6, pp. 396 & 436.
The Secret Out. 1859.
To Place Seven Counters upon an Eight-Pointed Star, pp. 373-374.
J. J. Cohen, New York. Star puzzle. Advertising card for Star Soap, Schultz & Co., Zanesville,
Ohio, Copyright May 1887. Reproduced
in: Bert Hochberg; As advertised
Puzzles from the collection of Will Shortz; Games Magazine 17:1 (No.
113) 10-13, on p. 11. Identical to
pentalpha - see Bell & Cornelius above.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
Card Puzzles No. IX: The reversi puzzle, pp. 8-10. Version with 13 cards in a circle and one
can move ahead by any number of steps.
If there are x cards and one moves ahead s
steps, then x and
s must have no common factor.
Hoffmann. 1893.
Chap. VI, pp. 267-268 & 280-281 = Hoffmann-Hordern,
pp. 180-181, with photos.
No.
13. No name. Basic octagram puzzle.
Photo on p. 181 shows: The Seven Puzzles, by W. & T. Darton, dated
1806-1811; a Tunbridge ware version dated 1825-1840; and Jeu de Zig‑Zag,
by M. D., Paris, 1891-1900.
No.
14. The "Okto" puzzle, pp.
268 & 281. Here the counters and
points are coloured. Photo on p. 181 of
The "Okto" Puzzle by McGaw, Stevenson & Orr, Ltd. for John
Stewart, dated 1880-1895.
Chap. X, no. 8: Crossette, pp. 337 & 374-375 =
Hoffmann-Hordern, pp. 229-230, with photo.
10 counters in a circle. Start
anywhere and move ahead three. Photo on
p. 230 shows The Mystic Seven, a seven counter version, by the Lord Roberts
Workshops, 1914-1920.
Mittenzwey. 1895?
Prob. 329, pp. 58 & 106;
1917: 329, pp. 52 & 101.
Heptagram.
Dudeney. Problem 58: A wreath puzzle. Tit‑Bits 33 (6 &
27 Nov 1897) 99 & 153.
Complex nonagram puzzle involving moves in either direction and
producing the original word again.
Clark. Mental Nuts. 1897, no.
54; 1904, no. 80; 1916, no. 69. A little puzzle. Usual
octagram.
Benson. 1904.
The eight points puzzle, pp. 250‑251. c= Hoffmann, no. 13.
Slocum. Compendium.
Shows the "Octo" Star Puzzle from Gamage's 1913 catalogue.
Williams. Home Entertainments. 1914.
Crossette,
pp. 115-116. Ten points, advancing
three places.
Eight
points puzzle, pp. 120-121. Usual
octagram.
"Toymaker". Top in Hole Puzzle. Work (23 Dec 1916) 200. 10 holes and one has to move to the third
position and reverse the top in that hole.
Blyth. Match-Stick Magic.
1921. Crossing the points, pp.
83-84.
Hummerston. Fun, Mirth & Mystery. 1924.
The
sacred seven, Puzzle no. 5, pp. 26 & 173.
Octagram puzzle on the outer points of the diagram shown in 5.A.
The
four rabbits, Puzzle no. 6, pp. 26 & 173.
Using the octagram shown in 5.A, put black counters on locations 1 and 2
and white counters on 7 and
8. The object is to interchange the colours. This is like the 4 knights problem except
the corresponding 8-cycle has men at positions
1, 2, 5, 6. He counts a sequence
of steps by the same man as a move and hence solves it in 6 moves (comprising
16 steps).
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926.
Turning the tails, pp. 66-69. 8
coins in a circle, tails up. Count from
a tail four ahead and reverse that coin.
Get 7 heads up. Counting four
ahead means that if you start at 1, you count
1, 2, 3, 4 and reverse 4.
King. Best 100. 1927. No. 64, pp. 26-27 & 54.
Rohrbough. Puzzle Craft. 1932.
Count
4, p. 6. 10 points on a circle, moving
ahead 3. (= Rohrbough; Brain Resters
and Testers; c1935, p. 21.)
Star
Puzzle, p. 8 (= p. 10 of 1940s?).
Consider the pentagram with its internal vertices. First puzzle is Pentalpha. Second is to place a counter and move ahead
three positions. The object is to get four
counters on the points, which is the same as the pentagram puzzle, moving one
position.
Jerome S. Meyer. Fun-to-do.
Op. cit. in 5.C. 1948. Prob. 18:
Odd man out, pp. 27 & 184. Version
with 7 positions in a circle and 6 men where one must place a man and then move
him three places ahead.
Putnam. Puzzle Fun.
1978. No. 63: Ten card turnover,
pp. 11 & 35. Ten face down cards in
a circle. Mark a card, count ahead
three and turnover.
The
usual version is to have 8 counters in a row which must be converted to 4 piles
of two, but each move must pass a counter over two others. Martin Gardner pointed out to me that the
problem for 10, 12, 14, ... counters is easily reduced to that for
8. The problem is impossible for 2, 4, 6.
There are many later appearances of the problem than given here. In describing solutions, 4/1
means move the 4th piece on top of the 1st piece.
There
are trick solutions where a counter moves to a vacated space or even lands
between two spaces. See: Mittenzwey; Haldeman-Julius; Hemme.
Berkeley & Rowland give a problem
where each move must pass a counter over two piles. This makes the problem easier and it is solvable for any even
number of counters ³ 6,
but it gives more solutions.
See: Berkeley & Rowland; Wood; Indoor Tricks & Games; Putnam;
Doubleday - 1.
One
could also permit passing over one pile, which is solvable for any even
number ³ 4.
Mittenzwey,
Double Five Puzzle, Hummerston, and Singmaster & Abbott deal with the
problem in a circle and with piles to be left in specific locations.
Mittenzwey,
Lucas and Putnam consider making piles of three by passing over 3, etc.
Kanchusen. Wakoku Chiekurabe. 1727. Pp. 38-39. Jukkoku-futatsu-koshi (Ten stones jumping
over two). Ten counters, one solution.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more
details. F. 4r is "Analysis of the
Essay of Games". F. 4v has
"The question of the shillings passing at each time over two or a certain
number 8 is the least number Any number being given and any law of transit Dr Roget" The layout suggests that Roget had posed the
general version. Adjacent is a diagram
with a row of 10 counters and the first move 1 to 4 shown, but with some unclear
later moves.
Endless Amusement II. 1826?
Prob. 10, p. 195. 10 halfpence. One solution: 4/1 7/3 5/9 2/6 8/10. = New Sphinx, c1840, pp. 135-135.
Nuts to Crack II (1833), no.
122. 10 counters, identical to Endless
Amusement II.
Nuts to Crack V (1836), no.
68. Trick of the eight sovereigns. Usual form.
Young Man's Book. 1839.
P. 234. Ingenious Problem. 10 halfpence. Identical to Endless Amusement II.
Family Friend 2 (1850) 178 &
209. Practical Puzzle, No. VI. Usual form with eight counters or
coins. One solution.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 2, p. 176 (1868:
187). Passing over coins. Gives two symmetric solutions.
Magician's Own Book. 1857.
Prob. 34: The counter puzzle, pp. 277 & 300. Identical to Book of 500 Puzzles, prob. 34.
The Sociable. 1858.
Prob. 16: Problem of money, pp. 291-292 & 308. Start with 10 half‑dimes, says to pass
over one, but solution has passing over two.
One solution. = Book of 500
Puzzles, 1859, prob. 16, pp. 9-10 & 26.
Book of 500 Puzzles. 1859.
Prob. 16:
Problem of money, pp. 9-10 & 26. As
in The Sociable.
Prob.
34: The counter puzzle, pp. 91 & 114.
Eight counters, two solutions given.
Identical to Magician's Own Book.
The Secret Out. 1859.
The Crowning Puzzle, p. 386.
'Crowning' is here derived from the idea of crowning in draughts or
checkers. One solution: 4/1 6/9 8/3 2/5 7/10.
Boy's Own Conjuring Book. 1860.
Prob.
33: The counter puzzle, pp. 240 & 264.
Identical to Magician's Own Book, prob. 34.
The
puzzling halfpence, p. 342. Almost identical
to The Sociable, prob. 16, with half-dimes replaced by halfpence.
Illustrated Boy's Own
Treasury. 1860. Prob. 17, pp. 398 & 438. Same as prob. 34 in Magician's Own Book but
only gives one solution.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 593, part 6,
pp. 299-411: Sechs Knacknüsse. 10
counters, one solution.
Hanky Panky. 1872.
Counter puzzle, p. 132. Gives
two solutions for 8 counters and one for 10 counters.
Kamp. Op. cit. in 5.B.
1877. No. 12, p. 325.
Mittenzwey. 1880.
Prob. 235-238, pp. 43-44 & 93-94;
1895?: 262-266, pp. 47-48 & 95-96;
1917: 262-266, pp. 43-44 & 91.
235
(262). Usual problem, with 10
counters. Two solutions.
---
(263). Added in 1895? Same with 8 counters. Two solutions.
236
(264). 12 numbered counters in a
circle. Pass over two to leave six
piles of two on the first six positions.
Solution is misprinted in all editions!
237
(265). 12 counters in a circle. Pass over three to leave six piles of two,
except the last move goes over six. The
solution allows landing counters on vacated locations!
238
(266). 15 counters in a row. Pass over 3 to leave five piles of
three. The solution allows landing a
counter on a vacated location and landing a counter between two locations!!
Lucas. RM2. 1883. Les huit pions, pp. 139-140. Solves for
8, 10, 12, ... counters. Says Delannoy has generalized to the problem
of mp
counters to be formed into
m piles (m ³ 4) of p by passing over p counters.
[More
generally, using one of Berkeley & Rowland's variations (see below), one
can ask when the following problem is solvable: form a line of n =
kp counters into k
piles of p by passing over q [counters or
piles]. Does q have to be £ p?]
Double Five Puzzle. c1890.
??NYS -- described by Slocum from his example. 10 counters in a circle,
but the final piles must alternate with gaps, e.g. the final piles are at the
even positions. This is also solvable
for 4, 8, 12, 16, ..., and I conjectured it was only solvable
for 4n
or 10 counters -- it is easy to see there is no solution for 2
or 6 counters and my computer gave no solutions for 14
or 18. For 4, 8, 10 or 12 counters, one can also leave the final piles
in consecutive locations, but there is no such solution for 6, 14, 16 or 18 counters. See Singmaster
& Abbott, 1992/93, for the resolution of these conjectures.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
Card Puzzles.
No.
VII: The halma puzzle, pp. 6-7. Arrange
the first ten cards of a suit in a row so that passing over two cards leaves
five piles whose cards total 11 and are in the odd places. Arrangement is 7,6,3,4,5,2,1,8,9,10.
Move 2 to 9, 4 to 7,
8 to 3, 6 to 5, 10 to 1.
No.
VIII: Another version, p. 7. With the
cards in order and passing over two piles, leave five piles of two. But this is so easy, he adds that one wants
to leave as low a total as possible on the tops of the piles. He moves
7 to 10, 6 to 3, 4 to 9,
1 to 5, 2 to 8, leaving a total of 20.
[However, this is not minimal --
there are six solutions leaving 18 exposed, e.g. 1 to 4, 3 to 6, 7 to 10,
5 to 9, 2 to 8. For 6 cards, the minimum is 6, achieved
once; for 8 cards, the minimum is 11,
achieved 3 times; for 12 cards, the
minimum is 27, achieved 10 times. For
the more usual case of passing over two cards, the minimum for 8 cards is 15,
achieved twice; for 10 cards, the
minimum is 22, achieved 4 times, e.g. by
7 to 10, 5 to 2, 3 to 8,
1 to 4, 6 to 9; for 12 cards, the minimum is 31, achieved 6
times; for 14 cards, the minimum is 42,
achieved 8 times. For passing over one
pile, the minimum for 4 cards is 3, achieved once; the minimum for 6 cards is 7, achieved twice; the minimum for 8 cards is 13, achieved 3 times; for 10 cards, the minimum is 21, achieved 4
times; the minimum for 12 cards is 31,
achieved 5 times. Maxima are obtained
by taking mirror images of the minimal solutions.]
Puzzles with draughtsmen. The Boy's Own Paper 17 or 18?? (1894??)
751. 8 men, passing over two men each
time. Notes that it can be extended to
any even number of counters.
Clark. Mental Nuts. 1897, no.
63: Toothpicks; 1904, no. 83 &
1916, no. 70: Place 8 toothpicks
in a row. One solution.
Parlour Games for
Everybody. John Leng, Dundee &
London, nd [1903 -- BLC], p. 32: The five pairs. 10 counter version, one solution.
Wehman. New Book of 200 Puzzles. 1908.
P. 15: The counter puzzle and Problem of money. 8 and 10 counter versions, the latter using pennies. Two and one solutions.
Ahrens. MUS I.
1910. Pp. 15-17. Essentially repeats Lucas.
Manson. 1911.
Decimal game, pp. 253-254. Ten
rings on pegs. "Children are
frequently seen playing the game out of doors with pebbles or other convenient
articles."
Blyth. Match-Stick Magic. 1921. Straights and
crosses, pp. 85-87. 10 matchsticks, one
must pass over two of them. Two
solutions, both starting with 4 to 1.
Hummerston. Fun, Mirth & Mystery. 1924.
The
pairing puzzle, Puzzle no. 8, pp. 27 & 173. Essential 8 counters in a circle, with four in a row being white,
the other four being black. Moving only
the whites, and passing over two, form four piles of two.
Pairing
the pennies, Puzzle no. 39, pp. 102 & 178.
Ten pennies, one solution.
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926.
Marrying the coins, pp. 113‑115. Ten coins or eight coins, passing over 2. Gives two solutions for 10, not noting that
the case of 10 is immediately reduced to 8.
Says there are several solutions for 8 and gives two.
Wood. Oddities. 1927. Prob. 45: Fish in the basket, pp.
39-40. 12 fish in baskets in a circle. Move a fish over two baskets, continuing
moving in the same direction, to get get two fish in each of six baskets, in
the fewest number of circuits.
Rudin. 1936. No. 121, pp. 43
& 103. 10 matches. Two solutions.
J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Straights and crosses, pp. 105-106. As in Blyth, 1926.
Indoor Tricks and Games. Success Publishing, London, nd
[1930s??].
How to
pair the pennies, p. 4. 8 pennies, one
solution.
The
ten rings. p. 4. 10 rings, passing over
two piles, one solution.
Haldeman-Julius. 1937.
No.
91: Jumping pennies, pp. 11 & 25.
Six pennies to be formed into two piles of three by jumping over three
pennies each time. Solution has a trick
move. Jump 1 to 5, 6 to 3 and 2 to 1/5,
which gives the position: 6/3 4
2/1/5. He then says: "No. 4
jumps over 5, 1 and 2 -- then jumps back over 5, 1 and 2 and lands upon 3 and
6, ...." Since the rules are not
clear about where a jumping piece can land, the trick move can be viewed as a
legitimate jump to the vacant 6 position, then a legitimate move over the 2/1/5
pile and the now vacant 4 position onto the 6/3 pile. If the pennies are considered as a cycle, this trick is not
needed.
No.
148: Half dimes, pp. 16 & 143. 10
half dimes, passing over one dime (i.e. two counters).
Sullivan. Unusual.
1947. Prob. 39: On the
line. Ten pennies.
Doubleday - 1. 1969.
Prob. 75: Money moves, pp. 91 & 170. Ten pennies. Jump over
two piles. Says there are several
solutions and gives one, which sometimes jumps over three or four pennies.
Putnam. Puzzle Fun.
1978.
No.
26: Pile up the coins, pp. 7 & 31.
12 in a row. Make four piles of
three, passing over three coins each time.
No.
27: Pile 'em up again, pp. 7 & 31.
16 in a row. Make four piles of
four, passing over four or fewer each time.
No.
60: Coin assembly, pp. 11 & 35. Ten
in a row, passing over two each time.
No.
61: Alternative coin assembly, pp. 11 & 35. Ten in a row, passing over two piles each time.
David Singmaster, proposer; H. L. Abbott, solver. Problem 1767. CM 18:7 (1992) 207 & 19:6 (1993). Solves the general version of the Double Five Puzzle, which the
proposer had not solved. One can leave
the counters on even numbered locations if and only if the number of counters
is a multiple of 4 or a multiple of 10.
One can leave the counters in consecutive locations if and only if the
number of counters is 4, 8, 10 or 12.
Heinrich Hemme. Email of 25 Feb 1999. Points out that the rules in the usual
version should say that the counter must land on a pile of a single coin. This would eliminate the trick solutions
given by Mittenzwey and Haldeman-Julius.
Hemme says that without this rule, the problem is easy and can be solved
for 4 and 6 counters!
5.S. CHAIN CUTTING AND REJOINING
The basic problem is to minimise the
cost or effort of reforming a chain from some fragments.
Loyd. Problem 25: A brace of puzzles -- No. 25: The chain puzzle. Tit‑Bits 31 (27 Mar 1897) &
32 (17 Apr 1897) 41. 13
lengths: 5, 6, 7, 7, 7, 7, 8, 8, 8, 8,
8, 9, 12. (Not in the Cyclopedia.)
Loyd. Problem 42: The blacksmith puzzle. Tit‑Bits 32 (10
& 31 Jul &
21 Aug 1897) 273, 327 &
385. Complex problem involving
10 pieces of lengths from 3 to 23 to be joined.
Clark. Mental Nuts. 1897, no.
7 & 1904, no. 14: The chain question; 1916, no. 59: The chain puzzle.
5 pieces of 3 links to make into a single length.
Mr. X [cf 4.A.1]. His Pages.
The Royal Magazine 9:4 (Feb 1903) 390-391 & 9:5 (Mar 1903) 490-491.
The five chains. 5 pieces of 3
links to make into a single length.
Pearson. 1907.
Part II, no. 67, pp. 128 & 205.
5 pieces of 3 links to make into a single length.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. He attributes such
puzzles to Loyd (Tit‑Bits prob. 25) and gives that problem.
Cecil H. Bullivant. Home Fun.
T. C. & E. C. Jack, London, 1910.
Part VI, Chap IV, No. 9: The broken chain, pp. 518 & 522. 5 3‑link pieces into an open chain.
Loyd. The missing link.
Cyclopedia, 1914, pp. 222 (no solution) (c= MPSL2, prob. 25,
pp. 19 & 129). 6 5‑link
pieces into a loop.
Loyd. The necklace puzzle.
Cyclopedia, 1914, pp. 48 & 345 (= MPSL1, prob. 47, pp. 45‑46
& 138). 12 pieces, with large and
small links which must alternate.
D. E. Smith. Number Stories. 1919. Pp. 119 & 143‑144. 5 pieces of 3 links to make into one length.
Hummerston. Fun, Mirth & Mystery. 1924.
Q.E.D. -- The broken chain, Puzzle no. 38, pp. 99 & 178. Pieces of lengths 2, 2, 3, 3, 4, 4, 6 to
make into a closed loop.
Ackermann. 1925.
Pp. 85‑86. Identical to
the Loyd example cited by Dudeney.
Dudeney. MP.
1926. Prob. 212: A chain puzzle,
pp. 96 & 181 (= 536, prob. 513, pp. 211‑212 & 408). 13 pieces, with large and small links which
must alternate.
King. Best 100. 1927. No. 7, pp. 9 & 40. 5 pieces of three links to make into one
length.
William P. Keasby. The Big Trick and Puzzle Book. Whitman Publishing, Racine, Wisconsin,
1929. A linking problem, pp. 161 &
207. 6 pieces comprising 2, 4, 4, 5, 5, 6 links to be made into one length.
5.S.1. USING CHAIN LINKS TO PAY FOR A ROOM
The
landlord agrees to accept one link per day and the owner wants to minimise the
number of links he has to cut. The
solution depends on whether the chain is closed in a cycle or open at the
ends. Some weighing problems in 7.L.2.c
and 7.L.3 are phrased in terms of making daily payment, but these are like having
the chain already in pieces. See the
Fibonacci in 7.L.2.c.
New
section. I recall that there are older
versions.
Rupert T. Gould. The Stargazer Talks. Geoffrey Bles, London, 1944. A Few Puzzles -- write up of a BBC talk on
10 Jan 1939, pp. 106-113. 63 link chain with three cuts. On p. 106, he says he believes it is
quite modern -- he first heard it in 1935.
On p. 113, he adds a postscript that he now believes it first appeared
in John O'London's Weekly (16 Mar 1935) ??NYS.
Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 3.
Man has closed chain of 182 links and wants to stay 182 days. What is the minimum number of links to be
opened?
Birtwistle. Math. Puzzles & Perplexities. 1971.
Pp. 13-16. Begins with seven
link open-ended bracelet. Then how big
a bracelet can be dealt with using only two cuts? Gets 23. Then does general case, getting n + (n+1)(2n+1 - 1).
Angela Fox Dunn. Second Book of Mathematical Bafflers. Dover, 1983. Selected from Litton's Problematical Recreations, which appeared
in 1959‑1971. Prob. 26, pp. 28
& 176. 23 link case.
Howson. Op. cit. in 5.R.4. 1988. Prob. 30. Says a
23 link chain need only be cut
twice, giving lengths 1, 1, 3, 6,
12, which make all values up to 23.
Asks for three cuts in a 63 link chain and the maximum length chain one
can deal with in n cuts.
Mittenzwey. 1880.
Prob. 200, pp. 37 & 89;
1895?: 225, pp. 41 & 91;
1917: 225, pp. 38 & 88.
Family of 4 adults and 4 children.
With three cuts, divide a cake so the adults and the children get equal
pieces. He makes two perpendicular
diametrical cuts and then a circular cut around the middle. He seems to mean the adults get equal pieces
and the children get equal pieces, not necessarily the same. But if the circular cut is at Ö2/2 of the radius, then the areas are all equal. Not clear where this should go -- also
entered in 5.Q.
B. Knaster. Sur le problème du partage pragmatique de H.
Steinhaus. Annales de la Société Polonaise
de Mathématique 19 (1946) 228‑230.
Says Steinhaus proposed the problem in a 1944 letter to Knaster. Outlines the Banach & Knaster method of
one cutting 1/n and each being allowed to diminish it --
last diminisher takes the piece. Also
shows that if the valuations are different, then everyone can get > 1/n
in his measure. Gives Banach's
abstract formulations.
H. Steinhaus. Remarques sur le partage pragmatique. Ibid., 230‑231. Says the problem isn't solved for irrational
people and that Banach & Knaster's method can form a game.
H. Steinhaus. The problem of fair division. Econometrica 16:1 (Jan 1948) 101‑104. This is a report of a paper given on 17
Sep. Gives Banach & Knaster's
method.
H. Steinhaus. Sur la division pragmatique. (With English summary) Econometrica 17
(Supplement) (1949) 315‑319.
Gives Banach & Knaster's method.
Max Black. Critical Thinking. Prentice‑Hall, Englewood Cliffs, (1946, ??NYS), 2nd ed.,
1952. Prob. 12, pp. 12 & 432. Raises the question but only suggests combining
two persons.
van Etten. 1624.
Prob. 89, part II, pp. 131‑132 (not in English editions). Two men have same number of hairs. Also:
birds & feathers, fish &
scales, trees & leaves, flowers or
fruit, pages & words -- if there
are more pages than words on any page.
E. Fourrey. Op. cit. in 4.A.1, 1899, section 213: Le
nombre de cheveux, p. 165. Two
Frenchmen have the same number of hairs.
"Cette question fut posée et expliquée par Nicole, un des auteurs
de la Logique de Port‑Royal, à la duchesse de Longueville." [This would be c1660.]
The same story is given in a
review by T. A. A. Broadbent in MG 25 (No. 264) (May 1941) 128. He refers to MG 11 (Dec 1922) 193,
??NYS. This might be the item
reproduced as MG 32 (No. 300) (Jul 1948) 159.
The question whether two trees
in a large forest have the same number of leaves is said to have been posed to
Emmanuel Kant (1724-1804) when he was a boy.
[W. Lietzmann; Riesen und Zwerge im Zahlbereich; 4th ed., Teubner,
Leipzig, 1951, pp. 23-24.] Lietzmann
says that an oak has about two million leaves and a pine has about ten million
needles.
Jackson. Rational Amusement. 1821.
Arithmetical Puzzles, no. 9, pp. 2-3 & 53. Two people in the world have the same number of hairs on their
head.
Manuel des Sorciers. 1825.
Pp. 84-85. ??NX Two men have the same number of hairs, etc.
Gustave Peter Lejeune
Dirichlet. Recherches sur les formes
quadratiques à coefficients et à indéterminées complexes. (J. reine u. angew. Math. (24 (1842) 291‑371) = Math. Werke, (1889‑1897), reprinted
by Chelsea, 1969, vol. I, pp. 533‑618.
On pp. 579‑580, he uses the principle to find good rational
approximations. He doesn't give it a
name. In later works he called it the
"Schubfach Prinzip".
Illustrated Boy's Own
Treasury. 1860. Arithmetical and Geometrical Problems, No.
34, pp. 430 & 434. Hairs on
head.
Pearson. 1907.
Part II, no. 51, pp. 123 & 201.
"If the population of Bristol exceeds by two hundred and thirty‑seven
the number of hairs on the head of any one of its inhabitants, how many of them
at least, if none are bald, must have the same number of hairs on their
heads?" Solution says 474!
Dudeney. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909) 670‑676 (= AM,
pp. 137‑141). Two people
have same number of hairs.
Ahrens. A&N, 1918, p. 94. Two Berliners have same number of hairs.
Abraham. 1933.
Prob. 43 -- The library, pp. 16 & 25 (12 & 113). All books have different numbers of words
and there are more books than words in the largest book. (My copy of the 1933 ed. is a presentation
copy inscribed 'For the Athenaeum Library No 43 p 16 R M Abraham Sept 19th 1933'.)
Perelman. FMP.
c1935? Socks and gloves. Pp. 277 & 283‑284. = FFF, 1957: prob. 25, pp. 41 &
43; 1977, prob. 27, pp. 53‑54
& 56. = MCBF, prob. 27, pp. 51
& 54. Picking socks and gloves to
get pairs from 10 pairs of brown and 10 pairs of black socks and gloves.
P. Erdös & G. Szekeres. Op. cit. in 5.M. 1935. Any permutation of
the first n2 ‑ 1 integers contains an increasing or a
decreasing subsequence of length >
n.
P. Erdös, proposer; M. Wachsberger & E. Weiszfeld, M.
Charosh, solvers. Problem 3739. AMM 42 (1935) 396 & 44 (1937) 120. n+1
integers from first 2n have one dividing another.
H. Phillips. Question Time. Dent, London, 1937. Prob.
13: Marbles, pp. 7 & 179. 12
black, 8 red &
6 white marbles -- choose enough
to get three of the same colour.
The Home Book of Quizzes, Games
and Jokes. Op. cit. in 4.B.1,
1941. Pp. 148‑149, prob. 6. Blind maid bringing stockings from a drawer
of white and black stockings.
I am surprised that the context
of picking items does not occur before Perelman, Phillips and Home Book.
Sullivan. Unusual.
1943. Prob. 18: In a dark
room. Picking shoes and socks to get
pairs.
H. Phillips. News Chronicle "Quiz" No. 3:
Natural History. News Chronicle,
London, 1946. Pp. 22 & 43. 12
blue, 9 red and
6 green marbles in a bag. Choose enough to have three of one colour
and two of another colour.
H. Phillips. News Chronicle "Quiz" No. 4:
Current Affairs. News Chronicle,
London, 1946. Pp. 17 & 40. 6
yellow, 5 blue and
2 red marbles in a bag. Choose enough to have three of the same
colour.
L. Moser, proposer; D. J. Newman, solver. Problem 4300 -- The identity as a product of
successive elements. AMM 55 (1948)
369 & 57 (1950) 47. n elements from a group of order n
have a a subinterval with product
= 1.
Doubleday - 2. 1971.
In the
dark, pp. 145-146. How many socks do you
have to pick from a drawer of white and black socks to get two pairs (possibly
different)?
Lucky
dip, pp. 147-148. How many socks do you
have to pick from a drawer of with many white and black socks to get nine pairs
(possibly different)? Gives the general
answer 2n+1 for n pairs.
[Many means that the drawer contains more than n pairs.]
Doubleday - 3. 1972.
In the dark, pp. 35-36. Four
sweaters and 5, 12, 4, 9 socks of the same colours as the
sweaters. Lights go out. He can only find two of the sweaters. How many socks must he bring down into the
light to be sure of having a pair matching one of the sweaters?
I
managed to acquire one of these without instructions or packaging some years
ago. Michael Keller provided an example
complete with instructions and packaging.
I have recently seen Dockhorn's article on variations of the idea. This is related to Binary Recreations, 7.M.
The
device was produced by E.S.R., Inc. The
box or instructions give an address of 34 Label St., Montclair, New
Jersey, 07042, USA, but the company has long been closed. In Feb 2000, Jim McArdle wrote that he
believed that this became the well known Edmund Scientific Co. (101 East
Gloucester Pike, Barrington, New Jersey, 08007, USA; tel: 609‑547 3488;
email: scientifics@edsci.com; web: http://www.edmundscientific.com). But he later wrote that investigation of the
manuals of DifiComp, one of their other products, reveals that there appears to
be no connection. E.S.R. = Education
Science Research. The inventors of
DigiComp, as listed in the patent for it, are: Irving J. Lieberman, William H,
Duerig and Charles D. Hogan, all of Montclair, and they were the founders of
the company. The DigiComp manuals say
Think‑A‑Dot was later invented by John Weisbacker. There is a website devoted to DigiComp which
contains this material and/or pointers to related sites and has a DigiComp
emulator: http://members.aol.com/digicomp1/DigiComp.html . www.yahoo.com has a Yahoo club called Friends of DigiComp. There is another website with the DigiComp
manual: http://galena.tj.edu.inter.net/digicomp/ .
E.S.R. Instructions, 8pp, nd -- but box says ©1965. No patent number anywhere but leaflet says
the name Think-A-Dot is trademarked.
E.S.R., Inc. Corporation. US trademark registration no. 822,770. Filed: 8 Dec 1965; registered: 24 Jan 1967.
First used 23 Aug 1965.
Expired. The US Patent and
Trademark Office website entry says the owner is the company and gives no
information about the inventor(s). The
name has been registered for a computer game on 23 Jul 2002.
Benjamin L. Schwartz. Mathematical theory of Think‑A‑Dot. MM 40:4 (Sep 1967) 187‑193. Shows there are two classes of patterns and
that one can transform any pattern into any other pattern in the same class in
at most 15 drops.
Ray Hemmings. Apparatus Review: Think‑a‑Dot.
MTg 40 (1967) 45.
Sidney Kravitz. Additional mathematical theory of Think‑A‑Dot. JRM 1:4 (Oct 1968) 247‑250. Considers problems of making ball emerge
from one side and of viewing only the back of the game.
Owen Storer. A think about Think‑a‑dot. MTg 45 (Winter 1968) 50‑55. Gives an exercise to show that any possible
transformation can be achieved in at most 9 drops.
T. H. O'Beirne. Letter:
Think‑a‑dot. MTg 48
(Autumn 1969) 13. Proves Steiner's
(Storer?? - check) assertion about 9 drops and gives an optimal algorithm.
John A. Beidler. Think-A-Dot revisited. MM 46:3 (May 1973) 128-136. Answers a question of Schwarz by use of
automata theory. Characterizes all minimal
sequences. Suggests some generalized
versions of the puzzle.
Hans Dockhorn. Bob's binary boxes. CFF 32 (Aug 1993) 4-6. Bob Kootstra makes boxes with the same sort
of T-shaped switch present in Think-A-Dot, but with just one entrance. One switch with two exits is the simplest
case. Kootstra makes a box with three
switches and four exits along the bottom, and the successive balls come out of
the exits in cyclic sequence. Using a
reset connection between switches, he also makes a two switch, three exit, box.
Boob Kootstra. Box seven.
CFF 32 (Aug 1993) 7. Says he has
managed to design and make boxes with
5, 6, 7, 8 exits, again with
successive balls coming out the exits in cyclic order, but he cannot see any
general method nor a way to obtain solutions with a minimal number of movable
parts (switches and reset levers).
Further his design for 7 exits is awkward and the design of an optimal
box for seven is posed as a contest problem.
5.W. MAKING THREE PIECES OF TOAST
This involves an old‑fashioned
toaster which does one side of two pieces at a time. An alternative version is frying steaks or hamburgers on a grill
which holds two objects, assuming each side has to be cooked the same length of
time. The problem is probably older
than these examples.
Sullivan. Unusual.
1943. Prob. 7: For the busy
housewife.
J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. P. 4 (26). Mentions problem and solution.
Simon Dresner. Science World Book of Brain Teasers. 1962.
Op. cit. in 5.B.1. Prob. 40:
Minute toast, pp. 18 & 93.
D. St. P. Barnard. 50 Daily Telegraph Brain‑Twisters. 1985.
Op. cit. in 4.A.4. Prob. 5: Well
done, pp. 16, 80, 103‑104.
Grilling three steaks on a grill which only holds two. He complicates the problem in two ways: a)
each side takes a minute to season before cooking; b)
the steaks want to be cooked 4,
3, 2 minutes per side.
Edward Sitarski. When do we eat? CM 27:2 (Mar 2001) 133-135.
Hamburgers which require time T per side.
After showing that three hamburgers take 3T, he asks how long it
will take to cook H hamburgers.
Easily shows that it can be done in
HT, except for H = 1,
which takes 2T. Then remarks that this is an easy version of
a scheduling problem -- in reality, the hamburgers would have different numbers
of sides, there would be several grills and each hamburger would have different
parts requiring different grills, but in a particular order!
New
section. These are essentially parodies
of the Cistern Problem, 7.H.
McKay. Party Night. 1940. No. 28, p. 182. "An egg takes
3½ minutes to boil. How long should 12 eggs take?"
Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 88: A
boiling problem, pp. 29 & 82.
"If it takes 3½ minutes to boil 2 eggs, how long will it
take to boil 4 eggs?"
John King, ed. John King
1795 Arithmetical Book. Published by the editor, who is the
great-great-grandson of the 1795 writer, Twickenham, 1995. P. 161, the editor mentions "If a girl
on a hilltop can see two miles, how far would two girls be able to see?"
5.X. COUNTING FIGURES IN A PATTERN
New
section -- there must be older examples.
There are two forms of such problems depending on whether one must use
the lattice lines or just the lattice points.
For
counting several shapes, see: Young
World (c1960); Gooding (1994) in 5.X.1.
Counting
triangles in a pattern is always fraught with difficulties, so I have written a
program to do this, but I haven't checked all the examples here.
Pearson. 1907.
Part II.
No.
74: A triangle of triangles, p. 74.
Triangular array with four on a side, but with all the altitudes also
drawn. Gets 653 triangles of various
shapes.
No. 75:
Pharaoh's seal, pp. 75 & 174.
Isosceles right triangles in a square pattern with some diagonals.
Anon. Prob. 76. Hobbies 31 (No. 791) (10 Dec 1910) 256 &
(No. 794) (31 Dec 1910) 318.
Make as many triangles as possible with six matches. From the solution, it seems that the
tetrahedron was expected with four triangles, but many submitted the figure of
a triangle with its altitudes drawn, but only one solver noted that this figure
contains 16 triangles! However, if the
altitudes are displaced to give an interior triangle, I find 17 triangles!!
Loyd. Cyclopedia. 1914. King Solomon's seal, pp. 284 & 378. = MPSL2, No. 142, pp. 100 & 165 c= SLAHP: Various triangles, pp. 25 &
91. How many triangles in the
triangular pattern with 4 on a side?
Loyd Sr. has this embedded in a larger triangle.
Collins. Book of Puzzles. 1927. The swarm of
triangles, pp. 97-98. Same as Pearson
No. 74. He says there are 653
triangles and that starting with 5 on a side gives 1196
and 10,000 on a side gives 6,992,965,420,382. When I
gave August's problem in the Weekend Telegraph, F. R. Gill wrote that this
puzzle with 5 on a side was given out as a competition problem by a furniture
shop in north Lancashire in the late 1930s, with a three piece suite as a prize
for the first correct solution.
Evelyn August. The Black-Out Book. Harrap, London, 1939. The eternal triangle, pp. 64 & 213. Take a triangle, ABC, with midpoints a, b, c,
opposite A, B, C. Take a point d between a
and B. Draw Aa, ab, bc, ca, bd,
cd. How many triangles? Answer is given as 24, but I (and my
program) find 27 and others have confirmed this.
Anon. Test your eyes.
Mathematical Pie 7 (Oct 1952) 51.
Reproduced in: Bernard Atkin,
ed.; Slices of Mathematical Pie; Math. Assoc., Leicester, 1991, pp. 15 & 71
(not paginated - I count the TP as p. 1).
Triangular pattern with 2 triangles on a side, with the three
altitudes drawn. Answer is 47 'obtained
by systematic counting'. This is
correct. Cf Hancox, 1978.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. How many triangles, pp. 43 & 130. Take a pentagon and draw the pentagram
inside it. In the interior pentagon,
draw another pentagram. How many
triangles are there? Answer is 85.
Young World. c1960.
P. 57: One for Pythagoras.
Consider a L-tromino. Draw all
the midlines to form 12 unit squares. Or take a 4 x 4 square array and remove a 2 x 2
array from a corner. Now draw
the two main diagonals of the 4 x
4 square - except half of one diagonal
would be outside our figure. How many
triangles and how many squares are present?
Gives correct answers of 26 &
17.
J. Halsall. An interesting series. MG 46 (No. 355) (Feb 1962) 55‑56. Larsen (below) says he seems to be the first
to count the triangles in the triangular pattern with n on a side, but he does
not give any proof.
Although there are few
references before this point, the puzzle idea was pretty well known and occurs
regularly. E.g. in the children's
puzzle books of Norman Pulsford which start c1965, he gives various irregular
patterns and asks for the number of triangles or squares.
J. E. Brider. A mathematical adventure. MTg (1966) 17‑21. Correct derivation for the number of
triangles in a triangle. This seems to
be the first paper after Halsall but is not in Larsen.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/12, pp. 23 & 75. Consider an isosceles right triangle with
legs along the axes from (0,0) to
(4,0) and (0,4).
Draw the horizontals and verticals through the integer lattice points,
except that the lines through
(1,1) only go from the legs to
this point and stop. Draw the diagonals
through even-integral lattice points, e.g. from (2,0) to (0,2).
How many triangles. Says he
found 27, but his secretary then found
29. I find 29.
Ripley's Puzzles and Games. 1966.
Pp. 72-73 have several problems of counting triangles.
Item
3. Consider a Star of David with the
diameters of its inner hexagon drawn. How
many triangles are in it? Answer: 20,
which I agree with.
Item
4. Consider a 3 x 3 array of squares
with their diagonals drawn. How many
triangles are there? Answer: 150,
however, there are only 124.
Item
5. Consider five squares, with their
midlines and diagonals drawn, formed into a Greek cross. How many triangles are there? Answer: 104, but there are 120.
Doubleday - 2. 1971.
Count down, pp. 127-128. How
many triangles in the pentagram (i.e. a pentagon with all its diagonals)? He says
35.
Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Triangular, pp. 27 & 63. Count triangles in an irregular pattern.
[Henry] Joseph & Lenore
Scott. Master Mind Brain Teasers. 1973.
Op. cit. in 5.E. An unusual
star, pp. 49-50. Consider a pentagram
and draw lines from each star point through the centre to the opposite crossing
point. How many triangles? They say 110.
[Henry] Joseph and Lenore
Scott. Master Mind Pencil Puzzles. 1973.
Op. cit. in 5.R.4.
Diamonds
are forever, pp. 35-36. Hexagon with
Star of David inside and another Star of David in the centre of that one. How many triangles? Answer is 76.
Count
the triangles, pp. 55-56. Ordinary
Greek cross of five squares, with all the diagonals and midlines of the five
squares drawn. How many
triangles> Answer is 104.
C. P. Chalmers. Note 3353:
More triangles. MG 58 (No. 403)
(Mar 1974) 52‑54. How many
triangles are determined by N points lying on M lines? (Not in Larsen.)
Nicola Davies. The 2nd Target Book of Fun and Games. Target (Universal-Tandem), London,
1974. Squares and triangles, pp. 18
& 119. Consider a chessboard of 4 x 4
cells. Draw all the diagonals, except
the two main ones. How many squares and
how many triangles?
Shakuntala Devi. Puzzles to Puzzle You. Op. cit. in 5.D.1. 1976. Prob. 136: The
triangles, pp. 85 & 133. How many
triangles in a Star of David made of 12 equilateral triangles?
Michael Holt. Figure It Out -- Book Two. Granada, London, 1978. Prob. 67, unpaginated. How many triangles in a Star of David made
of 12 equilateral triangles?
Putnam. Puzzle Fun.
1978. No. 91: Counting
triangles, pp. 12 & 37. Same as
Doubleday - 2.
D. J. Hancox, D. J. Number Puzzles For all The Family.
Stanley Thornes, London, 1978.
Puzzle
8, pp. 2 & 47. Draw a line with
five points on it, say A, B, C, D,
E, making four segments. Connect all these points to a point F on
one side of the line and to a point
G on the other side of the line,
with FCG collinear. How many
triangles are there? Answer is 24,
which is correct.
Puzzle
53, pp. 24 & 54. Same as Anon.;
Test Your Eyes, 1952. Answer is 36, but
there are 47.
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex,
1984.
Problem
40, with Solution at the back of the book.
Same as Doubleday - 2.
Problem
116, with Solution at the back of the book.
Count the triangles in a 'butterfly' pattern.
Sue Macy. Mad Math.
The Best of DynaMath Puzzles.
Scholastic, 1987. (Taken from
Scholastic's DynaMath magazine.)
Shape Up, pp. 5 & 56.
Take
a triangle, trisect one edge and join the points of trisection to the opposite
vertex. How many triangles? [More generally, if one has n
points on a line and joins them all to a vertex, there are 1 + 2 + ... + n-1 = n(n-1)/2
triangles.]
Take
a triangle, join up the midpoints of the edges, giving four smaller triangles,
and draw one altitude of the original triangle. How many triangles?
1980 Celebration of Chinese New
Year Contest Problem No. 5; solution by
Leroy F. Meyers. CM 17 (1991) 2 &
18 (1992) 272-273. n x n array of squares with all diagonals
drawn. Find the number of isosceles
right triangles. [Has this also been
done in half the diagram? That is, how
many isosceles right triangles are in the isosceles right triangle with legs
going from (0,0) to
(n,0) and (0,n)
with all verticals, horizontals and diagonals through integral points
drawn?]
Mogens Esrom Larsen. The eternal triangle -- a history of a
counting problem. Preprint, 1988. Surveys the history from Halsall on. The problem was proposed at least five times
from 1962 and solved at least ten times.
I have sent him the earlier references.
Marjorie Newman. The Christmas Puzzle Book. Hippo (Scholastic Publications), London,
1990. Star time, pp. 26 & 117. Consider a Star of David formed from 12
triangles, but each of the six inner triangles is subdivided into 4
triangles. How many triangles in
this pattern? Answer is 'at least 50'.
I find 58.
Erick Gooding. Polygon counting. Mathematical Pie No. 131 (Spring 1994) 1038 &
Notes, pp. 1-2. Consider the
pentagram, i.e. the pentagon with its diagonals drawn. How many triangles, quadrilaterals and
pentagons are there? Gets 35, 25, 92,
with some uncertainty whether the last number is correct.
When F. R. Gill (See Pearson and
Collins above) mentioned the problem of counting the triangles in the figure
with all the altitudes drawn, I decided to try to count them myself for the figure
with N
intervals on each side. The
theoretical counting soon gets really messy and I adapted my program for
counting triangles in a figure (developed to verify the number found for
August's problem). However, the number
of points involved soon got larger than my simple Basic could handle and I
rewrote the program for this special case, getting the answers of 653 and 1196
and continuing to N = 22. I expected
the answers to be like those for the simpler triangle counting problem so that
there would be separate polynomials for the odd and even cases, or perhaps for
different cases (mod 3 or 4 or 6 or 12 or ??).
However, no such pattern appeared for moduli 2, 3, 4 and I did not get
enough data to check modulus 6 or higher.
I communicated this to Torsten Sillke and Mogens Esrom Larsen. Sillke has replied with a detailed answer
showing that the relevant modulus is 60!
I haven't checked through his work yet to see if this is an empirical
result or he has done the theoretical counting.
Heather Dickson, Heather,
ed. Mind-Bending Challenging Optical
Puzzles. Lagoon Books, London, 1999,
pp. 40 & 91. Gives the version m = n = 4
of the following. I have seen
other versions of this elsewhere, but I found the general solution on 4 Jul
2001 and am submitting it as a problem to AMM.
Consider
a triangle ABC. Subdivide the side AB into m
parts by inserting m‑1 additional points. Connect these points to
C. Subdivide the side AC
into n parts by inserting n-1
additional points and connect them to
B. How many triangles are in
this pattern? The number is [m2n + mn2]/2. When
m = n, we get n3, but I cannot see any simple geometric interpretation for this.
5.X.2. COUNTING RECTANGLES OR SQUARES
I
have just seen M. Adams. There are probably
earlier examples of these types of problems.
Anon. Prob. 63. Hobbies 30 (No. 778) (10 Sep 1910) 488 &
31 (No. 781) (1 Oct 1910) 2. How
many rectangles on a 4 x 4 chessboard?
Solution says 100, which is correct, but then says they are of 17
different types -- I can only get
16 types.
Blyth. Match-Stick Magic.
1921. Counting the squares, p.
47. Count the squares on a 4 x 4
chessboard made of matches with an extra unit square around the central
point. The extra unit square gives 5
additional squares beyond the usual 1 +
4 + 9 + 16.
King. Best 100. 1927. No. 9, pp. 10 & 40. = Foulsham's, no. 5, pp. 6 & 10. 4 x 4
board with some diagonals yielding one extra square.
Loyd Jr. SLAHP.
1928. How many rectangles?, pp.
80 & 117. Asks for the number of
squares and rectangles on a 4 x 4 board (i.e. a 5 x 5 lattice of
points). Says answers are 1 + 4 + 9 + 16 and
(1 + 2 + 3 + 4)2 and that these generalise to any size of
board.
M. Adams. Puzzles That Everyone Can Do. 1931. o o
Prob.
37, pp. 22 & 134: 20 counter problem.
Given the pattern of o o
20 counters at the tight, 'how many perfect
squares are o
o o o o o
contained in the figure.' This means having their vertices o o o o o o
at counters.
There are surprisingly more than I expected. o o
Taking the basic spacing as one, one can have
squares of o o
edge
1, Ö2, Ö5, Ö8, Ö13,
giving 21 squares in all.
He then asks how many counters need
to be removed in order to destroy all the squares? He gives a solution deleting six counters.
Prob.
217, pp. 83 & 162: Match squares.
He gives 10 matches making a row of three equal squares and asks you to
add 14 matches to form 14 squares. The
answer is to make a 3 x 3 array of squares and count all of the
squares in it.
J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Counting the squares, pp. 84-85. As in Blyth.
Indoor Tricks and Games. Success Publishing, London, nd
[1930s??]. Square puzzle, p. 62. Start with a square and draw its diagonals
and midlines. Join the midpoints of the
sides to form a second level square inscribed in the first level original
square. Repeat this until the 9th
level. How many squares are there? Given answer is 16, but in my copy
someone has crossed this out and written
45, which seems correct to me.
Meyer. Big Fun Book. 1940. No. 9, pp. 162 & 752. Draw four equidistant horizontal lines and
then four equidistant verticals. How
many squares are formed? This gives
a 3 x 3 array of squares, but he counts all sizes of squares,
getting 9 + 4 + 1 = 14. (Also in 7.AU.)
Foulsham's New Party Book. Foulsham, London, nd [1950s?]. P. 103: How many squares? 4 x 4
board with some extra diagonals giving one extra square.
Although there are few
references before this point, the puzzle idea was pretty well known and occurs
regularly in the children's puzzle books of Norman Pulsford which start
c1965. He gives various irregular
patterns and asks for the number of triangles or squares.
Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 140: A
surprising answer, pp. 43 & 90. 4 x
4 chessboard with four corner cells
deleted. How many rectangles are there?
Anon. Puzzle page: Strictly for squares. MTg 30 (1965) 48
& 31 (1965) 39 &
32 (1965) 39. How many squares
on a chessboard? First solution gets
S(8) = 1 + 4 + 9 + ... + 64 =
204. Second solution observes that
there are skew squares if one thinks of the board as a lattice of points and
this gives
S(1) + S(2) + ... + S(8) = 540 squares.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966.
Prob.
2/11, pp. 22 & 74. 4 x 4 array of squares bordered on two sides by
bricks 1 x 2, 1 x 3, 2 x 1, 2 x 1. Count the squares and the rectangles. Gets 35 and 90.
Prob.
2/14, pp. 23 & 75. Pattern of
squares making the shape of a person -- how many squares in it?
Ripley's Puzzles and Games. 1966.
Pp. 72-73 have several problems of counting squares.
Item
4. Consider a 3 x 3 array of squares
with their diagonals drawn. The
solution says this has 30 squares. I
get 31, but perhaps they weren't counting the whole figure. I have computed the total number of squares for
an n x n array and get (2n3
+ n2)/2 squares for n
even and (2n3 + n2
-1)/2 squares for n
odd.
Unnumbered
item at lower right of p. 73. 4 x
4 array of squares with their diagonals
drawn, except that the four corner squares have only one diagonal -- the one
not pointing to an opposite corner -- and this reduces the number of squares by
eight, agreeing with the given answer of
64.
Doubleday - 2. 1971.
Bed of nails, pp. 129-130. 20
points in the form of a Greek cross with double-length arms (so that the axes
are five times the width of the central square, or the shape is a
9-omino). How many squares can be
located on these points? He finds 21.
W. Antony Broomhead. Note 3315:
Two unsolved problems. MG 55
(No. 394) (Dec 1971) 438. Find the
number of squares on an n x n array of dots, i.e. the second problem in
MTg (1965) above, and another problem.
W. Antony Broomhead. Note 3328:
Squares in a square lattice. MG
56 (No. 396) (May 1972) 129. Finds
there are n2(n2 ‑
1)/12 squares and gives a proof due to
John Dawes. Editorial note says the
problem appears in: M. T. L. Bizley;
Probability: An Intermediate Textbook; CUP, 1957, ??NYS. A. J. Finch asks the question for cubes.
Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Squares, pp. 26 & 63. Same as Briggs.
Nicola Davies. The 2nd Target Book of Fun and Games. 1974.
See entry in 5.X.1.
Putnam. Puzzle Fun.
1978.
No.
107: Square the coins, pp. 17 & 40.
20 points in the form of a Greek cross made from five 2 x 2
arrays of points. How many
squares -- including skew ones? Gets
21.
No.
108: Unsquaring the coins, pp. 17 & 40.
How many points must be removed from the previous pattern in order to
leave no squares? Gets 6.
M. Adams. Puzzle Book. 1939. Prob. C.157: Making
hexagons, pp. 163 & 190. The
hexagon on the triangular lattice which is two units along each edge contains 8
hexagons. [It is known that the hexagon
of side n contains n3 hexagons.
I recently discovered this but have found that it is known, though I
don't know who discovered it first.]
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex,
1984. Problem 32, with Solution at the
back of the book. Count the hexagons in
the hexagon of side three on the triangular lattice. They get 27.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/13, pp. 23 & 75. Pattern with hexagonal symmetry and lots of overlapping
circles, some incomplete.
5.Y. NUMBER OF ROUTES IN A LATTICE
The
common earlier form was to have the route spell a word or phrase from the
centre to the boundaries of a diamond.
I will call this a word diamond.
Sometimes the phrase is a palindrome and one reads to the centre and
then back to the edge. See Dudeney, CP,
for analysis of the most common cases.
I have seen such problems on the surface of a 3 x 3 x 3 cube. The problems of counting Euler or
Hamiltonian paths are related questions, but dealt with under 5.E and 5.F.
New
section -- in view of the complexity of the examples below, there must be
older, easier, versions, but I have only found the few listed below. The first entry gives some ancient lattices,
but there is no indication that the number of paths was sought in ancient
times.
Roger Millington. The Strange World of the Crossword. M. & J. Hobbs, Walton‑on‑Thames,
UK, 1974. (This seems to have been
retitled: Crossword Puzzles: Their History and Cult for a US ed from Nelson,
NY.)
On
pp. 38-39 & 162, he gives the cabalistic triangle shown below and says it
is thought to have been constructed from the opening letters of the Hebrew
words Ab (Father), Ben (Son),
Ruach Acadash (Holy Spirit). He
then asks how many ways one can read
ABRACADABRA in it, though there
is no indication that the ancients did this.
His answer is 1024 which is correct.
A
B R A C A D A B R A
A B R A C A D A B R
A B R A C A D A B
A B R A C A D A
A B R A C A D
A B R A C A
A B R A
A B R
A B
A
On
pp. 39-40 he describes and illustrates an inscription on the Stele of Moschion
from Egypt, c300. This is a 39 x 39
square with a Greek text from the middle to the corner, e.g. like the
example in the following entry. The
text reads: ΟΥIΡIΔIΜΟΥΧIΩΝΥΓIΑΥΘΕIΥΤΟΝΠΟΔΑIΑΤΠΕIΑIΥ
which means: Moschion to Osiris, for the treatment which
cured his foot. Millington does not ask
for the number of ways to read the inscription, which is 4 BC(38,19) = 14 13810 55200.
Curiosities for the Ingenious
selected from The most authentic Treasures of E
D C D E
Nature,
Science and Art, Biography, History, and General Literature. D C B C D
(1821);
2nd ed., Thomas Boys, London, 1822.
Remarkable epitaph, C
B A B C
p.
97. Word diamond extended to a square,
based on 'Silo Princeps Fecit', D
C B C D
with
the ts
at the corners. An example based
on 'ABCDEF' is shown E
D C D E
at
the right. Says this occurs on the tomb
of a prince named Silo at the
entrance
of the church of San Salvador in Oviedo, Spain. Says the epitaph can be read in 270 ways. I find there are 4 BC(16, 8) = 51490 ways.
In the churchyard of St. Mary's,
Monmouth, is the gravestone of John Rennie, died 31 May 1832, aged 33
years. This has the inscription shown
below. Further down the stone it gives
his son's name as James Rennie.
Apparently an N has been dropped to get a message with an
odd number of letters. I have good
photos. Nothing asks for the number of
ways of reading the inscription. I
get 4 BC(16,9) =
45760 ways.
eineRnhoJsJohnRenie
ineRnhoJsesJohnReni
neRnhoJseiesJohnRen
eRnhoJseiliesJohnRe
RnhoJseileliesJohnR
nhoJseilereliesJohn
hoJseilerereliesJoh
oJseilereHereliesJo
hoJseilerereliesJoh
nhoJseilereliesJohn
RnhoJseileliesJohnR
eRnhoJseiliesJohnRe
neRnhoJseiesJohnRen
ineRnhoJsesJohnReni
eineRnhoJsJohnRenie
Nuts to Crack I (1832), no.
200. The example from Curiosities for
the Ingenious with 'SiloPrincepsFecit', but no indication of what is wanted --
perhaps it is just an amusing picture.
W. Staniforth. Letter.
Knowledge 16 (Apr 1893) 74-75.
Considers 1 2
3 4 5 6
"figure
squares" as at the right. "In
how many different ways may 2 3
4 5 6 7
the
figures in the square be read from
1 to 11 consecutively?" 3
4 5 6 7 8
He
computes the answers for the n x n case for the first few 4 5 6
7 8 9
cases
and finds a recurrence. "Has such
a series of numbers any 5 6
7 8 9 10
mathematical
designation?" The editor notes
that he doesn't 6 7
8 9 10 11
know.
J. J. Alexander. Letter.
Knowledge 16 (May 1893) 89. Says
Staniforth's numbers are the sums of the squares of the binomial
coefficients BC(n, k), the formula for which is BC(2n, n). Editor say he has received more than one note pointing this out
and cites a paper on such figure squares by T. B. Sprague in the Transactions
of the Royal Society of Edinburgh -- ??NYS, no more details provided.
Loyd. Problem 12: The temperance puzzle. Tit‑Bits 31 (2
& 23 Jan 1897) 251 &
307. Red rum & murder. = Cyclopedia, 1914, The little brown jug,
pp. 122 & 355. c= MPSL2, no. 61,
pp. 44 & 141. Word diamond based on
'red rum & murder', i.e. the central line is redrum&murder. He
allows a diagonal move from an E back to an inner R and this gives 372
paths from centre to edge, making
3722 = 138,384 in
total.
Dudeney. Problem 57: The commercial traveller's
puzzle. Tit‑Bits 33 (30 Oct &
20 Nov 1897) 82 & 140.
Number of routes down and right on a
10 x 12 board. Gives a general solution for any board.
Dudeney. A batch of puzzles. The Royal Magazine 1:3 (Jan 1899)
269-274 & 1:4 (Feb 1899) 368-372.
A "Reviver" puzzle.
Complicated pattern based on 'reviver'.
544 solutions.
Dudeney. Puzzling times at Solvamhall Castle. London Magazine 7 (No. 42) (Jan 1902) 580‑584 &
8 (No. 43) (Feb 1902) 53-56. The
amulet. 'Abracadabra' in a triangle
with A at top, two B's
below, three R's
below that, etc. Answer:
1024. = CP, 1907, No. 38,
pp. 64-65 & 190. CF Millington at
beginning of this section.
Dudeney. CP. 1907.
Prob.
30: The puzzle of the canon's yeoman, pp. 55-56 & 181-182. Word diamond based on 'was it a rat I
saw'. Answer is 63504
ways. Solution observes that for
a diamond of side n+1, with no diagonal moves, the number of routes
from the centre to an edge is 4(2n-1) and the number of ways to spell the phrase
is this number squared. Analyses four
types with the following central lines:
A ‑ 'yoboy'; B -
'level'; C - 'noonoon'; D ‑ 'levelevel'.
In
A, one wants to spell 'boy', so there are 4(2n-1) solutions.
In B, one wants to spell
'level' and there are [4(2n-1)]2 solutions.
In
C, one wants to spell 'noon' and there are 8(2n-1)
solutions.
In
D, one wants to spell 'level' and there are complications as one can
start and finish at the edge. He
obtains a general formula for the number of ways. Cf Loyd, 1914.
Prob.
38: The amulet, pp. 64-65 & 190.
See: Dudeney, 1902.
Pearson. 1907.
Part II: A magic cocoon, p. 147.
Word diamond based on 'cocoon', so the central line is noocococoon. Because one can start at the non‑central Cs,
and can go in as well as out, I get
948 paths. He says
756.
Loyd. Cyclopedia. 1914. Alice in Wonderland, pp. 164 & 360. = MPSL1, no. 109, pp. 107 & 161‑162. Word diamond based on 'was it a cat I
saw'. Cf Dudeney, 1907.
Dudeney. AM.
1917.
Prob.
256: The diamond puzzle, pp. 74 & 202.
Word diamond based on 'dnomaidiamond'.
This is type A of his discussion in CP and he states the
general formula. 252 solutions.
Prob.
257: The deified puzzle, pp. 74-75 & 202.
Word diamond based on 'deifiedeified'.
This is type D in CP and has 1992 solutions. He says 'madamadam' gives 400
and 'nunun' gives 64, while 'noonoon' gives 56.
Prob.
258: The voter's puzzle, pp. 75 & 202.
Word diamond built on 'rise to vote sir'. Cites CP, no. 30, for the result, 63504, and the general
formula.
Prob.
259: Hannah's puzzle, pp. 75 & 202.
6 x 6 word square based on
'Hannah' with Hs on the outside, As adjacent to the Hs
and four Ns in the middle. Diagonal moves allowed.
3468 ways.
Wood. Oddities. 1927. Prob. 44: The amulet problem, p. 39. Like the original ABRACADABRA triangle, but
with the letters in reverse order.
Collins. Book of Puzzles. 1927. The magic cocoon
puzzle, pp. 169-170. As in Pearson.
Loyd Jr. SLAHP.
1928. A strolling pedagogue, pp.
38 & 97. Number of routes to
opposite corner of a 5 x 5 array of points.
D. F. Lawden. On the solution of linear difference
equations. MG 36 (No. 317) (Sep 1952)
193-196. Develops use of integral
transforms and applies it to find that the number of king's paths going down or
right or down‑right from (0,
0) to
(n, n) is Pn(3) where Pn(x) is
the Legendre polynomial.
Leo Moser. King paths on a chessboard. MG 39 (No. 327) (Feb 1955) 54. Cites Lawden and gives a simpler proof of
his result Pn(3).
Anon. Puzzle Page: Check this.
MTg 36 (1964) 61 & 27 (1964) 65. Find the number of king's routes from corner to corner when he
can only move right, down or right‑down.
Gets 48,639 routes on
8 x 8 board.
Ripley's Puzzles and Games. 1966.
P. 32. Word diamond laid out
differently so A A
A
one
has to read from one side to the opposite side. Rotating by 45o, one gets B B
the
pattern at the right for edge three.
One wants the number of ways to C
C C
read ABCDEF.
In general, when the first line of
As has n positions, D D
the
total number of ways to reach the first row is
n. For each successive E E E
row,
the total number is alternately twice the number for the previous row less F F
twice
the end term of that row or just twice the the number for the previous
row. In our example with n = 3,
the number of ways to reach the second row is 4 = 2x3 - 2x1. The number of ways to reach the third row is 8 = 2x4.
The number of ways to reach the fourth row is 12 = 2x8 - 2x2, then we
get 24 = 2 x 12; 36 = 2x24 ‑ 2x6. It happens that the first n
end terms are the central binomial coefficients BC(2k,k),
so this is easy to calculate. I
find the total number of routes, for n
= 2, 3, ..., 7, is 4,
18, 232, 1300,
6744, 33320, the last being the desired and given answer
for the given problem.
Pál Révész. Op. cit. in 5.I.1. 1969. On p. 27, he gives
the number of routes for a king moving forward on a chessboard and a man moving
forward on a draughtsboard.
Putnam. Puzzle Fun.
1978. No. 8: Level - level, pp.
3 & 26. Form a wheel of 16 points
labelled LEVELEVELEVELEVE. Place 4
Es inside, joined to two
consecutive Vs and the intervening L.
Then place a V in the middle, joined to these four Es.
How many ways to spell LEVEL? He
gets 80, which seems right.
5.Z. CHESSBOARD PLACING PROBLEMS
See
MUS I 285-318, some parts of the previous chapter and the Appendix in II
351-360. See also 5.I.1, 6.T.
There
are three kinds of domination problems.
In
strong domination, a piece dominates the square it is on.
In
weak domination, it does not, hence more pieces may be needed to dominate the board.
Non‑attacking
domination is strong domination with no piece attacking another. Graph theorists say the pieces are
independent. This also may require more
pieces than strong domination, but it may require more or fewer pieces than
weak domination.
The
words 'guarded' or 'protected' are used for weak domination, but 'unguarded' or
'unprotected' may mean either strong or non‑attacking domination.
Though
these results seem like they must be old, the ideas seem to have originated
with the eight queens problem, c1850, (cf 5.I.1) and to have been first really
been attacked in the late 19C. There
are many variations on these problems, e.g. see Ball, and I will not attempt to
be complete on the later variations. In
recent years, this has become a popular subject in graph theory, where the
domination number, γ(G), is the size of the smallest strongly
dominating set on the graph G and the independent domination number, i(G),
is the size of the smallest non-attacking (= independent) dominating set.
Mario Velucchi has a web site devoted
to the non-dominating queens problem and related sites for similar
problems. See: http://www.bigfoot.com/~velucchi/papers.html and
http://www.bigfoot.com/~velucchi/biblio.html.
Ball. MRE, 3rd ed., 1896, pp. 109-110: Other problems with queens. Says:
"Captain Turton has called my attention to two other problems of a
somewhat analogous character, neither of which, as far as I know, has been
hitherto published, ...." These
ask for ways to place queens so as to attack as few or as many cells as
possible -- see 5.Z.2.
Ball. MRE, 4th ed., 1905, pp. 119-120: Other problems with queens; Extension to other chess pieces. Repeats above quote, but replaces 'hitherto
published' by 'published elsewhere', extends the previous text and adds the new
section.
Ball. MRE, 5th ed., 1911.
Maximum pieces problem; Minimum
pieces problem, pp. 119‑122.
[6th ed., 1914 adds that Dudeney has written on these problems in The
Weekly Dispatch, but this is dropped in the 11th ed. of 1939.] Considerably generalizes the problems. On the
8 x 8 board, the maximum number
of non-attacking kings is 16, queens is 8, bishops is 14 [6th ed., 1914, adds there are 256 solutions], knights is 32 with 2 solutions and
rooks is 8 with 88 solutions [sic, but changed to 8! in the 6th ed.]. The minimum number of pieces to strongly
dominate the board is 9 kings, 5 queens with 91 inequivalent solutions [the
91 is omitted in the 6th ed., since it is stated later], 8 bishops,
12 knights, 8 rooks. The minimum number of pieces to weakly
dominate the board is 5 queens, 10 bishops,
14 knights, 8 rooks.
Dudeney. AM.
1917. The guarded chessboard,
pp. 95‑96. Discusses different
ways pieces can weakly or non‑attackingly dominate n x n
boards.
G. P. Jelliss. Multiple unguard arrangements. Chessics 13 (Jan/Jun 1982) 8‑9. One can have 16 kings, 8 queens, 14 bishops,
32 knights or 8 rooks
non‑attackingly placed on a
8 x 8 board. He considers mixtures of pieces -- e.g. one
can have 10 kings and 4 queens non‑attacking. He tries to maximize the product of the numbers of each type in a
mixture -- e.g. scoring 40 for the example.
Ball. MRE, 4th ed., 1905. Other
problems with queens; Extension to
other chess pieces, pp. 119-120.
Says problems have been proposed for
k kings on an n x n,
citing L'Inter. des math. 8 (1901) 140, ??NYS.
Gilbert Obermair. Denkspiele auf dem Schachbrett. Hugendubel, Munich, 1984. Prob. 27, pp. 29 & 58. 9
kings strongly, and 12 kings weakly, dominate an 8 x 8
board.
Here
the graph is denoted Qn, but I will denote γ(Qn)
by γ(n) and
i(Qn) by i(n).
Murray. Pp. 674 & 691. CB249 (c1475) shows
16 queens weakly dominating an 8 x 8
board, but the context is unclear to me.
de Jaenisch. Op. cit. in 5.F.1. Vol. 3, 1863. Appendice,
pp. 244-271. Most of this is due to "un
de nos anciens amis, Mr de R***". Finds and describes the 91 ways of placing 5 queens so as to
non-attackingly dominate the 8 x 8 board.
Then considers the n x n board for
n = 2, ..., 7 with strong and
non-attacking domination. Up through 5,
he gives the number of pieces being attacked in each solution which allows one
to determine the weak solutions.
For n < 6, he gets the answers in the table below, but
for n = 6, he gets 21 non-attacking solutions instead of 17?.
Ball. MRE, 3rd ed., 1896. Other
problems with queens, pp. 109-110.
"Captain [W. H.] Turton has called my attention to two other problems
of a somewhat analogous character, neither of which, as far as I know, has been
hitherto published, or solved otherwise than empirically." The first is to place 8 queens so as to
strongly dominate the fewest squares.
The minimum he can find is 53.
(Cf Gardner, 1999.) The second
is to place m queens, m £
5, so as to strongly dominate as many
cells as possible. With 4 queens, the
most he can find is 62.
Dudeney. Problem 54: The hat‑peg puzzle. Tit‑Bits 33 (9 &
30 Oct 1897) 21 & 82. Problem
involves several examples of strong domination by 5 queens on an 8 x 8
board leading to a non‑attacking domination. He says there are just 728 such. This
= 8 x 91. = Anon. &
Dudeney; A chat with the Puzzle King; The Captain 2 (Dec? 1899) 314-320; 2:6
(Mar 1900) 598-599 & 3:1 (Apr 1900) 89. = AM; 1917; pp. 93-94 & 221.
Ball. MRE, 4th ed., 1905, loc. cit. in 5.Z.1. Extends 3rd ed. by asking for the minimum number of queens to
strongly dominate a whole n x n board.
Says there seem to be 91 ways of having 5 non-attacking queens on
the 8 x 8, citing L'Inter. des Math. 8 (1901) 88, ??NYS.
Ball. MRE, 5th ed., 1911, loc. cit. in 5.Z. On pp. 120-122, he considers queens and states the minimum
numbers of queens required to strongly dominate the board and the numbers of
inequivalent solutions for 2 x 2, 3 x 3,
..., 7 x 7, citing the article cited in the 4th ed. and
Jaenisch, 1862, without a volume number.
For n = 7, he gives the same unique solution for
strongly dominating as for non-attacking dominating. [In the 6th ed., this is corrected and he says it is a
solution.] He says Jaenisch also posed
the question of the minimum number of non-attacking queens to dominate the
board and gives the numbers and the number of inequivalent ways for the 4 x 4,
.., 8 x 8, except that he follows Jaenisch in stating
that there are 21 solutions on the 6 x
6. [This is changed to 17 in the 6th
ed.]
Dudeney. AM.
1917. Loc. cit. in 5.Z. He uses 'protected' for 'weakly', but he
seems to copy the values for 'strongly' from Jaenisch or Ball. His 'not protected' seems to mean
'non-attacking'. However, some values are different and I consequently
am very uncertain as to the correct values??
Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 24‑25,
he shows 5 queens are sufficient to strongly dominate the board and says this
is minimal.
Below, min. denotes the minimum
number of queens to dominate and no. is the number of inequivalent ways to do
so.
STRONG WEAK NON-ATTACKING
n min. no. min.
no. min. no.
1 1 1 0 0 1 1
2 1 1 2 2 1 1
3 1 1 2 5 1 1
4 2 3 2 3 3 2
5 3 37 3 15 3 2
6 3 1 4 ³2 4
17?
7 4 4 5? 4 1
8 5 ³150 5 ³41 5
91
Rodolfo Marcelo Kurchan,
proposer; Henry Ibstedt & proposer,
solver. Prob. 1738 -- Queens in
space. JRM 21:3 (1989) 220 &
22:3 (1989) 237. How many queens
are needed to strongly dominate an n x
n x n cubical board? For
n = 3, 4, ..., 9, the best known
numbers are: 1, 4, 6, 8, 14, 20,
24. The solution is not clear if these
are minimal, but it seems to imply this.
Martin Gardner. Chess queens and maximum unattacked
cells. Math Horizons (Nov 1999). Reprinted in Workout, chap. 34. Considers the problem of Turton described in
Ball, 3rd ed, above: place 8
queens on an 8 x 8 board so as to strongly dominate the fewest
squares. That is, leave the maximum
number of unattacked squares. More
generally, place k queens on an n x n board to leave the
maximum number of unattacked squares.
He describes a simple problem by Dudeney (AM, prob. 316) and recent work
on the general problem. He cites Velucchi,
cf below, who provides the following table of maximum numbers of unattacked
cells and number of solutions for the maximum.
I'm not sure if some of these are still only conjectured.
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Max 0 0 0 1 3 5 7 11 18 22 30 36 47 56 72 82 97
Sols 0 0 0 25 1 3 38 7 1 1 2 7 1 4 3 1
Mario Velucchi has a web site
devoted to the non-dominating queens problem and related sites for similar
problems. See: http://www.bigfoot.com/~velucchi/papers.html and
http://www.bigfoot.com/~velucchi/biblio.html.
A. P. Burger & C. M.
Mynhardt. Symmetry and domination in
queens graphs. Bull. Inst.
Combinatorics Appl. 29 (May 2000) 11-24.
Extends results to n = 30, 45,
69, 77. Summarizes the field, with 14
references, several being earlier surveys.
The table below gives all known values.
It will be seen that the case n
= 4k + 1 seems easiest to deal
with. The values separated by
strokes, /, indicate cases where the value is one of the two given values,
but it is not known which.
n 4 5 6 7 8 9 10 11 12 13 14 15 16 17
γ(n) 2 3 3 4 5 5 5 5 6 7 7/8 8/9 8/9 9
i(n) 3 3 4 4 5 5 5 5 7 7 8 9 9 9
n 18 19 21 25 29 30 31 33 37 41 45 49 53 57
γ(n) 9 10 11 13 15 15 16 17 19 21 23 25 27 29
i(n) 11 13 17 23
n 61 69 77
γ(n) 31 35 39
Dudeney. AM.
1917. Prob. 299: Bishops in
convocation, pp. 89 & 215. There
are 2n ways to place 2n‑2 bishops non‑attackingly
on an n x n board. At loc. cit. in
5.Z, he says that for n = 2, ..., 8, there are
1, 2, 3, 6, 10, 20, 36
inequivalent placings.
Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 25‑26,
he shows the maximum number of non‑attacking bishops on one colour is 7
and there are 16 ways to place them.
Obermair. Op. cit. in 5.Z.1. 1984. Prob. 17, pp. 23
& 50. 8 bishops strongly, and 10
bishops weakly, dominate the 8 x 8 board.
Ball. MRE, 4th ed., 1905. Loc.
cit. in 5.Z.1. Says questions as to the
maximum number of non-attacking knights and minimum number to strongly dominate
have been considered, citing L'Inter. des math. 3 (1896) 58, 4 (1897) 15-17 & 254, 5 (1898) 87
[5th ed. adds 230‑231], ??NYS.
Dudeney. AM.
1917. Loc. cit. in 5.Z. Notes that if n is odd, one can
have (n2+1)/2 non‑attacking knights in one way,
while if n is even, one can have n2/2 in two equivalent ways.
Irving Newman, proposer; Robert Patenaude, Ralph Greenberg and Irving
Newman, solvers. Problem E1585 --
Nonattacking knights on a chessboard.
AMM 70 (1963) 438 & 71 (1964) 210-211. Three easy proofs that the maximum number of non-attacking
knights is 32. Editorial note cites
Dudeney, AM, and Ball, MRE, 1926, p. 171 -- but the material is on p. 171 only
in the 11th ed., 1939.
Gardner. SA (Oct 1967, Nov 1967 & Jan 1968) c= Magic Show, chap. 14. Gives Dudeney's results for the 8 x 8.
Golomb has noted that Greenberg's solution of E1585 via a knight's tour
proves that there are only two solutions.
For the k x k board,
k = 3, 4, ..., 10, the minimal number of knights to strongly dominate is: 4, 4, 5, 8, 10, 12,
14, 16. He says the table may
continue: 21, 24, 28, 32, 37. Gives numerous examples.
Obermair. Op. cit. in 5.Z.1. 1984. Prob. 16, pp. 21
& 47. 14 knights are necessary for
weak domination of the 8x8 board.
E. O. Hare & S. T.
Hedetniemi. A linear algorithm for
computing the knight's domination number of a
k x n chessboard. Technical report 87‑May‑1, Dept.
of Computer Science, Clemson University.
1987?? Pp. 1‑2 gives the
history from 1896 and Table 2 on p. 13 gives their optimal results for strong
domination on k x n boards,
4 £ k £
9, k £ n £ 12 and also for k = n = 10. For the k x k
board, k = 3, ..., 10, they confirm the results in Gardner.
Anderson H. Jackson & Roy P.
Pargas. Solutions to the N x N
knight's cover problem. JRM 23:4
(1991) 255-267. Finds number of knights
to strongly dominate by a heuristic method, which finds all solutions up
through N = 10. Improves the value given by Gardner for N = 15
to 36 and finds solutions for N
= 16, ..., 20 with 42, 48, 54, 60, 65 knights.
É. Lucas. Théorie des Nombres. Gauthier‑Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. Section 128, pp. 220‑223. Determines the number of inequivalent
placings of n nonattacking rooks on an n x n
board in general and gives values for
n £ 12. For n = 1, ..., 8, there are
1, 1, 2, 7, 23, 115, 694, 5282
inequivalent ways.
Dudeney. AM.
1917. Loc. cit. at 5.Z. Notes there are n! ways to place n
non‑attacking rooks and asks how many of these are
inequivalent. Gives values for n = 1, ..., 5. AM prob. 296, pp. 88 & 214, is the case n = 4.
D. F. Holt. Rooks inviolate. MG 58 (No. 404) (Jun 1974) 131‑134. Uses Burnside's lemma to determine the
number of inequivalent solutions in general, getting Lucas' result in a more
modern form.
Ball. MRE, 5th ed., 1911. Loc.
cit. in 5.Z. P. 122: "There are endless similar questions in
which combinations of pieces are involved." 4 queens and king or queen or bishop or knight or rook or pawn
can strongly dominate 8 x 8.
King. Best 100. 1927.
No.
77, pp. 30 & 57. 4 queens and a
rook strongly dominate 8 x 8.
No.
78, pp. 30 & 57. 4 queens and a
bishop strongly dominate 8 x 8.
New
section. I have been meaning to add
this sometime, but I have just come across an expository article, so I am now
starting. The mathematics of this gets
quite formidable. See 5.AD for a
somewhat related topic.
A
faro, weave, dovetail or perfect (riffle) shuffle starts by cutting the deck in
half and then interleaving the two halves.
When the deck has an even number of cards, there are two ways this can
happen -- the original top card can remain on top (an out shuffle) or it
can become the second card of the shuffled deck (an in shuffle). E.g. if our deck is 123456,
then the out shuffle yields
142536 and the in shuffle
yields 415263. Note that removing the first and last cards
converts an out shuffle on 2n cards to an in shuffle on 2n-2
cards. When the deck has an odd
number of cards, say 2n+1, we cut above or below the middle card and
shuffle so the top of the larger pile is on top, i.e. the larger pile straddles
the smaller. If the cut is below the
middle card, we have piles of n+1 and
n and the top card remains on
top, while cutting above the middle card leaves the bottom card on bottom. Removing the top or bottom card leaves an in
shuffle on 2n cards.
Monge's
shuffle takes the first card and then alternates the next cards over and under
the resulting pile, so 12345678 becomes
86421357.
At G4G2, 1996, Max Maven gave a talk
on some magic tricks based on card shuffling and gave a short outline of the
history. The following is an attempt to
summarise his material. The faro
shuffle, done by inserting part of the deck endwise into the other part, but
not done perfectly, began to be used in the early 18C and a case of cheating
using this is recorded in 1726. The
riffle shuffle, which is the common American shuffle, depends on mass produced
cards of good quality and began to be used in the mid 19C. However, magicians did not become aware of
the possibilities of the perfect shuffle until the mid 20C, despite the early
work of Stanyans C. O. Williams and Charles T. Jordan in the 1910s.
Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, pp. 78-85:
Of the combinations of the cards. This
describes a shuffle, where one takes the top two cards, then puts the next two
cards on top, then the next three cards underneath, then the next two on top,
then the next three underneath. For ten
cards 1234567890, it produces
8934125670, a permutation of
order 7. Tables of the first few
repetitions are given for 10, 24, 27
and 32 cards, having orders 7, 30, 30, 156.
The Secret Out. 1859.
Permutation table, pp. 394-395 (UK: 128-129). Describes Hooper's shuffle for ten cards.
Bachet-Labosne. Problemes.
3rd ed., 1874. Supp. prob. XV,
1884: 214-222. Discusses Monge's shuffle
and its period.
John Nevil Maskelyne. Sharps and Flats. 1894. ??NYS -- cited by
Gardner in the Addendum of Carnival.
"One of the earliest mentions". Called the "faro dealer's shuffle".
Ahrens. MUS I.
1910. Ein Kartenkunststück
Monges, pp. 152-145. Expresses the general
form of Monge's shuffle and finds its order for n = 1, 2, ..., 10.
Mentions the general question of finding the order of a shuffle.
Charles T. Jordan. Thirty Card Mysteries. The author, Penngrove, California, 1919
(??NYS), 2nd ed., 1920 (?? I have copy of part of this). Cited by Gardner in the Addendum to
Carnival. First magician to apply the
shuffle, but it was not until late 1950s that magicians began to seriously use
and study it. The part I have (pp.
7-10) just describes the idea, without showing how to perform it. The text clearly continues to some
applications of the idea. This material
was reprinted in The Bat (1948-1949).
Frederick Charles Boon. Shuffling a pack of cards and the theory of
numbers. MG 15 (1930) 17-20. Considers the Out shuffle and sees that it
relates to the order of 2 (mod 2n+1)
and gives some number theoretic observations on this. Also considers odd decks.
J. V. Uspensky & M. A.
Heaslet. Elementary Number Theory. McGraw-Hill, NY, 1939. Chap. VIII: Appendix: On card shuffling, pp.
244-248. Shows that an In shuffle of a
deck of 2n cards takes the card in position
i to position 2i (mod 2n+1), so the order of the
permutation is the exponent or order of
2 (mod 2n+1), which is 52 when
n = 26. [Though not discussed,
this shows that the order of the Out shuffle is the order of 2 (mod 2n-1), which is only 8
for n = 26. And the order of a shuffle of 2n+1
cards is the order of 2 (mod
2n+1).] Monge's shuffle is more
complex, but leads to congruences (mod 4n+1) and has order equal to the
smallest exponent e such that
2e º ±1 (mod 4n+1), which is
12 for n = 26.
T. H. R. Skyrme. A shuffling problem. Eureka 7 (Mar 1942) 17-18. Describes Monge's shuffle with the second
card going under or over the first.
Observes that in the under shuffle for an even number of cards, the last
card remains fixed, while the over shuffle for an odd number of cards also
leaves the last card fixed. By
appropriate choice, one always has the
n-th card becoming the first.
Finds the order of the shuffle essentially as in Uspensky &
Heaslet. Makes some further
observations.
N. S. Mendelsohn, proposer and
solver. Problem E792 -- Shuffling
cards. AMM 54 (1947) 545 ??NYS &
55 (1948) 430-431. Shows the
period of the out shuffle is at most
2n-2. Editorial notes cite
Uspensky & Heaslet and MG 15 (1930) 17-20 ??NYS.
Charles T. Jordan. Trailing the dovetail shuffle to its
lair. The Bat (Nov, Dec 1948; Jan, Feb, Mar, 1949). ??NYS -- cited by Gardner. I have
No. 59 (Nov 1948) cover & 431-432, which reprints some of the
material from his book.
Paul B. Johnson. Congruences and card shuffling. AMM 63 (1956) 718-719. ??NYS -- cited by Gardner.
Alexander Elmsley. Work in Progress. Ibidem 11 (Sep 1957) 222.
He had previously coined the terms 'in' and 'out' and represented them
by I
and O. He discovers and shows that to put the top card into the k‑th position, one writes k-1
in binary and reads off the sequence of
1s and 0s, from the most
significant bit, as I and
O shuffles. He asks but does not solve the question of
how to move the k-th card to the top --
see Bonfeld and Morris.
Alexander Elmsley. The mathematics of the weave shuffle, The Pentagram 11:9 (Jun 1957) 70‑71;
11:10 (Jul 1957) 77-79; 11:11 (Aug
1957) 85; 12 (May 1958) 62. ??NYR -- cited by Gardner in the
bibliography of Carnival, but he doesn't give the Ibidem reference in the
bibliography, so there may be some confusion here?? Morris only cites Pentagram.
Solomon W. Golomb. Permutations by cutting and shuffling. SIAM Review 3 (1961) 293‑297. ??NYS -- cited by Gardner. Shows that cuts and the two shuffles
generate all permutations of an even deck.
However, for an odd deck of
n cards, the two kinds of
shuffles can be intermixed and this only changes the cyclic order of the
result. Since cutting also only changes
the cyclic order, the number of possible permutations is n
times the order of the shuffle.
Gardner. SA (Oct 1966) = Carnival, chap. 10. Defines the in and out shuffles as above and
gives the relation to the order of 2.
Notes that it is easier to do the inverse operations, which consist of
extracting every other card. Describes
Elmsley's method. Addendum says no easy
method is known to determine shuffles to bring the k‑th card to the top.
Murray Bonfeld. A solution to Elmsley's problem. Genii 37 (May 1973) 195-196. Solves Elmsley's 1957 problem by use of an
asymmetric in-shuffle where the top part of the deck has 25 cards, so the first
top card becomes second and the last two cards remain in place. (If one ignores the bottom two cards this is
an in-shuffle of a 50 card deck.)
S. Brent Morris. The basic mathematics of the faro
shuffle. Pi Mu Epsilon Journal 6 (1975)
86-92. Obtains basic results, getting
up to Elmsley's work. His reference to
Gardner gives the wrong year.
Israel N. Herstein & Irving
Kaplansky. Matters Mathematical. 1974;
slightly revised 2nd ed., Chelsea, NY, 1978. Chap. 3, section 4: The interlacing shuffle, pp. 118-121. Studies the permutation of the in shuffle,
getting same results as Uspensky & Heaslet.
S. Brent Morris. Faro shuffling and card placement. JRM 8:1 (1975) 1-7. Shows how to do the faro shuffle. Gives Elmsley's and Bonfeld's results.
Persi Diaconis, Ronald L. Graham
& William M. Kantor. The
mathematics of perfect shuffles. Adv.
Appl. Math. 4 (1983) 175-196. ??NYS.
Steve Medvedoff & Kent
Morrison. Groups of perfect
shuffles. MM 60:1 (1987) 3-14. Several further references to check.
Walter Scott. Mathematics of card sharping. M500 125 (Dec 1991) 1-7. Sketches Elmsley's results. States a peculiar method for computing the
order of 2 (mod 2n+1) based on adding translates of the binary
expansion of 2n+1 until one obtains a binary number of
all 1s. The number of ones is the order
a and the method is thus
producing the smallest a such that
2a-1 is a multiple
of 2n+1.
John H. Conway & Richard K.
Guy. The Book of Numbers. Copernicus (Springer-Verlag), NY, 1996. Pp. 163-165 gives a brief discussion of
perfect shuffles and Monge's shuffle.
5.AB. FOLDING A STRIP OF STAMPS
É. Lucas. Théorie des Nombres. Gauthier‑Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. P. 120.
Exemple
II -- La bande de timbres-poste. -- De combien de manières peut-on replier, sur un
seul, une bande de p timbres-poste?
Exemple
III -- La feuille de timbres-poste. -- De combien de manières peut-on replir, sur un
seul, une feuille rectangulaire de pq timbres-poste?
"Nous
ne connaissons aucune solution de ces deux problèmes difficiles proposés par M.
Em. Lemoine."
M. A. Sainte-Laguë. Les Réseaux (ou Graphes). Mémorial des Sciences Mathématiques, fasc.
XVIII. Gauthier-Villars, Paris, 1926. Section 62: Problème des timbres-poste,
pp. 39‑41. Gets some basic
results and finds the numbers for a strip of
n,
n = 1, 2, ..., 10 as: 1, 2, 6, 16, 50, 144,
448, 7472, 17676, 41600.
Jacques Devisme. Contribution a l'étude du problème des
timbres-poste. Comptes-Rendus du
Deuxième Congrès International de Récréation Mathématique, Paris, 1937. Librairie du "Sphinx", Bruxelles,
1937, pp. 55-56. Cites Lucas (but in
the wrong book!) and Sainte-Laguë.
Studies the number of different forms of the result, getting
numbers: 1, 2, 3, 8, 18,
44, 115, 294, 783.
5.AC. PROPERTIES OF THE SEVEN BAR DIGITAL DISPLAY
┌─┐ 2
The
seven bar display, in the form of a figure
8, as at the right, is │ │ 1
3
now the standard form for
displaying digits on calculators, clocks, etc. ├─┤ 4
This lends itself to numerous
problems of a combinatorial/numerical │
│ 5 7
New
Section. └─┘ 6
For reference, we number the seven bars in the reverse-S pattern
shown. We can then refer to a pattern by its binary 7-tuple or its decimal equivalent. E.g. the number one is displayed by having
bars 3
and 7 on, which gives a binary pattern
1000100 corresponding to
decimal 68. NOTE that there is some ambiguity with the 6 / 9. Most versions use the upper / lower bar for
these, i.e. 1101111 / 1111011, but the bar is sometimes omitted,
giving 1001111 / 1111001. I will assume the first case unless
specified.
I have been interested in these for some time for several
reasons. First, my wife has such a
clock on her side of the bed and she often has a glass of water in front of it,
causing patterns to be reversed. At
other times the clock has been on the floor upside down, causing a different
reversal of patterns. Second, segments
often fail or get stuck on and I have tried to analyse which would be the worst
segment to fail or get stuck. As an
example, the clock in my previous car went from 16:59 to 15:00. Third, I
have analysed which segment(s) in a clock are used most/least often.
Birtwistle. Calculator Puzzle Book. 1978.
Prob. 35: New numbers, pp. 26-27 & 83. Asks for the number of new digits one can make, subject to their
being connected and full height. Says
it is difficult to determine when these are distinct -- e.g. calculators differ
as to the form of their 6s and 9s -- so he is not sure how to count, but he
gives 22 examples. I find there are 55
connected, full-height patterns.
Gordon Alabaster, proposer &
Robert Hill, solver. Problem
134.3 -- Clock watching. M500 134 (Aug
1993) 17 & 135 (Oct 1993) 14-15. Proposer notes that one segment of the units
digit of the seconds on his station clock was stuck on, but that the sequence
of symbols produced were all proper digits.
Which segment was stuck? Asks if
there are answers for 2, ..., 6
segments stuck on. Solver gives
systematic tables and discusses problems of how to determine which segment(s)
are stuck and whether one can deduce the correct time when the stuck segments
are known.
Martin Watson. Email to NOBNET, 17 Apr 2000 08:17:32 PDT
[NOBNET 2334]. Observes that the 10
digits have a total of 49 segments and asks if they can be placed on a 4 x 5
square grid. He calls these
forms 'digigrams'. He had been unable
to find a solution but Leonard Campbell has found 5 distinct solutions, though
they do no differ greatly. He has the
pieces and some discussion on his website:
http://martnal.tripod.com/puzzles.html .
Dario Uri [22 Apr 2000 14:44:35 +0200] found two extra solutions, but
Rick Eason [22 Apr 2000 09:37: -0400] also found these, but points out that
these have an error due to misreading the lattice which gives the two bars of
the 1 being parallel instead of end to end.
Eason's program also found the 5 solutions.
5.AD. STACKING A DECK TO PRODUCE A SPECIAL EFFECT
New section. This refers to the process of arranging a deck of cards or a
stack of coins so that dealing it by some rule produces a special effect. In many cases, this is just inverting the
permutation given by the rule and the Josephus problem (7.B) is a special
case. Other cases involve spelling out
the names of cards, etc.
Will Blyth. Money Magic. C. Arthur Pearson, London, 1926.
Alternate heads, pp. 61-63.
Stack of eight coins. Place one
on the table and the next on the bottom of the stack. The sequence of placed coins is to alternate heads and
tails. How do you arrange the
stack? Answer is HHTHHTTT.
This is the same process as counting out by 2s -- see 7.B.
Doubleday - 2. 1971.
Heads and tails, pp. 105-106.
Same as Blyth, but with six coins and solution HTTTHH.
New
section. There are several versions of
this and they usually involve parity.
The basic move is to reverse two of the cups. The classic problem seems to be to start with UDU
and produce DDD in three moves. A trick version is to demonstrate this several times to someone
and then leave him to start from
DUD. Another easy problem is to
leave three cups as they were after three moves. This is equivalent to a 3
x 3 array with an even number in each
row and column -- see 6.AO.2. These
problems must be much older than I have, but the following are the only
examples I have yet noted.
Anonymous. Social Entertainer and Tricks (thus on
spine, but running title inside is New Book of Tricks). Apparently a compilation with advertisements
for Johnson Smith (Detroit, Michigan) products, c1890?. P. 38a: Bottoms up. Given
UDU, produce DDD
in three moves.
Young World. c1960.
P. 39: Water switch. Full and
empty glasses: FFFEEE. Make them alternately full and empty in one
move.
Putnam. Puzzle Fun.
1978.
No. 3:
Tea for three, pp. 1 & 25. Cups
given as UDU. Produce DDD in three moves.
No.
16: Glass alignment, pp. 5 & 28.
Six cups arranged UUUDDD. Produce an alternating row. He gets
UDUDUD in three moves. I can get DUDUDU in four moves.
New
section. In the early 1980s, I asked
Richard Guy what was the 'standard' configuration for a die and later asked Ray
Bathke if he used a standard pattern. Assuming
opposite sides add to seven there are two handednesses. But also the spot pattern of the two, three
and six has two orientations, giving 16 different patterns of die. Ray said that when he furnished dice with
games, some customers had sent them back because they weren't the same. Within about three years, I had obtained
examples of all sixteen patterns!
Indeed, I often found several patterns in a single batch from one
manufacturer. Ray Bathke also pointed
out that the small dice that come from the oriental games have the two arranged
either horizontally or vertically rather than diagonally, giving another 16
patterns. I have only obtained five of
these, but with both handednesses included.
I used this idea in one of my Brain Twisters, cf below.
Since
the 2, 3 and 6 faces all meet at a corner, one has just to describe this
corner. The 2, 3, 6 can be clockwise
around the corner or anti-clockwise.
Note that 236 is clockwise if and only if 132
is clockwise. The position of
the 2 and 3 can be described by saying whether the pattern points toward or
away from the corner. If we place the 2
upward, then 6 will be a vertical face and we can describe it by saying whether
the lines of three spots are vertical or horizontal. Guy told me a system for describing a die, but it's not in
Winning Ways and I've forgotten it, so I'll invent my own.
We
write the sequence 236 if
236 is arranged clockwise at
the 236 corner and we write
263 otherwise. When looked at cornerwise, with the 2 on
top, the pattern of the 2 may appear vertical or horizontal. We write
2 when it is vertical and 2
when it is horizontal. (For
oriental dice, the 2 will appear on a diagonal and can be
indicated by 2 or
2. If we now rotate the cube to
bring the 3 on top, its pattern will appear either vertical or horizontal and
we write 3 or 3. Putting the
2 back on top, the 6
face will be upright and the lines of three spots will be either
vertical or horizontal, which we denote by
6 or 6.
David Singmaster. Dicing around. Weekend Telegraph (16 Dec 1989).
= Games & Puzzles No. 15 (Jun 1995) 22-23 & 16 (Jul 1995)
43-44. How many dice are there? Describes the normal 16 and mentions the
other 16.
Ian Stewart. The lore and lure of dice. SA (Nov 1997) ??. He asserts that the standard pattern has 132
going clockwise at a corner, except that the Japanese use the
mirror-image version in playing mah-jongg.
His picture has both 2 and 3 toward the 236 corner and the 6 being
vertical, i.e. in pattern 236. He
discusses crooked dice of various sorts and that the only way to make all
values from 1 to 12 equally likely is to have 123456 on one die and 000666 on
the other.
Ricky Jay. The story of dice. The New Yorker (11 Dec 2000) 90-95.
5.AG. RUBIK'S CUBE AND SIMILAR PUZZLES
I
have previously avoided this as being too recent to be covered in a historical
work, but it is now old enough that it needs to be covered, and there are some
older references. Much of the history
is given in my Notes on Rubik's Cube and my Cubic Circular. Jaap Scherphuis has sent me a file of puzzle
patents and several dozen of them could be entered here, but I will only enter
older or novel items. Scherphuis's file
has about a dozen patents for the 4 x 4
x 4 and 5 x 5 x 5 cubes! See Section 5.A for predecessors of the
idea. However, this Section will mostly
deal with puzzles where pieces are permuted without having any empty places, so
these are generally permutation puzzle.
New
section. Much to be added.
Richard E. Korf. Finding optimal solutions to Rubik's Cube
using pattern databases. Proc. Nat.
Conf. on Artificial Intelligence (AAAI-97), Providence, Rhode Island, Jul 1997,
pp. 700-705. Studies heuristic
methods of finding optimal solutions of the Cube. Claims to be the first to find optimal solutions for random
positions of the Cube -- but I think others such as Kociemba and Reid were
doing it up to a decade earlier. For
ten random examples, he found optimal solutions took 16 moves in one case, 17
moves in three cases, 18 moves in six cases, from which he asserts the median
optimal solution length seems to be 18.
He uses the idea of axial moves and obtains the lower bound of 18 for
God's Algorithm, as done in my Notes in 1980.
Cites various earlier work in the field, but only one reference to the
Cube literature.
Richard E. Korf &
Ariel Felner. Disjoint pattern
database heuristics. Artificial
Intelligence 134 (2002) 9-22. Discusses
heuristic methods of solving the Fifteen Puzzle, Rubik's Cube, etc. Asserts the median optimal solution length
for the Cube is only 18. Seems to say
one of the problems in the earlier paper took a couple of weeks running time,
but improved methods of Kociemba and Reid can find optimal solutions in about an
hour.
New
section. Much to be added.
William Churchill. US Patent 507,215 -- Puzzle. Applied: 28 May 1891; patented: 24 Oct 1893. 1p + 1p diagrams. Two rings of 22 balls, intersecting six spaces apart.
Hiester Azarus Bowers. US Patent 636,109 -- Puzzle. Filed: 16 Aug 1899; patented 31 Oct 1899. 2pp + 1p diagrams. 4 rotating discs which overlap in simple lenses.
Ivan Moscovich. US Patent 4,509,756 -- Puzzle with Elements
Transferable Between Closed‑loop Paths.
Filed: 18 Dec 1981; patented: 9
Apr 1985. Cover page + 3pp + 2pp
diagrams. Two rings of 18 balls, each
stretched to have two straight sections with semicircular ends. The rings cross in four places, at the ends
of the straight sections, so adjacent crossing points are separated by two
balls. I'm not sure this was ever
produced. Mentions three circular rings
version, but there each pair of rings only overlaps in two places so this is a
direct generalization of the Hungarian Rings.
David Singmaster. Hungarian Rings groups. Bull. Inst. Math. Appl. 20:9/10 (Sep/Oct
1984) 137-139. [The results were stated
in Cubic Circular 5 & 6 (Autumn & Winter 1982) 9‑10.] An article by Philippe Paclet [Des anneaux
et des groupes; Jeux et Stratégie 16 (Aug/Sep 1982) 30-32] claimed that all
puzzles of two rings have groups either the symmetric or the alternating group
on the number of balls. This article
shows this is false and determines the group in all cases. If we have rings of size m, n
and the intersections are distances
a, b apart on the two
rings. Then the group, G(m, n, a, b) is the symmetric group on
m+n-2 if mn
is even and is the alternating group if
mn is odd; except
that G(4, 4, 1, 1) is the exceptional group described in R. M. Wilson's
1974 paper: Graph puzzles, homotopy and the alternating group -- cited in
Section 5.A under The Fifteen Puzzle -- and is also the group generated by two
adjacent faces on the Rubik Cube acting on the six corners on those faces; and
except that G(2a, 2b, a, b) keeps antipodal pairs at antipodes and hence
is a subgroup of the wreath product Z2 wr Sa+b‑1,
with three cases depending on the parities of
a and b.
Bala Ravikumar. The Missing Link and the Top-Spin. Report TR94-228, Department of Computer
Science and Statistics, University of Rhode Island, Jan 1994. Top-Spin has a cycle of 20 pieces and a
small turntable which permits inverting a section of four pieces. After developing the group theory and doing
the Fifteen Puzzle and the Missing Link, he shows the state space of Top-Spin
is S20.
This is too big a topic to cover
completely. The first items should be
consulted for older material and the general history. Then I include material of particular interest. See also 6.BL which has some formulae which
are used to compute π. I have compiled a separate file on the
history of π.
Augustus De Morgan. A Budget of Paradoxes. (1872);
2nd ed., edited by D. E. Smith, (1915), Books for Libraries Press,
Freeport, NY, 1967.
J. W. Wrench Jr. The evolution of extended decimal
approximations to π. MTr 53 (Dec 1960) 644‑650. Good survey with 55 references, including
original sources.
Petr Beckmann. A History of π. The Golem Press,
Boulder, Colorado, (1970); 2nd ed.,
1971.
Lam Lay-Yong & Ang
Tian-Se. Circle measurements in ancient
China. HM 13 (1986) 325‑340. Good survey of the calculation of π
in China.
Dario Castellanos. The ubiquitous π. MM 61 (1988)
67-98 & 148-163. Good survey of
methods of computing π.
Joel Chan. As easy as pi. Math Horizons 1 (Winter 1993) 18-19. Outlines some recent work on calculating π and gives several
of the formulae used.
David Singmaster. A history of π. M500 168 (Jun 1999) 1-16. A chronology. (Thanks to Tony Forbes and Eddie Kent for carefully proofreading
and amending my file.)
Aristophanes. The Birds.
‑414. Lines 1001‑1005. In:
SIHGM I 308‑309. Refers to
'circle-squarers', possibly referring to the geometer/astronomer Meton.
E. J. Goodwin. Quadrature of the circle. AMM 1 (1894) 246‑247.
House Bill No. 246, Indiana
Legislature, 1897. "A bill for an
act introducing a new mathematical truth ..." In Edington's paper (below), p. 207, and in several of the
newspaper reports.
(Indianapolis)
Journal (19 Jan 1897) 3. Mentions the
Bill in the list of bills introduced.
Die
Quadratur des Zirkels. Täglicher
Telegraph (Indianapolis) (20 Jan 1897) ??.
Surveys attempts since -2000 and notes that Lindemann and Weierstrass
have shown that the problem is impossible, like perpetual motion.
A man
of 'genius'. (Indianapolis) Sun (6 Feb
1897) ??. An interview with Goodwin,
who says: "The astronomers have all been wrong. There's about 40,000,000 square miles on the surface of this earth
that isn't here." He says his
results are revelations and gives several rules for the circle and the sphere.
Mathematical
Bill passed. (Indianapolis) Journal (6
Feb 1897) 5. "This is the
strangest bill that has ever passed an Indiana Assembly." Gives whole text of the Bill.
Dr.
Goodwin's theaorem (sic) Resolution
adopted by the House of Representatives.
(Indianapolis) News (6 Feb 1897) 4.
Gives whole text of the Bill.
The
Mathematical Bill Fun-making in the
Senate yesterday afternoon -- other action.
(Indianapolis) News (13 Feb 1897) 11.
"The Senators made bad puns about it, ...." The Bill was indefinitely postponed.
House
Bills in the Senate. (Indianapolis)
Sentinel (13 Feb 1897) 2. Reports the
Bill was killed.
(No
heading??) (Indianapolis) Journal (13
Feb 1897) 3, col. 4. "...
indefinitely postponed, as not being a subject fit for legislation."
Squaring
the circle. (Indianapolis) Sunday
Journal (21 Feb 1897) 9. Says Goodwin
has solved all three classical impossible problems. Says π = 3.2, using
the fact that Ö2 = 10/7,
giving diagrams and a number of rules.
My
thanks to Underwood Dudley for locating and copying the above newspaper items.
C. A. Waldo. What might have been. Proc. Indiana Acad. Science 26 (1916) 445‑446.
W. E. Edington. House Bill No. 246, Indiana State
Legislature, 1897. Ibid. 45 (1935) 206‑210.
A. T. Hallerberg. House Bill No. 246 revisited. Ibid. 84 (1975) 374‑399.
Manuel H. Greenblatt. The 'legal' value of pi, and some related
mathematical anomalies. American
Scientist 53 (Dec 1965) 427A‑434A.
On p. 427A he tries to interpret the bill and obtains three different
values for π.
David Singmaster. The legal values of pi. Math. Intell. 7:2 (1985) 69‑72. Analyses Goodwin's article, Bill and other
assertions to find 23 interpretable statements giving 9 different values
of π !
Underwood Dudley. Mathematical Cranks. MAA, 1992.
Legislating pi, pp. 192-197.
C. T. Heisel. The Circle Squared Beyond Refutation. Published by the author, 657 Bolivar Rd.,
Cleveland, Ohio, 1st ed., 1931, printed by S. J. Monck, Cleveland; 2nd ed., 1934, printed by Lezius‑Hiles
Co., Cleveland, ??NX + Supplement: "Fundamental Truth",
1936, ??NX, distributed by the author
from 2142 Euclid Ave., Cleveland. This
is probably the most ambitious publication of a circle-squarer -- Heisel
distributed copies all around the world.
Underwood Dudley. πt: 1832-1879.
MM 35 (1962) 153-154. He plots
45 values of π as a function of time over the period
1832-1879 and finds the least-squares straight line which fits the data,
finding that πt =
3.14281 + .0000056060 t, for t
measured in years AD. Deduces
that the Biblical value of 3 was a good approximation for the time and that
Creation must have occurred when πt
= 0, which was in -560,615.
Underwood Dudley. πt. JRM 9 (1976-77) 178 & 180. Extends his previous work to 50 values
of π over 1826-1885, obtaining
πt = 4.59183 - .000773 t. The fact that πt
is decreasing is worrying -- when
πt = 1, all
circles will collapse into straight lines and this will certainly be the end of
the world, which is expected in 4646 on 9 Aug at 20:55:33 -- though this is
only the expected time and there is considerable variation in this
prediction. [Actually, I get that this
should be on 11 Aug. However, it seems
to me that circles will collapse once
πt = 2, as then
the circumference corresponds to going back and forth along the diameter. This will occur when t = 3352.949547, i.e. in 3352, on 13 Dec at 14:01:54 -- much earlier than Dudley's
prediction, so start getting ready now!]
See
Yates for a good survey of the field.
James Watt. UK Patent 1432 -- Certain New Improvements upon
Fire and Steam Engines, and upon Machines worked or moved by the same. Granted: 28 Apr 1784; complete specification: 24 Aug 1784. 14pp + 1 plate. Pp. 4-6 & Figures 7‑12 describe Watt's parallel
motion. Yates, below, p. 170 quotes one
of Watt's letters: "... though I
am not over anxious after fame, yet I am more proud of the parallel motion than
of any other invention I have ever made."
P. F. Sarrus. Note sur la transformation des mouvements
rectilignes alternatifs, en mouvements circulaires; et reciproquement. C. R. Acad. Sci. Paris 36 (1853) 1036‑1038. 6 plate linkage. The name should be Sarrus, but it is printed Sarrut on this and
the following paper.
Poncelet. Rapport sur une transformation nouvelle des
mouvements rectilignes alternatifs en mouvements circulaires et reciproquement,
par Sarrut. Ibid., 36 (1853) 1125‑1127.
A. Peaucellier. Lettre au rédacteur. Nouvelles Annales de Math. (2) 3 (1864) 414‑415. Poses the problem.
A. Mannheim. Proces‑Verbaux des sceances des 20 et
27 Juillet 1867. Bull. Soc.
Philomathique de Paris (1867) 124‑126.
??NYS. Reports Peaucellier's
invention.
Lippman Lipkin. Fortschritte der Physik (1871) 40‑?? ??NYS
L. Lipkin. Über eine genaue Gelenk‑Geradführung. Bull. Acad. St. Pétersbourg [=? Akad. Nauk,
St. Petersburg, Bull.] 16 (1871) 57‑60.
??NYS
L. Lipkin. Dispositif articulé pour la transformation
rigoureuse du mouvement circulaire en mouvement rectiligne. Revue Univers. des Mines et de la
Métallurgie de Liége 30:4 (1871) 149‑150. ??NYS. (Now spelled
Liège.)
A. Peaucellier. Note sur un balancier articulé a mouvement
rectiligne. Journal de Physique 2
(1873) 388‑390. (Partial English
translation in Smith, Source Book, vol. 2, pp. 324‑325.) Says he
communicated it to Soc. Philomath. in 1867 and that Lipkin has since also found
it. There is also an article in Nouv.
Annales de Math. (2) 12 (1873) 71‑78 (or 73?), ??NYS.
E. Lemoine. Note sur le losange articulé du Commandant
du Génie Peaucellier, destiné a remplacer le parallélogramme de Watt. J. de Physique 2 (1873) 130‑134. Confirms that Mannheim presented
Peaucellier's cell to Soc. Philomath. on 20 Jul 1867. Develops the inversive geometry of the cell.
[J. J. Sylvester.] Report of the Annual General Meeting of the
London Math. Soc. on 13 Nov 1873. Proc.
London Math. Soc. 5 (1873) 4 & 141.
On p. 4 is: "Mr. Sylvester
then gave a description of a new instrument for converting circular into
general rectilinear motion, and into motion in conics and higher plane curves,
and was warmly applauded at the close of his address." On p. 141 is an appendix saying that
Sylvester spoke "On recent discoveries in mechanical conversion of
motion" to a Friday Evening's Discourse at the Royal Institution on 23 Jan
1874. It refers to a paper 20 pages
long but is not clear if or where it was published.
H. Hart. On certain conversions of motion. Cambridge Messenger of Mathematics 4 (1874)
82‑88 and 116‑120 & Plate I.
Hart's 5 bar linkage. Obtains
some higher curves.
A. B. Kempe. On some new linkages. Messenger of Mathematics 4 (1875) 121‑124
& Plate I. Kempe's linkages for
reciprocating linear motion.
H. Hart. On two models of parallel motions. Proc. Camb. Phil. Soc. 3 (1876‑1880)
315‑318. Hart's parallelogram (a
5 bar linkage) and a 6 bar one.
V. Liguine. Liste des travaux sur les systèms
articulés. Bull. d. Sci. Math. 18 (or
(2) 7) (1883) 145‑160. ??NYS ‑
cited by Kanayama. Archibald; Outline
of the History of Mathematics, p. 99, says Linguine is entirely included in
Kanayama.
Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and
Novelties of Construction. A
second volume to accompany his previous Mechanical Movements, Powers and
Devices. Norman W. Henley Publishing
Co, NY, (1904), 2nd ed., 1910. This is
filled with many types of mechanisms.
Pp. 245-247 show five straight-line linkages and some related
mechanisms.
(R. Kanayama). (Bibliography on linkages. Text in Japanese, but references in roman
type.) Tôhoku Math. J. 37 (1933) 294‑319.
R. C. Archibald. Bibliography of the theory of linkages. SM 2 (1933‑34) 293‑294. Supplement to Kanayama.
Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge,
1941); revised ed., Educational Publishers,
St. Louis, 1949. Pp. 82-101 &
168-191. Gets up to outlining Kempe's
proof that any algebraic curve can be drawn by a linkage.
R. H. Macmillan. The freedom of linkages. MG 34 (No. 307) (Feb 1960) 26‑37. Good survey of the general theory of
linkages.
Michael Goldberg. Classroom Note 312: A six‑plate linkage in three
dimensions. MG 58 (No. 406) (Dec 1974)
287‑289.
Such
curves play an essential role in some ways to drill a square hole, etc.
L. Euler. Introductio in Analysin Infinitorum. Bousquet, Lausanne, 1748. Vol. 2, chap. XV, esp. § 355, p. 190 &
Tab. XVII, fig. 71. = Introduction to
the Analysis of the Infinite; trans. by John D. Blanton; Springer, NY,
1988-1990; Book II, chap. XV: Concerning curves with one or several diameters,
pp. 212-225, esp. § 355, p. 221 & fig. 71, p. 481. This doesn't refer to constant width, but
fig. 71 looks very like a Reuleaux triangle.
L. Euler. De curvis triangularibus. (Acta Acad. Petropol. 2 (1778(1781)) 3‑30) = Opera Omnia (1) 28 (1955) 298‑321. Discusses triangular versions.
M. E. Barbier. Note sur le problème de l'aiguille et jeu du
joint couvert. J. Math. pures appl. (2)
5 (1860) 273‑286. Mentions
that perimeter = π * width.
F. Reuleaux. Theoretische Kinematik; Vieweg,
Braunschweig, 1875. Translated: The Kinematics of Machinery. Macmillan, 1876; Dover, 1964. Pp. 129‑147.
Gardner D. Hiscox. Mechanical Appliances Mechanical Movements and
Novelties of Construction. A
second volume to accompany his previous Mechanical Movements, Powers and
Devices. Norman W. Henley Publishing
Co, NY, (1904), 2nd ed., 1910.
Item
642: Turning a square by circular motion, p. 247. Plain face, with four pins forming a centred square, is turned by
the lathe. A triangular follower is
against the face, so it is moved in and out as a pin moves against it. This motion is conveyed by levers to the
tool which moves in and out against the work which is driven by the same lathe.
Item
681: Geometrical boring and routing chuck, pp. 257-258. Shows it can make rectangles, triangles,
stars, etc. No explanation of how it
works.
Item
903A: Auger for boring square holes, pp. 353-354. Uses two parallel rotating cutting wheels.
H. J. Watts. US Patents 1,241,175‑7 -- Floating
tool‑chuck; Drill or boring
member; Floating tool‑chuck. Applied: 30 Nov 1915; 1 Nov 1916;
22 Nov 1916; all patented:
25 Sep 1917. 2 + 1, 2 + 1,
4 + 1 pp + pp diagrams. Devices for drilling square holes based on
the Reuleaux triangle.
T. Bonnesen & W.
Fenchel. Theorie der konvexen
Körper. Berlin, 1934; reprinted by Chelsea, 1971. Chap. 15: Körper konstanter Breite, pp. 127‑141. Surveys such curves with references to the
source material.
G. D. Chakerian & H.
Groemer. Convex bodies of constant
width. In: Convexity and Its Applications; ed. by Peter M. Gruber & Jörg
M. Wills; Birkhäuser, Boston, 1983.
Pp. 49‑96. (??NYS --
cited in MM 60:3 (1987) 139.)
Bibliography of some 250 items since 1930.
These
were discovered by Arthur H. Stone, an English graduate student at Princeton in
1939. American paper was a bit wider
than English and would not fit into his notebooks, so he trimmed the edge off
and had a pile of long paper strips which he played with and discovered the
basic flexagon. Fellow graduate
students Richard P. Feynman, Bryant Tuckerman and John W. Tukey joined in the
investigation and developed a considerable theory. One of their fathers was a patent attorney and they planned to
patent the idea and began to draw up an application, but the exigencies of the
1940s led to its being put aside, though knowledge of it spread as mathematical
folklore. E.g. Tuckerman's father,
Louis B. Tuckerman, lectured on it at the Westinghouse Science Talent Search in
the mid 1950s.
S&B, pp. 148‑149, show
several versions. Most square versions
(tetraflexagons or magic books) don't fold very far and are really just
extended versions of the Jacob's Ladder -- see 11.L
Martin Gardner. Cherchez la Femme [magic trick]. Montandon Magic Co., Tulsa, Okla.,
1946. Reproduced in: Martin Gardner Presents; Richard Kaufman and
Alan Greenberg, 1993, pp. 361-363.
[In: Martin Gardner Presents, p.
404, this is attributed to Gardner, but Gardner told me that Roger Montandon
had the copyright -- ?? I have learned
a little more about Gardner's early life -- he supported himself by inventing
and selling magic tricks about this time, so it may be that Gardner devised the
idea and sold it to Montandon.]. A
hexatetraflexagon.
"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and
Sciences, Croydon, 1947. A trick book,
pp. 42-43. Same hexatetraflexagon.
Sidney Melmore. A single‑sided doubly collapsible
tessellation. MG 31 (No. 294) (1947)
106. Forms a Möbius strip of three
triangles and three rhombi. He sees it
has two distinct forms, but doesn't see the flexing property!!
Margaret Joseph. Hexahexaflexagrams. MTr 44 (Apr 1951) 247‑248. No history.
William R. Ransom. A six‑sided hexagon. SSM 52 (1952) 94. Shows how to number the 6 faces.
No history.
F. G. Maunsell. Note 2449:
The flexagon and the hexahexaflexagram.
MG 38 (No. 325) (Sep 1954) 213‑214. States that Joseph is first article in the field and that this is
first description of the flexagon.
Gives inventors' names, but with Tulsey for Tukey.
R. E. Rogers & Leonard L.
D'Andrea. US Patent 2,883,195 -- Changeable
Amusement Devices and the Like.
Applied: 11 Feb 1955; patented:
21 Apr 1959. 2pp + 1p correction + 2pp
diagrams. Clearly shows the 9 and 18
triangle cases and notes that one can trim the triangles into hexagons so the
resulting object looks like six small hexagons in a ring.
M. Gardner. Hexa‑hexa‑flexagon and
Cherchez la femme. Hugard's
MAGIC Monthly 13:9 (Feb 1956) 391.
Reproduced in his: Encyclopedia
of Impromptu Magic; Magic Inc., Chicago, 1978, pp. 439-442. Describes hexahexa and the hexatetra of
Gardner/Montandon & Willane.
M. Gardner. SA (Dec 1956) = 1st Book, chap. 1. His first article in SA!!
Joan Crampin. Note 2672:
On note 2449. MG 41 (No. 335)
(Feb 1957) 55‑56. Extends to a
general case having 9n triangles of 3n colours.
C. O. Oakley & R. J.
Wisner. Flexagons. AMM 64:3 (Mar 1957) 143‑154.
Donovan A. Johnson. Paper Folding for the Mathematics
Class. NCTM, 1957, section 61,
pp. 24-25: Hexaflexagons.
Describes the simplest case, citing Joseph.
Roger F. Wheeler. The flexagon family. MG 42 (No. 339) (Feb 1958) 1‑6. Improved methods of folding and colouring.
M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Tetraflexagons and flexatube.
P. B. Chapman. Square flexagons. MG 45 (1961) 192‑194.
Tetraflexagons.
Anthony S. Conrad &
Daniel K. Hartline.
Flexagons. TR 62-11, RIAS, (7212
Bellona Avenue, Baltimore 12, Maryland,) 1962, 376pp. This began as a Science Fair project in 1956 and was then
expanded into a long report. The
authors were students of Harold V. McIntosh who kindly sent me one of the
remaining copies in 1996. They discover
how to make any chain of polygons into a flexagon, provided certain relations
among angles are satisfied. The
bibliography includes almost all the preceding items and adds the references to
the Rogers & D'Andrea patent, some other patents (??NYS) and a number of
ephemeral items: Conrad produced an
earlier RIAS report, TR 60-24, in 1960;
Allan Phillips wrote a mimeographed paper on hexaflexagons; McIntosh wrote an unpublished paper on flexagons; Mike Schlesinger wrote an unpublished paper
on Tuckerman tree theory.
Sidney H. Scott. How to construct hexaflexagons. RMM 12 (Dec 1962) 43‑49.
William R. Ransom. Protean shapes with flexagons. RMM 13 (Feb 1963) 35‑37. Describes 3‑D shapes that can be
formed. c= Madachy, below.
Robert Harbin. Party Lines. Op. cit. in 5.B.1.
1963. The magic book, pp.
124-125. As in Gardner's Cherchez la
Femme and Willane.
Pamela Liebeck. The construction of flexagons. MG 48 (No. 366) (Dec 1964) 397‑402.
Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Other flexagon diversions, pp. 76‑81. Describes 3‑D shapes that one can
form. Based on Ransom, RMM 13.
Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 66-75.
Describes various tetra- and hexa-flexagons.
Douglas A. Engel. Hexaflexagon + HFG = slipagon! JRM 25:3 (1993) 161-166. Describes his slipagons, which are linked
flexagons.
Robert E. Neale (154 Prospect
Parkway, Burlington, Vermont, 05401, USA).
Self-designing tetraflexagons.
12pp document received in 1996 describing several ways of making
tetraflexagons without having to tape or paste. He starts with a creased square sheet, then makes some internal
tears or cuts and then folds things through to miraculously obtain a
flexagon! A slightly rearranged version
appeared in: Elwyn R. Berlekamp
& Tom Rodgers, eds.; The Mathemagician and Pied Puzzler A Collection in Tribute to Martin
Gardner; A. K. Peters, Natick,
Massachusetts, 1999, pp. 117-126.
Jose R. Matos. US Patent 5,735,520 -- Fold-Through Picture
Puzzle. Applied: 7 Feb 1997; patented: 7 Apr 1998. Front page + 6pp diagrams + 13pp text. Robert Byrnes sent an example of the
puzzle. This is a square in thin
plastic, 100mm on an edge. Imagine
a 2 x 2 array of squares with their diagonals drawn. Fold along all the diagonals and between the
squares. This gives an array of 16
isosceles right triangles. Now cut from
the centres of the four squares to the centre of the whole array. This produces an X cut in the middle. This object can now be folded through itself
in various ways to produce a double thickness square of half the area with
various logos. The example is 100mm
along the edge of the large square and has four logos advertising Beanoland (at
Chessington, 3 versions) and Strip Cheese.
The patent is assigned to Lulirama International, but Byrnes says it has
not been a commercial success as it is too complicated. The patent cites 19 earlier patents, back to
1881, and discusses the history of such puzzles. It also says the puzzle can form three dimensional objects.
This
is the square cylindrical tube that can be inverted by folding. It was also invented by Arthur H. Stone,
c1939, cf 6.D.
J. Leech. A deformation puzzle. MG 39 (No. 330) (Dec 1955) 307. Doesn't know source. Says there are three solutions.
M. Gardner. Flexa-tube puzzle. Ibidem 7 (Sep 1956) 129.
Cites the inventors of the flexagons and the articles of Maunsell and
Leech (but he doesn't have its details).
(I have a note that this came with attached sample, but the copy I have
doesn't indicate such.)
T. S. Ransom. Flexa-tube solution. Ibidem 9 (Mar 1957) 174.
M. Gardner. SA (May 1958) = 2nd Book, chap. 2. Says Stone invented it and shows Ransom's
solution.
H. Steinhaus. Mathematical Snapshots. Not in the Stechert, NY, 1938, ed. nor the
OUP, NY, 1950 ed. OUP, NY: 1960: pp. 189‑193 & 326; 1969 (1983): pp. 177-181 & 303. Erroneous attribution to the Dowkers. Shows a different solution than Ransom's.
John Fisher. John Fisher's Magic Book. Muller, London, 1968. Homage to Houdini, pp. 152‑155. Detailed diagrams of the solution, but no
history.
Highland Games (2 Harpers Court,
Dingwall, Ross-Shire, IV15 9HT) makes a version called Table Teaser, made in a
strip with end pieces magnetic. Pieces
are coloured so to produce several folding and inverting problems other than
the usual one. Bought in 1995.
See
S&B, pp. 15‑18. See 6.F.1,
6.F.3, 6.F.4 & 6.F.5 for early occurrences of polyominoes. See Lammertink, 1996 & 1997 for many
examples in two and three dimensions.
NOTATION. Each of the types of puzzle considered has a
basic unit and pieces are formed from a number of these units joined edge to
edge. The notation N: n1, n2, .... denotes a puzzle with N
pieces, of which ni pieces consist of i basic units. If ni
are single digit numbers, the intervening commas and spaces will be omitted,
but the digits will be grouped by fives, e.g. 15: 00382 11.
Polyiamonds: Scrutchin;
John Bull; Daily Sketch; Daily Mirror; B. T.s Zig-Zag;
Daily Mail;
Miller (1960); Guy (1960); Reeve & Tyrrell;
O'Beirne (2 & 9 Nov 1961); Gardner (Dec 1964 & Jul 1965); Torbijn;
Meeus; Gardner (Aug 1975); Guy (1996, 1999); Knuth,
Polycubes: Rawlings (1939); Editor (1948); Niemann
(1948); French (1948); Editor (1948); Niemann (1948);
Gardner (1958); Besley
(1962); Gardner (1972)
Solid Pentominoes: Nixon (1948); Niemann (1948); Gardner
(1958); Miller (1960); Bouwkamp (1967, 1969, 1978); Nelson (2002);
Cylindrical Pentominoes: Yoshigahara (1992);
Polyaboloes: Hooper (1774); Book of 500 Puzzles (1859);
J. M. Lester (1919); O'Beirne
(21 Dec 61 &
18 Jan 62)
Polyhexes: Gardner (1967); Te Riele & Winter
Polysticks: Benjamin;
Barwell; General
Symmetrics; Wiezorke & Haubrich; Knuth;
Jelliss;
Polyrhombs or
Rhombiominoes: Lancaster (1918); Jones (1992).
Polylambdas: Roothart.
Polyspheres -- see Section 6.AZ.
GENERAL
REFERENCES
G. P. Jelliss. Special Issue on Chessboard
Dissections. Chessics 28 (Winter 1986)
137‑152. Discusses many problems
and early work in Fairy Chess Review.
Branko Grünbaum & Geoffrey
C. Shephard. Tilings and Patterns. Freeman, 1987. Section 9.4: Polyiamonds, polyominoes and polyhexes,
pp. 497-511. Good outline of the
field with a number of references otherwise unknown.
Michael Keller. A polyform timeline. World Game Review 9 (Dec 1989)
4-5. This outlines the history of
polyominoes and other polyshapes.
Keller and others refer to polyaboloes as polytans.
Rodolfo Marcelo Kurchan (Parana
960 5 "A", 1017 Buenos Aires, Argentina). Puzzle Fun, starting with No. 1 (Oct 1994). This is a magazine entirely devoted to
polyomino and other polyform puzzles.
Many of the classic problems are extended in many ways here. In No. 6 (Aug 1995) he presents a labelling
of the 12 hexiamonds by the letters A,
C, H, I, J, M, O, P, S, V, X, Y, which
he obtained from Anton Hanegraaf. I
have never seen this before.
Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, recreation
23, pp. 64-66. Considers figures formed
of isosceles right triangles. He has
eight of these, coloured with eight colours, and uses some of them to form
"chequers or regular four-sided figures, different either in form or
colour".
Book of 500 Puzzles. 1859.
Triangular problem, pp. 74-75.
Identical to Hooper, dropping the last sentence.
Dudeney. CP.
1907. Prob. 74: The broken
chessboard, pp. 119‑121 & 220‑221. The 12 pentominoes and a
2 x 2.
A. Aubry. Prob. 3224.
Interméd. Math 14 (1907) 122-124.
??NYS -- cited by Grünbaum & Shephard who say Aubry has something of
the idea or the term polyominoes.
G. Quijano. Prob. 3430.
Interméd. Math 15 (1908) 195. ??NYS
-- cited by Grünbaum & Shephard, who say he first asked for the number
of n‑ominoes.
Thomas Scrutchin. US Patent 895,114 -- Puzzle. Applied: 20 Feb 1908; patented: 4 Aug 1908. 2pp + 1p diagrams. Mentioned in S&B, p. 18.
A polyiamond puzzle -- triangle of side 8, hence with 64 triangles,
apparently cut into 10 pieces (my copy is rather faint -- replace??).
Thomas W. Lancaster. US Patent 1,264,944 -- Puzzle. Filed: 7 May 1917; patented: 7 May 1918. 2pp
+ 1p diagrams. For a general polyrhomb
puzzle making a rhombus. His diagram
shows an 11 x 11 rhombus filled with 19
pieces formed from 4 to
10 rhombuses.
John Milner Lester. US Patent 1,290,761 -- Game Apparatus. Filed: 6 Feb 1918; patented: 7 Jan 1919.
2pp + 3pp diagrams. Fairly general
assembly puzzle claims. He specifically
illustrates a polyomino puzzle and a polyabolo puzzle. The first has a Greek cross of edge 3
(hence containing 45 unit cells) to be filled with polyominoes
-- 11: 01154. The second has an 8-pointed star formed by superimposing two 4 x 4
squares. This has area 20
and hence contains 40 isosceles right triangles of edge 1, which is the basic unit of this type of
puzzle. There are 11: 0128 pieces.
Blyth. Match-Stick Magic.
1921. Spots and squares, pp.
68-73. He uses matchsticks broken in
thirds, so it is easier to describe with units of one-third. 6 units,
4 doubles and
2 triples. Some of the
pieces have black bands or spots.
Object is to form polyomino shapes without pieces crossing, but every
intersection must have a black spot.
19 polyomino shapes are given to
construct, including 7 of the pentominoes, though some of the
shapes are only connected at corners.
"John Bull" Star of
Fortune Prize Puzzle. 1922. This is a puzzle with 20
pieces, coloured red on one side, containing 6 through 13
triangles to be assembled into a star of David with 4
triangles along each edge (hence
12 x 16 = 192 triangles). Made by Chad Valley. Prize of £250 for a red star matching the
key solution deposited at a bank; £150
for solution closest to the key; £100
for a solution with 10 red and
10 grey pieces, or as nearly as
possible. Closing date of competition
is 27 Dec 1922. Puzzle made by Chad
Valley Co. as a promotional item for John Bull magazine, published by
Odhams Press. A copy is in the toy shop
of the Buckleys Shop Museum, Battle, East Sussex, to whom I am indebted for the
chance to examine the puzzle and a photocopy of the puzzle, box and solution.
Daily Sketch Jig-Saw
Puzzle. By Chad Valley. Card polyiamonds. 39: 0,0,1,5,6, 12,9,6,
with a path printed on one side, to assemble into a shape of 16
rows of 15 with four corners removed and so the printed
sides form a continuous circuit. In box
with shaped bottom. Instructions on
inside cover and loose sheet to submit solution. No dates given, but appears to be 1920s, though it is somewhat
similar to the Daily Mail Crown Puzzle of 1953 -- cf below -- so it might be
much later.
Daily Mirror Zig-Zag £500 Prize
Puzzle. By Chad Valley. Card polyiamonds. 29: 0,0,1,1,4, 5,6,3,1,4, 1,2,1.
One-sided pieces to fit into frame in card box. Three pieces are duplicated and one is
triplicated. Solution and claim
instructions appeared in Daily Mirror (17 Jan 1930) 1-2. See: Tom Tyler & Felicity Whiteley; Chad
Valley Promotional Jig-Saw Puzzles; Magic Fairy Publishing, Petersfield,
Hampshire, 1990, p. 55.
B. T.s Zig-Zag. B.T. is a Copenhagen newspaper. Polyiamond puzzle. 33: 0,0,1,2,5, 6,7,2,2,4, 1,1,1, Some repetitions, so I only see 20 different
shapes. To be fit into an irregular
frame. Solution given on 23 Nov 1931,
pp. 1-2. (I have a photocopy of the form
to fill in; an undated set of rules, apparently from the paper, saying the
solutions must be received by 21 Nov; and the pages giving the solution;
provided by Jan de Geus.)
Herbert D. Benjamin. Problem 1597: A big cutting-out design --
and a prize offer. Problemist Fairy
Chess Supplement (later called Fairy Chess Review) 2:9 (Dec 1934) 92. Finds the
35 hexominoes and asks if they
form a 14 x 15 rectangle.
Cites Dudeney (Tribune (20 Dec 1906)); Loyd (OPM (Apr-Jul 1908)) (see 6.F.1); Dudeney (CP, no. 74) (see above)
and some other chessboard dissections.
Jelliss says this is the first dissection problem in this journal.
F. Kadner. Solution 1597. Problemist Fairy Chess Supplement (later called Fairy Chess
Review) 2:10 (Feb 1935) 104-105. Shows
the 35
hexominoes cannot tile a rectangle by two arguments, both essentially
based on two colouring. Gives some other
results and some problems are given as 1679-1681 -- ??NYS.
William E. Lester. Correction to 1597. Problemist Fairy Chess Supplement (later
called Fairy Chess Review) 2:11 (Apr 1935) 121. Corrects an error in Kadner.
Finds a number of near-solutions.
Editor says Kadner insists the editor should take credit for the
two-colouring form of the previous proof.
Frans Hansson, proposer &
solver?. Problem 1844. Problemist Fairy Chess Supplement (later
called Fairy Chess Review) 2:12 (Jun 1935) 128
& 2:13 (Aug 1935) 135. Finds both
3 x 20 pentomino
rectangles.
W. E. Lester & B. Zastrow,
proposers. Problem 1923. Problemist Fairy Chess Supplement (later
called Fairy Chess Review) 2:13 (Aug 1935) 138. Take an 8 x 8 board and remove its corners. Fill this with the 12 pentominoes.
H. D. Benjamin, proposer. Problem 1924. Problemist Fairy Chess Supplement (later called Fairy Chess
Review) 2:13 (Aug 1935) 138. Dissect
an 8 x 8 into the 12 pentominoes and the I-tetromino. Need solution -- ??NYS.
Thomas Rayner Dawson &
William E. Lester. A notation for
dissection problems. Fairy Chess Review
3:5 (Apr 1937) 46‑47. Gives
all n‑ominoes up to n = 6.
Describes the row at a time notation.
Shows the pentominoes and a 2 x
2 cover the chessboard with the 2 x 2
in any position. Asserts there
are 108 7‑ominoes and
368 8‑ominoes -- citing F.
Douglas & W. E. L[ester] for the hexominoes and J. Niemann for the
heptominoes.
H. D. Benjamin, proposer. Problem 3228. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect a 5 x 5 into the five tetrominoes and a pentomino so
that the pentomino touches all the tetrominoes along an edge. Asserts the solution is unique. Refers to problems 3026‑3030 -- ??NYS.
H. D. Benjamin, proposer. Problem 3229. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect an 8 x 8 into the
12 pentominoes and a tetromino
so that all pieces touch the edge of the board. Asserts only one tetromino works.
T. R. D[awson], proposer. Problems 3230-1. Fairy Chess Review 3:12 (Jun 1938) 129. Extends prob. 3229 to ask for solutions with 12
pieces on the edge, using two other tetrominoes. Thinks it cannot be done with the remaining
two tetrominoes.
Editorial note: The colossal count. Fairy Chess Review 3:12 (Jun 1938) 131. Describes progress on enumerating 8-ominoes (four people get 368
but Niemann gets 369) and
9‑ominoes (numbers vary from
1237 to 1285).
All workers are classifying them by the size of the smallest containing
rectangle.
W. H. Rawlings, proposer. Problem 3930. Fairy Chess Review 4:3 (Nov 1939) 28. How many pentacubes are there?
Ibid. 4:4 (Feb 1940) 75, reports
that both 25 or 26 are claimed, but the editor has only seen 24.
Ibid. 4:5 (Apr 1940) 85, reports that R. J. F[rench] has
clearly shown there are 23 -- but this considers reflections as equal
-- cf the 1948 editorial note.
R. J. French, proposer and
solver. Problem 4149. Fairy Chess Review 4:3 (Nov 1939) 43 &
4:6 (Jun 1940) 93. Asks for
arrangement of the pentominoes with the largest hole and gives one with 127
squares in the hole. (See: G. P. Jelliss; Comment on Problem 1277; JRM
22:1 (1990) 69. This reviews various
earlier solutions and comments on Problem 1277.)
J. Niemann. Item 4154: "The colossal
count". Fairy Chess Review 4:3
(Nov 1939) 44-45. Announces that there
are 369 8-ominoes, 1285 9-ominoes and 4654 10-ominoes, but
Keller and Jelliss note that he missed a
10‑omino which was not corrected until 1966.
H. D. Benjamin. Unpublished notes. ??NYS -- cited and briefly described in G. P. Jelliss; Prob. 48
-- Aztec tetrasticks; G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss says Benjamin studied polysticks,
which he called 'lattice dissections' around 1946-1948 and that some results by
him and T. R. Dawson were entered in W. Stead's notebooks but nothing is known
to have been published. For orders 1, 2, 3, 4,
there are 1, 2, 5, 16 polysticks.
Benjamin formed these into a 6 x
6 lattice square. Jelliss then mentions Barwell's rediscovery
of them and goes on to a new problem -- see Knuth, 1999.
D. Nixon, proposer and
solver. Problem 7560. Fairy Chess Review 6:16 (Feb 1948) 12 &
6:17 (Apr 1948) 131.
Constructs 3 x 4 x 5 from solid pentominoes.
Editorial discussion: Space
dissection. Fairy Chess Review 6:18 (Jun
1948) 141-142. Says that several people
have verified the 23 pentacubes but that 6 of
them have mirror images, making 29 if these are considered distinct. Says F. Hansson has found 77 6‑cubes
(these exclude mirror images and the
35 solid 6-ominoes).
Gives many problems using
n-cubes and/or solid polyominoes, which he calls flat n-cubes -- some are corrected in 7:2 (Oct 1948)
16 (erroneously printed as 108).
J. Niemann. The dissection count. Item 7803.
Fairy Chess Review 7:1 (Aug 1948) 8 (erroneously printed as 100). Reports on counting n‑cubes. Gets the following.
n = 4 5 6 7
flat
pieces 5 12 35 108
non-flats 2 11 77 499
TOTAL 7 23 112 607
mirror
images 1 6 55 416
GRAND TOTAL 8 29 167 1023
R. J. French. Space dissections. Fairy Chess Review 7:2 (Oct 1948) 16 (erroneously printed as
108). French writes that he and A. W.
Baillie have corrected the number of
6-cubes to
35 + 77 + 54 = 166. Baillie notes that every 6-cube lies in two layers -- i.e. has some
width £ 2 -- and asks for the result for
n‑cubes as prob. 7879. [I
suspect the answer is that n £
3k implies that an n-cube has some width £ k.] Editor adds some corrections to the discussion in 6:18.
Editorial note. Fairy Chess Review 7:3 (Dec 1948) 23. Niemann and Hansson confirm the number 166
given in 7:2.
Daily Mail Crown Puzzle. Made by Chad Valley Co. 1953.
26 pieces, coloured on one side, to be fit into a crown shape. 11 are border pieces and easily placed. The other 15 are polyiamonds: 15: 00112 24012 11. Prize of £100 for solution plus best slogan,
entries due on 8 Jun 1953.
S. W. Golomb. Checkerboards and polyominoes. AMM 61 (1954) 675‑682. Mostly concerned with covering the 8 x 8
board with copies of polyominoes. Shows one covering with the
12 pentominoes and the square
tetromino. Mentions that the idea can
be extended to hexagons. S&B, p.
18, and Gardner (Dec 1964) say he mentions triangles, but he doesn't.
Walter S. Stead. Dissection.
Fairy Chess Review 9:1 (Dec 1954) 2‑4. Gives many pentomino and hexomino patterns -- e.g. one of each
pattern of 8 x 8 with a
2 x 2 square deleted. "The possibilities of the 12
fives are not infinite but they will provide years of
amusement." Includes 3 x 20,
4 x 15, 5 x 12 and
6 x 10 rectangles. No reference to Golomb. In 1955, Stead uses the 108
heptominoes to make a 28 x
28 square with a symmetric hole of
size 28 in the centre -- first printed as cover of Chessics 28 (1986).
Jules Pestieau. US Patent 2,900,190 -- Scientific
Puzzle. Filed: 2 Jul 1956; patented: 18 Aug 1959. 2pp + 1p diagrams. For the 12 pentominoes! Diagram shows the 6 x
10 solution with two 5 x 6
rectangles and shows the two-piece non-symmetric equivalence of the N
and F pieces. Pieces have
markings on one side which may be used -- i.e. pieces may not be turned
over. Mentions possibility of using n-ominoes.
Gardner. SA (Dec 1957) = 1st Book, chap. 13. Exposits Golomb and Stead. Gives number of n-ominoes for n = 1, ...,
7. 1st Book describes Scott's
work. Says a pentomino set called
'Hexed' was marketed in 1957. (John
Brillhart gave me and my housemates an example in 1960 -- it took us two weeks
to find our first solution.)
Dana Scott. Programming a Combinatorial Puzzle. Technical Report No. 1, Dept. of Elec. Eng.,
Princeton Univ., 1958, 20pp. Uses
MANIAC to find 65 solutions for pentominoes on an 8 x 8
board with square 2 x 2 in the centre. Notes that the 3 x
20 pentomino rectangle has just two
solutions. In 1999, Knuth notes that
the total number of solutions with the
2 x 2 being anywhere does not
seem to have ever been published and he finds
16146.
M. Gardner. SA (Sep 1958) c= 2nd Book, chap. 6. First general mention of solid pentominoes,
pentacubes, tetracubes. In the Addendum
in 2nd Book, he says Theodore Katsanis of Seattle suggested the eight
tetracubes and the 29 pentacubes in a letter to Gardner on
23 Sep 1957. He also says that
Julia Robinson and Charles W. Stephenson both suggested the solid pentominoes.
C. Dudley Langford. Note 2793:
A conundrum for form VI. MG 42
(No. 342) (Dec 1958) 287. 4 each of the
L, N, and T (= Y) tetrominoes make a 7 x 7 square with the
centre missing. Also nine pieces make a 6 x 6
square but this requires an even number of Ts.
M. R. Boothroyd &
J. H. Conway. Problems drive,
1959. Eureka 22 (Oct 1959) 15-17 &
22-23. No. 8. Use the pentominoes to make two
5 x 5 squares at the same time. Solution just says there are several ways to
do so.
J. C. P. Miller. Pentominoes. Eureka 23 (Oct 1960) 13‑16. Gives the Haselgroves' number of
2339 solutions for the 6 x 10
and says there are 2 solutions for the 3 x 20. Says Lehmer
suggests assembling 12 solid pentominoes into a 3 x 4 x 5
and van der Poel suggests assembling the 12 hexiamonds into a
rhombus.
C. B. & Jenifer
Haselgrove. A computer program for
pentominoes. Ibid., 16‑18. Outlines program which found the 2339
solutions for the 6 x 10. It is usually said that they also found all
solutions of the 3 x 20, 4 x 15
and 5 x 12, but I don't see it mentioned here and in JRM
7:3 (1974) 257, it is reported that Jenifer (Haselgrove) Leech stated that only
the 6 x 10 and 3 x 20 were done
in 1960, but that she did the 5
x 12 and 4 x 15 with a
new program in c1966. See Fairbairn,
c1962, and Meeus, 1973.
Richard K. Guy. Some mathematical recreations I &
II. Nabla [= Bull. Malayan Math.
Soc.] 7 (Oct & Dec 1960) 97-106 & 144-153. Considers handed polyominoes, i.e. polyominoes
when reflections are not considered equivalent. Notes that neither the
5 plain nor the 7
handed tetrominoes can form a rectangle. The 10 chequered handed tetrominoes form 4 x 10
and 5 x 8 rectangles and he has several solutions of
each. There is no 2 x 20
rectangle. Discusses MacMahon
pieces -- cf 5.H.2 -- and polyiamonds.
He uses the word 'hexiamond', but not 'polyiamond' -- in an email of 8
Apt 2000, Guy says that O'Beirne invented all the terms. He considers making a 'hexagon' from the 19
hexiamonds. Part II considers solid
problems and uses the term 'solid pentominoes'.
Solomon W. Golomb. The general theory of polyominoes: part 2 --
Patterns and polyominoes. RMM 5 (Oct
1961) 3-14. ??NYR.
J. E. Reeve & J. A.
Tyrrell. Maestro puzzles. MG 45 (No. 353) (Oct 1961) 97‑99. Discusses hexiamond puzzles, using the 12
reversible pieces. [The puzzle
was marketed under the name 'Maestro' in the UK.]
T. H. O'Beirne. Pell's equation in two popular
problems. New Scientist 12 (No. 258)
(26 Oct 1961) 260‑261.
T. H. O'Beirne. Pentominoes and hexiamonds. New Scientist 12 (No. 259) (2 Nov 1961) 316‑317. This is the first use of the word
'polyiamond'. He considers the 19
one‑sided pieces. He says
he devised the pieces and R. K. Guy has already published many solutions in
Nabla. He asks for the number of ways
the 18 one-sided pentominoes can fill a
9 x 10. In 1999, Knuth found
this would take several months.
T. H. O'Beirne. Some hexiamond solutions: and an introduction to a set of 25
remarkable points. New Scientist
12 (No. 260) (9 Nov 1961) 378‑379.
Maurice J. Povah. Letter.
MG 45 (No. 354) (Dec 1961) 342.
States Scott's result of 65 and the Haselgroves' result of 2339
(computed at Manchester). Says
he has over 7000 solutions for the 8 x 8 board using a 2 x 2.
T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751‑752.
T. H. O'Beirne. Some tetrabolic difficulties. New Scientist 13 (No. 270) (18 Jan 1962) 158‑159. These two columns are the first mention of
tetraboloes, so named by S. J. Collins.
R. A. Fairbairn. Pentomino Problems: The 6 x 10,
5 x 12, 4 x 15, and
3 x 20 Rectangles -- The
Complete Drawings. Unpublished MS,
undated, but c1962, based on the Haselgroves' work of 1960. ??NYS -- cited by various authors, e.g. Madachy (1969), Torbijn (1969), Meeus
(1973). Madachy says Fairbairn is from
Willowdale, Ontario, and takes some examples from his drawings. However, the dating is at variance with
Jenifer Haselgrove's 1973 statement - cf Haselgrove, 1960. Perhaps this MS is somewhat later?? Does anyone know where this MS is now? Cf Meeus, 1973.
Serena Sutton Besley. US Patent 3,065,970 -- Three Dimensional
Puzzle. Filed: 6 Jul 1960; issued: 27 Nov 1962. 2pp + 4pp diagrams. For the
29 pentacubes, with one piece
duplicated giving a set of 30. Klarner had already considered omitting
the 1 x 1 x 5 and found that he could make two separate 2 x 5 x 7s. Besley says the following can be made: 5 x 5 x 6, 3 x 5 x
10, 2 x 5 x 15, 2 x 3 x 25;
3 x 5 x 6, 3 x 3 x 10, 2 x 5 x 9,
2 x 3 x 15; 3 x 4 x 5, 2 x 5 x 6, 2 x 3 x 10
(where the latter three are made with the 12 solid pentominoes and
the previous four are made with the
18 non-planar pentacubes) but
detailed solutions are only given for the
5 x 5 x 6, 3 x 5 x 6, 3 x 4 x 5.
Mentions possibility of
n-cubes.
M. Gardner. Polyiamonds. SA (Dec 1964) = 6th Book, chap. 18. Exposits basic ideas and results for the 12 double sided
hexiamonds. Poses several problems
which are answered by readers. The
six-pointed star using 8 pieces has a unique solution. John G. Fletcher and Jenifer (Haselgrove)
Leech both showed the 3 x 12 rhombus is impossible. Fletcher found the 3 x 11 rhombus has 24
solutions, all omitting the 'bat'.
Leech found 155 solutions for the 6 x 6 rhombus and 74
solutions for the 4 x 9. Mentions there are 160 9-iamonds, one with a
hole.
John G. Fletcher. A program to solve the pentomino problem by
the recursive use of macros. Comm. ACM
8 (1965) 621-623. ??NYS -- described by
Knuth in 1999 who says that Fletcher found the 2339 solutions for the 6 x 10
in 10 minutes on an IBM 7094 and that the program remains the fastest
known method for problems of placing the 12 pentominoes.
M. Gardner. Op art.
SA (Jul 1965) = 6th Book, chap. 24.
Shows the 24 heptiamonds and discusses which will tile the plane.
Solomon W. Golomb. Tiling with polyominoes. J. Combinatorial Theory 1 (1966)
280-296. ??NYS. Extended by his 1970 paper.
T. R. Parkin. 1966.
??NYS -- cited by Keller.
Finds 4655 10-ominoes.
M. Gardner. SA (Jun 1967) = Magic Show, chap. 11. First mention of polyhexes.
C. J. Bouwkamp. Catalogue of Solutions of the
Rectangular 3 x 4 x 5 Solid Pentomino Problem. Dept. of Math., Technische Hogeschool
Eindhoven, July 1967, reprinted 1981, 310pp.
C. J. Bouwkamp. Packing a rectangular box with the twelve
solid pentominoes.
J. Combinatorial Thy. 7 (1969) 278‑280. He gives the numbers of solutions for
rectangles as 'known'.
2 x 3 x 10 can be packed in
12 ways, which are given.
2 x 5 x
6 can be packed in 264
ways.
3 x 4 x
5 can be packed in 3940
ways. (See his 1967 report.)
T. R. Parkin, L. J. Lander & D. R. Parkin. Polyomino enumeration results. Paper presented at the SIAM Fall Meeting,
Santa Barbara, 1 Dec 1967. ??NYS --
described by Madachy, 1969. Gives
numbers of n-ominoes, with and without
holes, up to n = 15, done two independent ways.
Joseph S. Madachy. Pentominoes -- Some solved and unsolved
problems. JRM 2:3 (Jul 1969)
181-188. Gives the numbers of Parkin,
Lander & Parkin. Shows various
examples where a rectangle splits into two congruent halves. Discusses various other problems, including
Bouwkamp's 3 x 4 x 5 solid pentomino problem. Bouwkamp reports that the final total of
3940 was completed on 16 Mar 1967 after about three years work using three
different computers, but that a colleague's program would now do the whole
search in about three hours.
P. J. Torbijn. Polyiamonds. JRM 2:4 (Oct 1969) 216-227.
Uses the double sided hexiamonds and heptiamonds. A few years before, he found, by hand, that
there are 156 ways to cover the 6 x 6 rhombus with the 12 hexiamonds and 74
ways for the 4 x 9, but could find no way to cover the 3 x 12.
The previous year, John G. Fletcher confirmed these results with a
computer and he displays all of these -- but this contradicts Gardner (Dec 64)
-- ?? He gives several other problems
and results, including using the 24 heptiamonds to form 7 x 12,
6 x 14, 4 x 21 and
3 x 28 rhombuses.
Solomon W. Golomb. Tlling with sets of polyominoes. J. Combinatorial Theory 9 (1970) 60‑71. ??NYS.
Extends his 1966 paper. Asks
which heptominoes tile rectangles and says there are two undecided cases -- cf
Marlow, 1985. Gardner (Aug 75) says
Golomb shows that the problem of determining whether a given finite set of
polyominoes will tile the plane is undecidable.
C. J. Boukamp & D. A.
Klarner. Packing a box with Y-pentacubes. JRM 3:1 (1970) 10-26.
Substantial discussion of packings with
Y‑pentominoes and
Y-pentacubes. Smallest boxes
are 5 x 10 and 2 x 5 x 6 and
3 x 4 x 5.
Fred Lunnon. Counting polyominoes. IN:
Computers in Number Theory, ed. by A. O. L. Atkin &
B. J. Birch; Academic Press, 1971, pp. 347-372. He gets up through 18‑ominoes, but the larger ones can have
included holes. The numbers for n = 1, 2, ..., are as follows: 1, 1, 2,
5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107911, 192622052. These values have been quoted numerous
times.
Fred Lunnon. Counting hexagonal and triangular
polyominoes. IN: Graph Theory and Computing, ed. by R. C.
Read; Academic Press, 1972, pp. 87-100.
??NYS -- cited by Grünbaum & Shephard.
M. Gardner. SA (Sep 1972). c= Knotted, chap. 3. Says
the 8
tetracubes were made by E. S. Lowe Co. in Hong Kong and
marketed as "Wit's End". Says
an MIT group found 1390 solutions for the 2 x 4 x 4 box packed with
tetracubes. He reports that several
people found that there are 1023 heptacubes -- but see Niemann, 1948, above. Klarner reports that the heptacubes fill
a 2 x 6 x 83.
Jean Meeus. Some polyomino and polyamond [sic]
problems. JRM 6:3 (1973) 215-220. (Corrections in 7:3 (1974) 257.) Considers ways to pack a 5 x n
rectangle with some n pentominoes. A. Mank found the number of ways for n = 2, 3, ..., 11 as
follows, and the number for n = 12 was already known:
0,
7, 50, 107, 541,
1387, 3377, 5865, 6814,
4103, 1010.
Says
he drew out all the solutions for the area 60 rectangles in 1972 (cf Fairbairn,
c1962). Finds that 520
of the 6 x 10 rectangles can be divided into two congruent
halves, sometimes in two different ways.
For 5 x 12, there are
380; for 4 x 15,
there are 94. Gives some
hexomino rectangles by either deleting a piece or duplicating one, and an
'almost 11 x 19'. Says there are 46 solutions to the 3 x 30
with the 18 one-sided pentominoes and attributes this to Mrs
(Haselgrove) Leech, but the correction indicates this was found by A. Mank.
Jenifer Haselgrove. Packing a square with Y-pentominoes. JRM 7:3 (1974) 229. She finds and shows a way to pack 45
Y-pentominoes into a 15 x 15, but is unsure if there are more
solutions. In 1999, Knuth found 212
solutions. She also reports the
impossibility of using the Y-pentominoes to fill various other rectangles.
S. W. Golomb. Trademark for 'PENTOMINOES'. US trademark 1,008,964 issued
15 Apr 1975; published 21 Jan
1975 as SN 435,448. (First use: November 1953.) [These appear in the Official Gazette of the United States Patent
Office (later Patent and Trademark Office) in the Trademarks section.]
M. Gardner. Tiling with polyominoes, polyiamonds and
polyhexes. SA (Aug 75) (with slightly
different title) = Time Travel, chap. 14.
Gives a tiling criterion of Conway.
Describes Golomb's 1966 & 1970 results.
C. J. Bouwkamp. Catalogue of
solutions of the rectangular 2 x 5 x
6 solid pentomino problem. Proc. Koninklijke Nederlandse Akad. van
Wetenschappen A81:2 (1978) 177‑186.
Presents the 264 solutions which were first found in Sep
1967.
H. Redelmeier. Discrete Math.
36 (1981) 191‑203. ??NYS --
described by Jelliss. Obtains number of
n‑ominoes for n £ 24.
Karl Scherer. Problem 1045: Heptomino tessellations. JRM 14:1 (1981‑82) 64. XX
Says he has found that the heptomino at the right fills
a 26 x 42 rectangle. XXXXX
See
Dahlke below.
David Ellard. Poly-iamond enumeration. MG 66 (No. 438) (Dec 1982) 310‑314. For
n = 1, ..., 12,
he gets 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3342
n-iamonds. One of the 8-iamonds has a hole and there are many
later cases with holes.
Anon. 31: Polyominoes. QARCH
1:8 (June 1984) 11‑13. [This is
an occasional publication of The Archimedeans, the student maths society at
Cambridge.] Good survey of counting and
asymptotics for the numbers of polyominoes, up to n = 24, polycubes,
etc. 10 references.
T. W. Marlow. Grid dissections. Chessics 23 (Autumn, 1985) 78‑79.
X XX
Shows XXXXX
fills a 23 x 24 and
XXXXX fills a 19 x 28.
Herman J. J. te Riele & D.
T. Winter. The tetrahexes puzzle. CWI Newsletter [Amsterdam] 10 (Mar 1986) 33‑39. Says there are: 7 tetrahexes, 22
pentahexes, 82 hexahexes,
333 heptahexes, 1448 octahexes.
Studies patterns of 28 hexagons.
Shows the triangle cannot be constructed from the 7
tetrahexes and gives 48 symmetric patterns that can be made.
Karl A. Dahlke. Science News 132:20 (14 Nov 1987) 310. (??NYS -- cited in JRM 21:3 and
XX
22:1
and by Marlow below.) Shows XXXXX
fills a 21 x 26 rectangle.
The
results of Scherer and Dahlke are printed in JRM 21:3 (1989) 221‑223 and
Dahlke's solution is given by Marlow below.
Karl A. Dahlke. J. Combinatorial Theory A51 (1989) 127‑128. ??NYS -- cited in JRM 22:1. Announces a
19 x 28 solution for the above heptomino
problem, but the earlier
21 x 26 solution is
printed by error. The 19 x 28
solution is printed in JRM 22:1 (1990) 68‑69.
Tom Marlow. Grid dissections. G&PJ 12 (Sep/Dec 1989) 185.
Prints Dahlke's result.
Brian R. Barwell. Polysticks.
JRM 22:3 (1990) 165-175.
Polysticks are formed of unit lengths on the square lattice. There are:
1, 2, 5, 16, 55 polysticks
formed with 1, 2, 3, 4,
5 unit lengths. He forms
5 x 5 squares with one 4-stick omitted, but he permits pieces to
cross. He doesn't consider the
triangular or hexagonal cases. See also
Blyth, 1921, for a related puzzle. Cf
Benjamin, above, and Wiezorke & Haubrich, below.
General Symmetrics (Douglas
Engel) produced a version of polysticks, ©1991, with 4 3‑sticks and 3
4-sticks to make a 3 x 3 square array with no crossing of pieces.
Nob Yoshigahara. Puzzlart.
Tokyo, 1992. Ip-pineapple
(pineapple delight), pp. 78-81. Imagine
a cylindrical solution of the 6 x
10 pentomino rectangle and wrap it
around a cylinder, giving each cell a depth of one along the radius. Hence each cell is part of an annulus. He reduces the dimensions along the short
side to make the cells look like tenths of a slice of pineapple. Nob constructed and example for Toyo Glass's
puzzle series and it was later found to have a unique solution.
Kate Jones, proposer; P. J. Torbijn, Jacques Haubrich,
solvers. Problem 1961 --
Rhombiominoes. JRM 24:2 (1992) 144-146 &
25:3 (1993) 223‑225. A
rhombiomino or polyrhomb is a polyomino formed using rhombi instead of
squares. There are 20
pentarhombs. Fit them into a 10 x 10
rhombus. Various other
questions. Haubrich found many
solutions. See Lancaster, 1918.
Bernard Wiezorke & Jacques
Haubrich. Dr. Dragon's polycons. CFF 33 (Feb 1994) 6-7. Polycons (for connections) are the same as
the polysticks described by Barwell in 1990, above. Authors describe a Taiwanese version on sale in late 1993,
using 10 of the 4‑sticks
suitably shortened so they fit into the grooves of a 4 x 4 board -- so crossings
are not permitted. (An n x n
board has n+1 lines of
n edges in each direction.) They fit
15 of the 4-sticks onto a 5 x 5 board and determine
all solutions.
CFF
35 (Dec 1994) 4 gives a number of responses to the article. Brain Barwell wrote that he devised them as
a student at Oxford, c1970, but did not publish until 1990. He expected someone to say it had been done
before, but no one has done so. He also
considered using the triangular and hexagonal lattice. He had just completed a program to consider
fitting 15 of the 4-sticks onto a 5
x 5 board and found over 180,000
solutions, with slightly under half having no crossings, confirming the
results of Wiezorke & Haubrich.
Dario
Uri also wrote that he had invented the idea in 1984 and called them polilati
(polyedges). Giovanni Ravesi wrote
about them in Contromossa (Nov 1984) 23 -- a defunct magazine.
Chris Roothart. Polylambdas. CFF 34 (Oct 1994) 26-28.
A lambda is a 30o-60o-90o triangle.
These may be joined along corresponding legs, but not along
hypotenuses. For n = 1, 2, 3, 4, 5, there are 1, 4, 4, 11, 12 n-lambdas.
He gives some problems using various sets of these pieces.
Richard Guy. Letters of 29 May and 13 Jun 1996. He is interested in using the 19
one-sided hexiamonds. Hexagonal
rings of hexagons contain 1, 6, 12 hexagons, so the hexagon with three hexagons
on a side has 19 hexagons. If these
hexagons are considered to comprise six equilateral triangles, we have a board
with 19 x 6 triangles. O'Beirne asked
for the number of ways to fill this board with the one-sided hexiamonds. Guy has collected over 4200
solutions. A program by Marc
Paulhus found 907 solutions in eight hours, from which it
initially estimated that there are about
30,000 solutions. The second letter gives the final results --
there are 124,518 solutions.
This is modulo the 12 symmetries of the hexagon. In 1999, Knuth found 124,519
and Paulhus has rerun his program and found this number.
Ferdinand Lammertink. Polyshapes.
Parts 1 and 2. The author,
Hengelo, Netherlands, 1996 & 1997.
Part 1 deals with two dimensional puzzles. Good survey of the standard polyform shapes and many others.
Hilarie Korman. Pentominoes: A first player win. IN: Games of No Chance; ed. by Richard
Nowakowski; CUP, 1997??, ??NYS - described in
William Hartston; What mathematicians get up to; The Independent Long
Weekend (29 Mar 1997) 2. This studies
the game proposed by Golomb -- players alternately place one of the pentominoes
on the chess board, aligned with the squares and not overlapping the previous
pieces, with the last one able to play being the winner. She used a Sun IPC Sparcstation for five
days, examining about 22 x 109 positions to show the game is a first player
win.
Nob Yoshigahara found in 1994
that the smallest box which can be packed with W-pentacubes is 5 x 6 x 6.
In 1997, Yoshya (Wolf) Shindo found that one can pack the 6 x 10 x 10
with Z-pentacubes, but it is not known if this is the smallest such
box. These were the last unsolved
problems as to whether a box could be packed with a planar pentacube
(= solid pentomino).
Marcel Gillen &
Georges Philippe. Twinform 462 Puzzles in one. Solutions for Gillen's puzzle exchange at
IPP17, 1997, 32pp + covers. Take 6 of
the pentominoes and place them in a 7 x
5 rectangle, then place the other six
to make the same shape on top of the first shape. There are 462 (= BC(12,6)/2) possible puzzles and all of them have solutions. Taking
F, T, U, W, X, Z for the first layer,
there is just one solution; all other cases have multiple solutions,
totalling 22,873 solutions, but only one solution for each
case is given here.)
Richard K. Guy. O'Beirne's hexiamond. In:
The Mathemagican and Pied Puzzler; ed. by Elwyn Berlekamp & Tom
Rodgers, A. K. Peters, Natick, Massachusetts, 1999, pp. 85‑96. He relates that O'Beirne discovered the 19
one-sided hexiamonds in c1959 and found they would fill a hexagonal shape in
Nov 1959 and in Jan 1960 he found a solution with the hexagonal piece in the
centre. He gives Paulhus's results (see
Guy's letters of 1996), broken down in various ways. He gives the number of double-sided (i.e. one can turn them over)
and single-sided n-iamonds for n = 1, ..., 7. Cf Ellard, 1982, for many more values for the double-sided case.
n 1 2
3 4 5 6 7
double 1
1 1 3 4 12
24
single 1
1 1 4 6 19
44
In
1963, Conway and Mike Guy considered looking for 'symmetric' solutions for
filling the hexagonal shape with the 19 one-sided hexiamonds. A number of these are described.
Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by the
author. Available as:
http://www-cs-faculty.stanford.edu/~knuth/preprints.html . In this he introduces a new technique for
backtrack programming which runs faster (although it takes more storage) and is
fairly easy to adapt to different problems.
In this approach, there is a symmetry between pieces and cells. He applies it to several polyshape problems,
obtaining new, or at least unknown, results.
He extends Scott's 1958 results to get
16146 ways to pack the 8 x 8
with the 12 pentominoes and the
2 x 2. He describes
Fletcher's 1965 work. He extends
Haselgrove's 1974 work and finds 212 ways to fit 15 Y-pentominoes in a 15 x 15.
Describes Torbijn's 1969 work and Paulhus' 1996 work on hexiamonds,
correcting the latter's number to
124,519. He then looks for the
most symmetric solutions for filling the hexagonal shape with the 19 one-sided
hexiamonds, in the sense discussed by Guy (1999). He then considers the 18 one-sided pentominoes (cf Meeus (1973))
and tries the 9 x 10, but finds it would take a few months on his
computer (a 500 MHz Pentium III), so he's abandoned it for now. He then considers polysticks, citing an
actual puzzle version that I've not seen.
He adapts his program to them.
He considers the 'welded tetrasticks' which have internal junction
points. There are six of these and ten
if they are taken as one-sided. The ten
can be placed in a 4 x 4 grid. There are
15 unwelded, one-sided,
tetrasticks, but they do not form a square, nor indeed any nice shape. He considers all 25 one-sided tetrasticks
and asks if they can be fit into what he calls an Aztec Diamond, which is the
shape looking like a square tilted 45o on the square lattice. The rows contain 1, 3, 5, 7, 9, 7, 5, 3, 1
cells. He thinks an exhaustive
search is beyond present computing power.
G. P. Jelliss. Prob. 48 -- Aztec tetrasticks. G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss first discusses Benjamin's work on
polysticks (see at 1946-1948 above) and Barwell's rediscovery of them (see
above). He then describes Knuth's Dancing
Links and gives the Aztec Diamond problem.
Jelliss has managed to get all but one of the polysticks into the shape,
but feels it is impossible to get them all in.
Harry L. Nelson. Solid pentomino storage, Question and
answer. 1p HO at G4G5, 2002. 1: Can one put all the solid
pentominoes into a cube of edge
4.5? What is the smallest cube
into which they can all be placed? He
gives 2 solutions to 1 and a solution due to Wei-Hwa Huang for a cube of edge
4.405889..., which is conjectured to be minimal. In fact, one edge of the packing is actually 4, so the volume is
less than (4.405889...)3. This leads me to ask what is the smallest
volume of a cuboid, with edges less than 5, that contains all the solid
pentominoes. In Summer 2002, Harry gave
me a set of solid pentominoes in a box with a list of various rectangles and
boxes to fit them into: 3 x 22; 3 x 21;
3 x 20; 4 x 16; 4 x 15;
5 x 13; 5 x 12; 6 x 11; 6 x 10; 7 x 9; 8 x 8;
2 x 4 x 8; 2
x 5 x 7; 2 x 5 x 6; 2 x 6 x 6;
3 x 4 x 6; 3 x 4 x
5; 3 x 5 x 5; 4 x 4 x 5; the given
box: 4.4 x 4.4 x 4.9.)
6.F.1. OTHER CHESSBOARD DISSECTIONS
See
S&B, pp. 12‑14. See also
6.F.5 for dissections of uncoloured boards.
Jerry Slocum. Compendium of Checkerboard Puzzles. Published by the author, 1983. Outlines the history and shows all manufactured
versions known then to him: 33 types in
61 versions. The first number in
Slocum's numbers is the number of pieces.
Jerry Slocum &
Jacques Haubrich. Compendium of
Checkerboard Puzzles. 2nd ed.,
published by Slocum, 1993. 90 types in
161 versions, with a table of which pieces are in which puzzles, making it much
easier to see if a given puzzle is in the list or not. This gives many more pictures of the puzzle
boxes and also gives the number of solutions for each puzzle and sometimes prints
all of them. The Slocum numbers are
revised in the 2nd ed. and I use the 2nd ed. numbers below. (There was a 3rd ed. in 1997, with new
numbering of 217 types in 376 versions.
NYR. Haubrich is working on an
extended version with Les Barton providing information.)
Henry Luers. US Patent 231,963 -- Game Apparatus or
Sectional Checker Board. Applied:
7 Aug 1880; patented: 7 Sep 1880. 1p + 1p diagrams. 15: 01329. Slocum
15.5.1. Manufactured as: Sectional
Checker Board Puzzle, by Selchow & Righter. Colour photo of the puzzle box cover is on the front cover of the
1st ed. of Slocum's booklet. B&W
photo is on p. 14 of S&B.
?? UK patent application 16,810.
1892. Not granted, so never
published. I have spoken to the UK
Patent Office and they say the paperwork for ungranted applications is
destroyed after about three to five years.
(Edward Hordern's collection has an example with this number on it, by
Feltham & Co. In the 2nd ed., the
cover is reproduced and it looks like the number may be 16,310, but that number
is for a locomotive vehicle.)
14: 00149. Slocum
14.20.1. Manufactured as: The Chequers
Puzzle, by Feltham & Co.
Hoffmann. 1893.
Chap. III, no. 16: The chequers puzzle, pp. 97‑98 & 129‑130
= Hoffmann‑Hordern, pp. 88-89, with photos. 14: 00149.
Slocum 14.20.1. Says it is made
by Messrs. Feltham, who state it has over 50 solutions. He gives two solutions. Photo on p. 89 of a example by Feltham &
Co., dated 1880-1895.
At
the end of the solution, he says Jacques & Son are producing a series of
three "Peel" puzzles, which have coloured squares which have to be
arranged so the same colour is not repeated in any row or column. Photo on p. 89 shows an example, 9: 023,
with the trominoes all being L-trominoes. This makes a 5 x 5 square, but the colours have almost faded
into indistinguishability.
Montgomery Ward & Co. Catalog No 57, Spring &
Summer, 1895. Facsimile by Dover, 1969,
??NX. P. 237 describes item 25470: "The "Wonder" Puzzle. The object is to place 18 pieces of 81
squares together, so as to form a square, with the colors running
alternately. It can be done in several
different ways."
Dudeney. Problem 517 -- Make a chessboard. Weekly Dispatch (4 & 18 Oct 1903), both
p. 10. 8: 00010 12111 001. Slocum 8.3.1.
Benson. 1904.
The chequers puzzle, pp. 202‑203.
As in Hoffmann, with only one solution.
Dudeney. The Tribune (20 & 24 Dec 1906) both
p. 1. ??NX Dissecting a chessboard.
Dissect into maximum number of different pieces. Gets 18: 2,1,4,10,0, 0,0,1. Slocum 18.1, citing later(?) Loyd versions.
Loyd. Sam Loyd's Puzzle Magazine (Apr-Jul 1908) -- ??NYS, reproduced
in: A. C. White; Sam Loyd and His Chess
Problems; 1913, op. cit. in 1; no. 58, p. 52.
= Cyclopedia, 1914, pp. 221 & 368, 250 & 373. = MPSL2, prob. 71, pp. 51 &
145. = SLAHP: Dissecting the
chessboard, pp. 19 & 87. Cut into
maximum number of different pieces -- as in Dudeney, 1906.
Burren Loughlin &
L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909.
The rug, pp. 7-13 & 65. 14:
00149. Not in Slocum.
Loyd. A battle royal.
Cyclopedia, 1914, pp. 97 & 351 (= MPSL1, prob. 51, pp. 49 &
139). Same as Dudeney's prob. 517 of
1903.
Dudeney. AM. 1917.
Prob. 293: The Chinese chessboard, pp. 87 & 213‑214. Same as Loyd, p. 221.
Western Puzzle Works, 1926
Catalogue. No. 79: "Checker Board
Puzzle, in 16 pieces", but the picture only shows 14 pieces. 14: 00149.
Picture doesn't show any colours, but assuming the standard colouring of
a chess board, this is the same as Slocum 14.15.
John Edward Fransen. US Patent 1,752,248 -- Educational
Puzzle. Applied: 19 Apr 1929; patented: 25 Mar 1930. 1p + 1p diagrams. 'Cut thy life.'
11: 10101 43001.
Slocum 11.3.1.
Emil Huber-Stockar. Patience de l'echiquier. Comptes-Rendus du Premier Congrès
International de Récréation Mathématique, Bruxelles, 1935. Sphinx, Bruxelles, 1935, pp. 93-94. 15: 01329. Slocum 15.5. Says there
must certainly be more than 1000 solutions.
Emil Huber-Stockar. L'echiquier du diable. Comptes-Rendus du Deuxième Congrès
International de Récréation Mathématique, Paris, 1937. Librairie du "Sphinx", Bruxelles,
1937, pp. 64-68. Discusses how one
solution can lead to many others by partial symmetries. Shows several solutions containing about 40
altogether. Note at end says he has now
got 5275 solutions. This article
is reproduced in Sphinx 8 (1938) 36-41, but without the extra pages of
diagrams. At the end, a note says he
has 5330 solutions. Ibid., pp.
75-76 says he has got 5362 solutions and ibid. 91-92 says he has 5365.
By use of Bayes' theorem on the frequency of new solutions, he
estimates c5500 solutions.
Haubrich has found 6013.
Huber-Stocker intended to produce a book of solutions, but he died in
May 1939 [Sphinx 9 (1939) 97].
F. Hansson. Sam Loyd's 18-piece dissection -- Art. 48
& probs. 4152‑4153. Fairy
Chess Review 4:3 (Nov 1939) 44. Cites
Loyd's Puzzles Magazine. Asserts there
are many millions of solutions! He
determines the number of chequered handed
n-ominoes for
n = 1, 2, ..., 8
is 2, 1, 4, 10, 36, 110, 392,
1371. The first 17 pieces total 56
squares. Considers 8 ways to dissect
the board into 18 different pieces.
Problems ask for the number of ways to choose the pieces in each of
these ways and for symmetrical solutions.
Solution in 4:6 (Jun 1940) 93-94 (??NX of p. 94) says there are a total
of 3,309,579 ways to make the choices.
C. Dudley Langford. Note 2864:
A chess‑board puzzle. MG
43 (No. 345) (Oct 1959) 200. 15:
01248. Not in Slocum. Two diagrams followed by the following
text. "The pieces shown in the
diagrams can be arranged to form a square with either side uppermost. If the squares of the underlying grid are
coloured black and white alternately, with each white square on the back of a
black square, then there is at least one more way of arranging them as a
chess-board by turning some of the pieces over." I thought this meant that the pieces were double-sided with the
underside having the colours being the reverse of the top and the two diagrams
were two solutions for this set of pieces.
Jacques Haubrich has noted that the text is confusing and that the
second diagram is NOT using the set of double-sided pieces which are implied by
the first diagram. We are not sure if
the phrasing is saying there are two different sets of pieces and hence two
problems or if we are misinterpreting the description of the colouring.
B. D. Josephson. EDSAC to the rescue. Eureka 24 (Oct 1961) 10‑12 &
32. Uses the EDSAC computer to find two
solutions of a 12 piece chessboard dissection.
12: 00025 41. Slocum 12.9.
Leonard J. Gordon. Broken chessboards with unique
solutions. G&PJ 10 (1989) 152‑153. Shows Dudeney's problem has four
solutions. Finds other colourings which
give only one solution. Notes some
equivalences in Slocum.
6.F.2. COVERING DELETED CHESSBOARD WITH DOMINOES
See
also 6.U.2.
There
is nothing on this in Murray.
Pál Révész. Op. cit. in 5.I.1. 1969. On p. 22, he says
this problem comes from John [von] Neumann, but gives no details.
Max Black. Critical Thinking, op. cit. in 5.T. 1946 ed., pp. 142 & 394, ??NYS. 2nd ed., 1952, pp. 157 & 433. He simply gives it as a problem, with no
indication that he invented it.
H. D. Grossman. Fun with lattice points: 14 -- A chessboard
puzzle. SM 14 (1948) 160. (The problem is described with 'his clever
solution' from M. Black, Critical Thinking, pp. 142 & 394.)
S. Golomb. 1954.
Op. cit. in 6.F.
M. Gardner. The mutilated chessboard. SA (Feb 1957) = 1st Book, pp. 24 & 28.
Gamow & Stern. 1958.
Domino game. Pp. 87‑90.
Robert S. Raven, proposer; Walter P. Targoff, solver. Problem 85 -- Deleted checkerboard. In: L.
A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959, pp. 52
& 227.
R. E. Gomory. (Solution for deletion of any two squares of
opposite colour.) In: M. Gardner, SA (Nov 1962) = Unexpected, pp.
186‑187. Solution based on a
rook's tour. (I don't know if this was
ever published elsewhere.)
Michael Holt. What is the New Maths? Anthony Blond, London, 1967. Pp. 68 & 97. Gives the
4 x 4 case as a
problem, but doesn't mention that it works on other boards. (I include this as I haven't seen earlier
examples in the educational literature.)
David Singmaster. Covering deleted chessboards with
dominoes. MM 48 (1975) 59‑66. Optimum extension to n‑dimensions. For an
n-dimensional board, each dimension must be ³ 2. If the board has an even number of cells, then one can delete
any n-1 white cells and any n-1 black cells and still cover the board with dominoes
(i.e.
2 x 1 x 1 x ... x 1 blocks).
If the board has an odd number of cells, then let the corner cells be
coloured black. One can then delete
any n
black cells and any n-1 white cells and still cover the board with
dominoes.
I-Ping Chu & Richard
Johnsonbaugh. Tiling deficient boards
with trominoes. MM 59:1 (1986)
34-40. (3,n) = 1 and
n ¹ 5 imply that
an n x n board with one cell deleted can be covered with L
trominoes. Some 5 x 5
boards with one cell deleted can be tiled, but not all can.
6.F.3. DISSECTING A CROSS INTO Zs AND Ls
The L
pieces are not always drawn carefully, and in some cases the unit pieces
are not all square. I have enlarged and
measured those which are not clear and approximated them as n-ominoes.
Minguet. 1733.
Pp. 119-121 (1755: 85-86; 1822: 138-139; 1864: 116-117). The problem has two parts. The first is a cross into 5 pieces:
L-tetromino, 2 Z-pentominoes, L‑hexomino, Z-hexomino. The two hexominoes are like the
corresponding pentominoes lengthened by one unit. Similar to Les Amusemens, but one Z is longer and one L is
shorter. The diagram shows 8
L and Z shaped pieces formed
from squares, but it is not clear what the second part of the problem is doing
-- either a piece or a label is erroneous or missing. Says one can make different figures with the pieces.
Les Amusemens. 1749.
P. xxxi. Cross into 3 Z
pentominoes and 2 L pieces.
Like Minguet, but the Ls are much lengthened and are approximately a
L-heptomino and an L‑octomino.
Catel. Kunst-Cabinet. 1790. Das mathematische Kreuz, p. 10 & fig. 27
on plate I. As in Les Amusemens, but
the Ls are approximately a 9-omino and a 10-omino.
Bestelmeier. 1801.
Item 274 -- Das mathematische Kreuz.
Cross into 6 pieces, but the picture has an erroneous extra line. It should be the reversal of the picture in
Catel.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more
details. F. 4r is "Analysis of the
Essay of Games". F. 4v has the
dissection of the cross into 3 Z pentominoes and two L
pieces. I don't have a copy of
this, but my sketch looks like the Ls are a tetromino and a pentomino, or
possibly a pentomino and a hexomino.
Manuel des Sorciers. 1825.
Pp. 204-205, art. 21. ??NX. Dissect a cross into three Zs
and two Ls. My notes don't indicate the size of the Ls.
Boy's Own Book. 1843 (Paris): 435 & 440, no. 3. As in Les Amusemens, but with the Ls
apparently intended to be a pentomino and a hexomino. = Boy's Treasury, 1844, pp. 424-425 & 428. = de Savigny, 1846, pp. 353 & 357, no.
2, except the solution has been redrawn with some slight changes and so the
proportions are less clear.
Family Friend 3 (1850) 330 &
351. Practical puzzle, No. XXI. As in Les Amusemens.
Magician's Own Book. 1857.
Prob. 31: Another cross puzzle, pp. 276 & 299. As in Les Amusemens.
Landells. Boy's Own Toy-Maker. 1858.
P. 152. As in Les Amusemens.
Book of 500 Puzzles. 1859.
Prob. 31: Another cross puzzle, pp. 90 & 113. As in Les Amusemens. = Magician's Own Book.
Indoor & Outdoor. c1859.
Part II, p. 127, prob. 5: The puzzle of the cross. As in Les Amusemens.
Illustrated Boy's Own
Treasury. 1860. Practical Puzzles, No. 24, pp. 399 &
439. Identical to Magician's Own Book.
Boy's Own Conjuring book. 1860.
Prob. 30: Another cross puzzle, pp. 239 & 263. = Magician's Own Book, 1857.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob.
584-2, pp. 286 & 404.
4 Z pentominoes to
make a (Greek) cross. (Also entered in
6.F.5.)
Prob.
584-8, pp. 287 & 405. 3 Z
pentominoes, L tetromino and L pentomino to make a
Greek cross. Despite specifically
asking for a Greek cross, the answer is a standard Latin cross with
height : width = 4 : 3.
Mittenzwey. 1880.
Prob. 173-174, pp. 33 & 85;
1895?: 198-199, pp. 38 & 87;
1917: 198‑199, pp. 35 & 84.
The first is 3 Z pentominoes, L tetromino and L
pentomino to make a cross. The
second is 4 Z pentominoes to
make a (Greek) cross. (Also entered in
6.F.5.)
Cassell's. 1881.
P. 93: The magic cross. =
Manson, 1911, p. 139. Same pattern as
Les Amusemens, but one end of the Zs is decidedly longer than the other and the
middle 'square' of the Zs is decidedly not square. The Ls are approximately a pentomino and a heptomino, But the middle 'square' of the Zs is almost
a domino and that makes the Zs into heptominoes, with the Ls being a hexomino
and a nonomino.
S&B, p. 20, shows a 7 piece
cross dissection, Jeu de La Croix, into 3
Zs, 2 Ls and 2 straights, from
c1890. The Zs are pentominoes, with the
centre 'square' lengthened a bit. The
Ls appear to be a heptomino and an octomino and the straights appear to be a
hexomino and a tetromino. Cf
Hoffmann-Hordern for a version without the straight pieces.
Handy Book for Boys and
Girls. Showing How to Build and
Construct All Kinds of Useful Things of Life.
Worthington, NY, 1892. Pp. 320-321:
The cross puzzle. As in Cassell's.
Hoffmann. 1893.
Chap. III, no. 29: Another cross puzzle, pp. 103 & 136
= Hoffmann‑Hordern, pp. 100-101, with photo. States that the two Ls are the same shape,
but the solution is as in Les Amusemens, with the Ls approximately a hexomino
and a heptomino. Hordern has corrected
the problem statement. Photo on
p. 100 shows an ivory version, dated 1850-1900, of the same
proportions. Hordern Collection, p. 65,
shows two wood versions, La Croix Brisée and Jeu de la Croix, dated 1880-1905,
both with Ls being approximately a heptomino and an octomino.
Benson. 1904.
The Latin cross puzzle, p. 200.
As in Hoffmann, but the solution is longer, as in Les Amusemens.
Wehman. New Book of 200 Puzzles. 1908.
Another cross puzzle, p. 32. As
in Les Amusemens, with the Ls being a pentomino and a hexomino.
S. Szabo. US Patent 1,263,960 -- Puzzle. Filed: 20 Oct 1917; patented: 23 Apr 1918. 1p + 1p diagrams. As in Les Amusemens, with even longer Ls, approximately a 10-omino and an 11-omino.
6.F.4. QUADRISECT AN L‑TROMINO, ETC.
See
also 6.AW.1 & 4.
Mittenzwey
and Collins quadrisect a hollow square obtained by removing a 2 x 2
from the centre of a 4 x 4.
Bile
Beans quadrisects a 5 x 5 after deleting corners and centre.
Minguet. 1733.
Pp. 114-115 (1755: 80; 1822: 133-134; 1864: 111-112). Quadrisect
L‑tromino.
Alberti. 1747.
Art. 30: Modo di dividere uno squadro di carta e di legno in quattro
squadri equali, p. ?? (131) & fig. 56, plate XVI, opp. p. 130.
Les Amusemens. 1749.
P. xxx. L-tromino
("gnomon") into 4 congruent pieces.
Vyse. Tutor's Guide. 1771? Prob. 9, 1793: p. 305, 1799: p. 317 &
Key p. 358. Refers to the land as a
parallelogram though it is drawn rectangular.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more
details. F. 4r is "Analysis of the
Essay of Games". F. 4v has an
entry "8½ a Prob of figure" followed by the L‑tromino. 8½ b is
the same with a mitre and there are other dissection problems adjacent -- see
6.F.3, 6.AQ, 6.AW.1, 6.AY, so it seems clear that he knew this problem.
Jackson. Rational Amusement. 1821.
Geometrical Puzzles, no. 3, pp. 23 & 83 & plate I, fig. 2.
Manuel des Sorciers. 1825.
Pp. 203-204, art. 20. ??NX. Quadrisect L-tromino.
Family Friend 2 (1850) 118 &
149. Practical Puzzle -- No. IV. Quadrisect L-tromino of land with four
trees.
Family Friend 3 (1850) 150 &
181. Practical puzzle, No. XV. 15/16
of a square with 10 trees to be divided equally. One tree is placed very close to another, cf
Magician's Own Book and Hoffmann, below.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 8, p. 179 (1868:
190). Land in the shape of an L-tromino to be cut into four congruent
parts, each with a cherry tree.
Magician's Own Book. 1857.
Prob.
3: The divided garden, pp. 267 & 292.
15/16 of a square to be divided
into five (congruent) parts, each with two trees. The missing 1/16 is in the middle. One tree is placed very close to another, cf Family Friend 3,
above, and Hoffmann below.
Prob.
22: Puzzle of the four tenants, pp. 273 & 296. Same as Parlour Pastime, but with apple trees. (= Illustrated Boy's Own Treasury, 1860, No.
10, pp. 397 & 437.)
Prob.
28: Puzzle of the two fathers, pp. 275-276 & 298. Each father wants to divide
3/4 of a square. One has
L‑tromino, other has the mitre shape. See 6.AW.1.
Landells. Boy's Own Toy-Maker. 1858.
P.
144. = Magician's Own Book, prob. 3.
Pp.
148-149. = Magician's Own Book, prob.
27.
Book of 500 Puzzles. 1859.
Prob.
3: The divided garden, pp. 81 & 106.
Identical to Magician's Own Book.
Prob.
22: Puzzle of the four tenants, pp. 87 & 110. Identical to Magician's Own Book.
Prob.
28: Puzzle of the two fathers, pp. 89-90 & 112. Identical to Magician's Own Book. See also 6.AW.1.
Charades, Enigmas, and
Riddles. 1860: prob. 28, pp. 59 &
63; 1862: prob. 29, pp. 135 &
141; 1865: prob. 573, pp. 107 & 154. Quadrisect
L-tromino, attributed to Sir F. Thesiger.
Boy's Own Conjuring book. 1860.
Prob.
3: The divided garden, pp. 229 & 255.
Identical to Magician's Own Book.
Prob.
21: Puzzle of the four tenants, pp. 235 & 260. Identical to Magician's Own Book.
Prob.
27: Puzzle of the two fathers, pp. 237‑238 & 262. Identical to Magician's Own Book.
Illustrated Boy's Own
Treasury. 1860. Prob. 21, pp. 399 & 439. 15/16
of a square to be divided into five (congruent) parts, each with two
trees. c= Magician's Own Book,
prob. 3.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 175, p.
88. L-tromino into four congruent
pieces, each with two trees. The
problem is given in terms of the original square to be divided into five parts,
where the father gets a quarter of the whole in the form of a square and the
four sons get congruent pieces.
Hanky Panky. 1872. The divided orchards, p. 130.
L‑tromino into 4 congruent pieces, each with two trees.
Boy's Own Book. The divided garden. 1868: 675.
= Magician's Own Book, prob. 3.
Mittenzwey. 1880.
Prob.
192, pp. 36 & 89; 1895?: 217, pp.
40 & 91; 1917: 217, pp. 37 &
87. Cut 1 x 1 out of the centre
of a 4 x 4. Divide the rest into five parts of equal area with four being
congruent. He cuts a 2 x 2
out of the centre, which has a 1
x 1 hole in it, then divides the rest
into four L-trominoes.
Prob.
213, pp. 38 & 90; 1895?: 238, pp.
42 & 92; 1917: 238, pp. 39 &
88. Usual quadrisection of an
L-tromino.
Prob.
214, pp. 38 & 90; 1895?: 239, pp.
42 & 92; 1917: 239, pp. 39 &
88. Square garden with mother receiving
1/4 and the rest being divided into four congruent parts.
Cassell's. 1881.
P. 90: The divided farm. =
Manson, 1911, pp. 136-137. = Magician's
Own Book, prob. 3.
Lemon. 1890.
The
divided garden, no. 259, pp. 38 & 107.
= Magician's Own Book, prob. 3.
Geometrical
puzzle, no. 413, pp. 55 & 113 (= Sphinx, no. 556, pp. 76 & 116). Quadrisect
L-tromino.
Hoffmann. 1893.
Chap. X, no. 41: The divided farm, pp. 352‑353 & 391
= Hoffmann‑Hordern, p. 250.
= Magician's Own Book, prob. 3.
[One of the trees is invisible in the original problem, but Hoffmann-Hordern
has added it, in a more symmetric pattern than in Magician's Own Book.]
Loyd. Origin of a famous puzzle -- No. 18: An ancient puzzle. Tit‑Bits 31 (13 Feb &
6 Mar 1897) 363
& 419. Nearly 50 years ago he was told of the
quadrisection of 3/4 of a square, but drew the mitre shape
instead of the L‑tromino. See 6.AW.1.
Clark. Mental Nuts. 1897, no.
73; 1904, no. 31. Dividing the land. Quadrisect an L‑tromino. 1904 also has the mitre -- see 6.AW.1.
Benson. 1904.
The farmer's puzzle, p. 196.
Quadrisect an L‑tromino.
Wehman. New Book of 200 Puzzles. 1908.
The
divided garden, p. 17. = Magician's Own
Book, prob. 3
Puzzle
of the two fathers, p. 43. = Magician's
Own Book, prob. 28.
Puzzle
of the four tenants, p. 46. =
Magician's Own Book, prob. 22.
Dudeney. Some much‑discussed puzzles. Op. cit. in 2. 1908. Land in shape of
an L‑tromino to be
quadrisected. He says this is supposed
to have been invented by Lord Chelmsford (Sir F. Thesiger), who died in 1878 --
see Charades, Enigmas, and Riddles (1860).
But cf Les Amusemens.
M. Adams. Indoor Games. 1912. The clever farmer,
pp. 23‑25. Dissect L‑tromino into four congruent pieces.
Blyth. Match-Stick Magic.
1921. Dividing the inheritance,
pp. 20-21. Usual quadrisection of L-tromino set out with matchsticks.
Collins. Book of Puzzles. 1927. The surveyor's
puzzle, pp. 2-3. Quadrisect 3/4
of a square, except the deleted
1/4 is in the centre, so we are
quadrisecting a hollow square -- cf Mittenzwey,
The Bile Beans Puzzle Book. 1933.
No.
22: Paper squares. Quadrisect a
P-pentomino into P-pentominoes. One
solution given, I find another. Are
there more? How about quadrisecting
into congruent pentominoes? Which
pentominoes can be quadrisected into four copies of themself?
No.
41: Five lines. Consider a 5 x 5
square and delete the corners and centre. Quadrisect into congruent pentominoes. One solution given. I
find three more. Are there more? One can extend this to consider
quadrisecting the 5 x 5 with just the centre removed into congruent
hexominoes. I find seven ways.
Depew. Cokesbury Game Book.
1939. A plot of ground, p.
227. 3/4 of XX
a
square to be quadrisected, but the shape is as shown at the right. XXX
X XX
XXXX
Ripley's Puzzles and Games. 1966.
Pp. 18 & 19, item 8. Divide
an L-tromino into eight congruent pieces.
F. Göbel. Problem 1771: The L‑shape dissection
problem. JRM 22:1 (1990) 64‑65. The
L‑tromino can be dissected into
2, 3, or 4 congruent parts. Can it be divided into 5 congruent parts?
Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Apple-eating monsters, pp. 40 & 63. Trisect into equal parts, the shape consisting of a 2 x 4
rectangle with a 1 x 1 square attached to one of the central
squares of the long side. [Actually,
this can be done with the square attached to any of the squares, though if it
is attached to the end of the long side, the resulting pieces are straight
trominoes.]
6.F.5. OTHER DISSECTIONS INTO POLYOMINOES
Catel. Kunst-Cabinet. 1790.
Das
Zakk- und Hakenspiel, p. 10 & fig. 11 on plate 1. 4 Z‑pentominoes and
4 L‑tetrominoes make a
6 x 6 square.
Die
zwolf Winkelhaken, p. 11 & fig. 26 on plate 1. 8 L‑pentominoes and
4 L‑hexominoes make a
8 x 8 square.
Bestelmeier. 1801.
Item 61 -- Das Zakken und Hakkenspiel.
As in Catel, p. 10, but not as regularly drawn. Text copies some of Catel.
Manuel des Sorciers. 1825.
Pp. 203-204, art. 20. ??NX Use four L-trominoes to make a 3 x 4 rectangle or a 4 x 4 square with four corners deleted.
Family Friend 3 (1850) 90 &
121. Practical puzzle -- No. XIII. 4 x 4
square, with 12 trees in the corners, centres of sides and four at the
centre of the square, to be divided into 4 congruent parts each with 3 trees. Solution uses 4 L-tetrominoes. The same problem is repeated as Puzzle 17 --
Twelve-hole puzzle in (1855) 339 with solution in (1856) 28.
Magician's Own Book. 1857.
Prob. 14: The square and circle puzzle, pp. 270 & 295. Same as Family Friend. = Book of 500 Puzzles, 1859, prob. 14, pp.
84 & 109. = Boy's Own Conjuring
book, 1860, prob. 13, pp. 231-232 & 257.
c= Illustrated Boy's Own Treasury, 1860, prob. 8, pp. 396 &
437. c= Hanky Panky, 1872, A square of
four pieces, p. 117.
Landells. Boy's Own Toy-Maker. 1858.
Pp. 146-147. Identical to Family
Friend.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob.
584-2, pp. 286 & 404.
4 Z‑pentominoes to make a Greek cross. (Also entered in 6.F.3.)
Prob.
584-3, pp. 286 & 404. 4 L-tetrominoes to make a square.
Prob.
584-5, pp. 286 & 404.
8 L‑pentominoes and 4 L‑hexominoes
make a 8 x 8 square. Same as Catel,
but diagram is inverted.
Prob.
584-7, pp. 287 & 405. 4 Z‑pentominoes and 4 L‑tetrominoes
make a 6 x 6 square. Same as Catel, but
diagram is inverted.
Mittenzwey. 1880.
Prob.
174, pp. 33 & 85; 1895?: 199, pp.
38 & 87; 1917: 199, pp. 35 &
84. 4 Z pentominoes to make a (Greek) cross. (Also entered in 6.F.5.)
Prob.
186, pp. 35 & 88; 1895?: 211, pp.
40 & 90; 1917: 211, pp. 36 &
87. 4 x 4 square into 4
L-tetrominoes.
Prob.
187, pp. 35 & 88; 1895?: 212, pp.
40 & 90; 1917: 212, pp. 36 &
87. 6 x 6 square into 4 Z‑pentominoes
and 4 L‑tetrominoes, as in Catel, p. 10.
Prob.
215, pp. 38 & 90; 1895?: 240, pp.
42 & 92; 1917: 240, pp. 39 &
88. Square garden with 12 trees
quadrisected into four L-tetrominoes.
S&B, p. 20, shows a 7 piece
cross dissection into 3 Zs,
2 Ls and 2 straights, from c1890.
Hoffmann. 1893.
Chap. X, no. 37: The orchard puzzle, pp. 350 & 390 = Hoffmann-Hordern, pp. 247, with
photo. Same as Family Friend 3. Photo on p. 247 shows St. Nicholas Puzzle
Card, © 1892 in the USA.
Tom Tit, vol 3. 1893.
Les quatre Z et des quatre L, pp. 181-182. = K, No. 27: The four Z's
and the four L's, pp. 70‑71. = R&A, Squaring the
L's and Z's,
p. 102. 6 x 6 square as in Catel, p. 10.
Sphinx. 1895.
The Maltese cross, no. 181, pp. 28 & 103. Make a Maltese cross (actually a Greek cross of five equal
squares) from 4 P-pentominoes. Also:
quadrisect a P‑pentomino.
Wehman. New Book of 200 Puzzles. 1908.
The square and circle puzzle, p. 5.
= Family Friend.
Burren Loughlin &
L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909.
The Zoltan's orchard, pp. 24-28 & 64. = Family Friend.
Anon. Prob. 84. Hobbies 31 (No.
799) (4 Feb 1911) 443. Use at least one
each of: domino; L-tetromino; P and X pentominoes to
make the smallest possible square Due
to ending of this puzzle series, no solution ever appeared. I find numerous solutions for 5 x 5,
6 x 6, 8 x 8, of which the first is easily seen to be the
smallest possibility.
A. Neely Hall. Carpentry & Mechanics for Boys. Lothrop, Lee & Shepard, Boston, nd
[1918]. The square puzzle, pp. 20‑21. 7 x 7
square cut into 1 straight tromino, 1 L‑tetromino and
7 L‑hexominoes.
Collins. Book of Puzzles. 1927. The surveyor's
puzzle, pp. 2-3. Quadrisect 3/4
of a square, except the deleted
1/4 is in the centre, so we are
quadrisecting a hollow square.
Arthur Mee's Children's
Encyclopedia 'Wonder Box'. The
Children's Encyclopedia appeared in 1908, but versions continued until the
1950s. This looks like 1930s?? 4 Z‑pentominoes and
4 L‑tetrominoes make a
6 x 6 square and a 4 x 9
rectangle.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. All square, pp. 42 & 129. Make a
6 x 6 square from the staircase
hexomino, 2 Y-pentominoes, an N‑tetromino, an L-tetromino
and 3 T-tetrominoes. None of the pieces is turned over in the solution,
though this restriction is not stated.
Piet Hein invented the Soma Cube
in 1936. (S&B, pp. 40‑41.) ??Is there any patent??
M. Gardner. SA (Sep 1958) = 2nd Book, Chap 6.
Richard K. Guy. Loc. cit. in 5.H.2, 1960. Pp. 150-151 discusses cubical solutions
-- 234
found so far. He proposes the
'bath' shape -- a 5 x 3 x 2 cuboid with a 3 x 1 x 1 hole in the top
layer. In a 1985 letter, he said that
O'Beirne had introduced the Soma to him and his family. in 1959 and they found
234 solutions before Mike Guy went to Cambridge -- see below.
P. Hein, et al. Soma booklet. Parker Bros., 1969, 56pp.
Asserts there are 240 simple solutions and 1,105,920
total solutions, found by J. H. Conway & M. J. T. Guy with a a
computer (but cf Gardner, below) and by several others. [There seem to be several versions of this
booklet, of various sizes.]
Thomas V. Atwater, ed. Soma Addict. 4 issues, 1970‑1971, produced by Parker Brothers. (Gardner, below, says only three issues
appeared.) ??NYS -- can anyone provide
a set or photocopies??
M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. States there are 240 solutions for the
cube, obtained by many programs, but first found by J. H. Conway & M. J. T.
Guy in 1962, who did not use a computer, but did it by hand "one wet
afternoon". Richard Guy's 1985
letter notes that Mike Guy had a copy of the Guy family's 234 solutions with
him.
SOMAP ??NYS -- ??details.
(Schaaf III 52)
Winning Ways, 1982, II, 802‑803
gives the SOMAP.
Jon Brunvall et al. The computerized Soma Cube. Comp. & Maths. with Appl. 12B:1/2
(1986 [Special issues 1/2
& 3/4 were separately printed as: I. Hargittai, ed.; Symmetry -- Unifying
Human Understanding; Pergamon, 1986.] 113‑121. They cite Gardner's 2nd Book which says the number of solutions
is unknown and they use a computer to find them.
See
also 6.N, 6.U.2, 6.AY.1 and 6.BJ. The
predecessors of these puzzles seem to be the binomial and trinomial cubes
showing (a+b)3 and
(a+b+c)3. I have an
example of the latter from the late 19C.
Here I will consider only cuts parallel to the cube faces -- cubes with
cuts at angles to the faces are in 6.BJ.
Most of the problems here involve several types of piece -- see 6.U.2
for packing with one kind of piece.
Catel. Kunst-Cabinet. 1790. Der algebraische Würfel, p. 6 & fig 50
on plate II. Shows a binomial
cube: (a + b)3 = a3 + 3a2b
+ 3ab2 + b3.
Bestelmeier. 1801.
Item 309 is a binomial cube, as in Catel. "Ein zerschnittener Würfel, mit welchem die Entstehung eines
Cubus, dessen Seiten in 2 ungleiche Theile a + b getheilet ist,
gezeigt ist."
Hoffmann. 1893.
Chap. III, no. 39: The diabolical cube, pp. 108 & 142
= Hoffmann-Hordern, pp. 108-109, with photos. 6: 0, 1, 1, 1, 1, 1, 1,
i.e. six pieces of volumes 2, 3,
4, 5, 6, 7. Photos on p. 108 shows Cube
Diabolique and its box, by Watilliaux, dated 1874-1895.
J. G.‑Mikusiński. French patent. ??NYS -- cited by Steinhaus.
H. Steinhaus. Mikusiński's
Cube. Mathematical Snapshots. Not in Stechert, 1938, ed. OUP, NY:
1950: pp. 140‑142 & 263;
1960, pp. 179‑181 & 326;
1969 (1983): pp. 168-169 & 303.
John Conway. In an email of 7 Apr 2000, he says he
developed the dissection of the 3 x 3 x
3 into
3 1 x 1 x 1 and
6 1 x 2 x 2 in c1960 and then adapted it to the 5 x 5 x 5 into
3 1 x 1 x 3, 1 2 x 2 x 2,
1 1 x 2 x 2 and
13 1 x 2 x 4 and the
5 x 5 x 5 into 3 1 x 1 x 3 and
29 1 x 2 x 2. He says his first publication of it was in
Winning Ways, 1982 (cf below).
Jan Slothouber &
William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS.
3 x 3 x 3 into 3 1
x 1 x 1 and 6 1 x 2 x 2. [Jan de Geus has sent a photocopy of some of
this but it does not cover this topic.]
M. Gardner. SA (Sep 1972) c= Knotted, chap. 3. Discusses Hoffmann's Diabolical Cube and
Mikusiński's cube. Says he has 8
solutions for the first and that there are just 2
for the second. The Addendum
reports that Wade E. Philpott showed there are just 13 solutions of the
Diabolical Cube. Conway has confirmed
this. Gardner briefly describes the
solutions. Gardner also shows the Lesk
Cube, designed by Lesk Kokay (Mathematical Digest [New Zealand] 58 (1978)
??NYS), which has at least 3 solutions.
D. A. Klarner. Brick‑packing puzzles. JRM 6 (1973) 112‑117. Discusses 3 x 3 x 3 into
3 1 x 1 x 1
and 6 1 x 2 x 2 attributed to
Slothouber‑Graatsma;
Conway's 5 x 5 x 5 into
3 1 x 1 x 3
and 29 1 x 2 x 2; Conway's 5 x 5 x 5 into
3 1 x 1 x 3, 1 2
x 2 x 2, 1 1 x 2 x
2 and
13 1 x 2 x 4. Because of the attribution to Slothouber
& Graatsma and not knowing the date of Conway's work, I had generally
attributed the 3 x 3 x 3 puzzle to them and Stewart Coffin followed
this in his book. However, it now seems
that it really is Conway's invention and I must apologize for misleading
people.
Leisure Dynamics, the US
distributor of Impuzzables, a series of
6 3 x 3 x 3 cube dissections identified by colours,
writes that they were invented by Robert Beck, Custom Concepts Inc.,
Minneapolis. However, the Addendum to
Gardner, above, says they were designed by Gerard D'Arcey.
Winning Ways. 1982.
Vol. 2, pp. 736-737 & 801.
Gives the 3 x 3 x 3 into
3 1 x 1 x 1 and
6 1 x 2 x 2 and the
5 x 5 x 5
into 3 1 x 1 x
3, 1
2 x 2 x 2, 1 1 x 2 x 2
and 13 1 x 2 x
4, which is called
Blocks-in-a-Box. No mention of the
other 5 x 5 x 5. Mentions Foregger & Mather, cf in 6.U.2.
Michael Keller. Polycube update. World Game Review 4 (Feb 1985) 13. Reports results of computer searches for solutions. Hoffmann's Diabolical Cube has 13;
Mikusinski's Cube has 2; Soma Cube has 240; Impuzzables: White -- 1;
Red -- 1; Green -- 16; Blue -- 8; Orange -- 30; Yellow --
1142.
Michael Keller. Polyform update. World Game Review 7 (Oct 1987) 10‑13. Says that Nob Yoshigahara has solved a
problem posed by O'Beirne: How many
ways can 9 L‑trominoes
make a cube? Answer is 111.
Gardner, Knotted, chap. 3, mentioned this. Says there are solutions with
n L‑trominoes and 9‑n
straight trominoes for n ¹ 1
and there are 4 solutions for n = 0. Says the Lesk Cube
has 4
solutions. Says Naef's Gemini
Puzzle was designed by Toshiaki Betsumiya.
It consists of the 10 ways to join two 1 x 2 x 2 blocks.
H. J. M. van Grol. Rik's Cube Kit -- Solid Block Puzzles. Analysis of all 3 x 3 x 3 unit solid
block puzzles with non‑planar 4‑unit
and 5‑unit shapes. Published by the author, The Hague, 1989,
16pp. There are 3
non‑planar tetracubes and
17 non‑planar
pentacubes. A 3 x 3 x 3 cube will
require the 3 non‑planar tetracubes and
3 of the non‑planar
pentacubes -- assuming no repeated pieces.
He finds 190 subsets which can form cubes, in 1
to 10 different ways.
Nob Yoshigahara. (Title in Japanese: (Puzzle in Wood)). H. Tokuda, Sowa Shuppan, Japan, 1987. Pp. 68-69 is a 3^3 designed by Nob
-- 6: 01005.
6.G.2. DISSECTION OF 63 INTO 33, 43 AND 53, ETC.
H. W. Richmond. Note 1672:
A geometrical problem. MG 27 (No.
275) (Jul 1943) 142. AND Note 1704:
Solution of a geometrical problem (Note 1672). MG 28 (No. 278) (Feb 1944) 31‑32. Poses the problem of making such a dissection, then gives a
solution in 12 pieces: three 1 x 3 x 3;
4 x 4 x 4; four 1 x 5 x 5; 1 x 4 x 4;
two 1 x 1 x 2 and a
V‑pentacube.
Anon. [= John Leech, according
to Gardner, below]. Two dissection
problems, no. 2. Eureka 13 (Oct 1950) 6 & 14 (Oct 1951)
23. Asks for such a dissection using at
most 10 pieces. Gives an 8 piece solution
due to R. F. Wheeler. [Cundy &
Rollett; Mathematical Models; 2nd ed., pp. 203‑205, say Eureka is the
first appearance they know of this problem.
See Gardner, below, for the identity of Leech.]
Richard K. Guy. Loc. cit. in 5.H.2, 1960. Mentions the 8 piece solution.
J. H. Cadwell. Some dissection problems involving sums of
cubes. MG 48 (No. 366) (Dec 1964)
391‑396. Notes an error in Cundy
& Rollett's account of the Eureka problem.
Finds examples for 123
+ 13 = 103 + 93 with 9 pieces and 93 = 83 + 63 + 13 with 9 pieces.
J. H. Cadwell. Note 3278:
A three‑way dissection based on Ramanujan's number. MG 54 (No. 390) (Dec 1970) 385‑387. 7 x 13 x 19
to 103 + 93 and
123 + 13
using 12 pieces.
M. Gardner. SA (Oct 1973) c= Knotted, chap. 16. He says that the problem was posed by John
Leech. He gives Wheeler's initials as
E. H. ?? He says that J. H.
Thewlis found a simpler 8‑piece solution, further simplified by T. H.
O'Beirne, which keeps the
4 x 4 x 4
cube intact. This is shown in Gardner. Gardner also shows an 8‑piece solution
which keeps the 5 x 5 x 5 intact, due to E. J. Duffy, 1970. O'Beirne showed that an 8‑piece
dissection into blocks is impossible and found a 9‑block solution in
1971, also shown in Gardner.
Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1984. Section 24.1, pp. 118‑120 gives
Wheeler's solution and admires it.
Richard K. Guy, proposer; editors & Charles H. Jepson [should be
Jepsen], partial solvers. Problem
1122. CM 12 (1987) 50 &
13 (1987) 197‑198. Asks
for such dissections under various conditions, of which (b) is the form given
in Eureka. Eight pieces is minimal in
one case and seems minimal in two other cases.
Eleven pieces is best known for the first case, where the pieces must be
blocks, but this appears to be the problem solved by O'Beirne in 1971, reported
in Gardner, above.
Charles H. Jepsen. Additional comment on Problem 1122. CM 14 (1988) 204‑206. Gives a ten piece solution of the first
case.
Chris Pile. Cube dissection. M500 134 (Aug 1993) 2-3.
He feels the 1 x 1 x 2 piece occurring in Cundy & Rollett is too
small and he provides another solution with 8 pieces, the smallest of which
contains 8 unit cubes. Asks how uniform
the piece sizes can be.
6.G.3. DISSECTION OF A DIE INTO NINE 1 x 1 x 3
Hoffmann. 1893.
Chap. III, no. 17: The "Spots" puzzle, pp. 98‑99 &
130‑131 = Hoffmann‑Hordern, pp. 90-91, with photo. Says it is made by Wolff & Son. Photo on p. 91 shows an example made by E.
Wolff & Son, London.
Benson. 1904.
The spots puzzle, pp. 203‑204.
As in Hoffmann.
Collins. Book of Puzzles. 1927. Pp. 131‑134:
The dissected die puzzle. The solution
is different than Hoffmann's.
Rohrbough. Puzzle Craft. 1932. P. 21 shows a
dissected die, but with no text. The
picture is the same as in Hoffmann's solution.
Slocum. Compendium.
Shows Diabolical Dice from Johnson Smith catalogue, 1935.
Harold Cataquet. The Spots puzzle revisited. CFF 33 (Feb 1994) 20-21. Brief discussion of two versions.
David Singmaster. Comment on the "Spots"
puzzle. 29 Sep 1994, 2pp. Letter in response to the above. I note that there is no standard pattern for
a die other than the opposite sides adding to seven. There are 23 =
8 ways to orient the spots forming 2, 3, and 6. There are two handednesses, so there are 16 dice altogether. (This was pointed out to me perhaps 10 years
before by Richard Guy and Ray Bathke. I
have since collected examples of all 16 dice.)
However, Ray Bathke showed me Oriental dice with the two spots of the 2
placed horizontal or vertically rather than diagonally, giving another 16 dice
(I have 5 types), making 32 dice in all.
A die can be dissected into 9 1 x 1 x 3
pieces in 6 ways if the layers have to alternate in direction, or in 21
ways in general. I then pose a number
of questions about such dissections.
6.G.4. USE OF OTHER POLYHEDRAL PIECES
S&B. 1986.
P. 42 shows Stewart Coffin's 'Pyramid Puzzle' using pieces made from
truncated octahedra and his 'Setting Hen' using pieces made from rhombic
dodecahedra. Coffin probably devised
these in the 1960s -- perhaps his book has some details of the origins of these
ideas. ??check.
Mark Owen & Matthew
Richards. A song of six splats. Eureka 47 (1987) 53‑58. There are six ways to join three truncated
octahedra. For reasons unknown, these
are called 3‑splats. They give
various shapes which can and which cannot be constructed from the six 3‑splats.
Georg Pick. Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen
naturwissenschaftlich‑medicinischen Vereines für Böhmen "Lotos"
in Prag (NS) 19 (1899) 311‑319.
Pp. 311‑314 gives the proof, for an oblique lattice. Pp. 318‑319 gives the extension to
multiply connected and separated regions.
Rest relates to number theory.
[I have made a translation of the material on Pick's Theorem.]
Charles Howard Hinton. The Fourth Dimension. Swan Sonnenschein & Co., London,
1906. Metageometry, pp. 46-60. [This material is in Speculations on the Fourth
Dimension, ed. by R. v. B. Rucker; Dover, 1980, pp. 130-141. Rucker says the book was published in 1904,
so my copy may be a reprint??] In the
beginning of this section, he draws quadrilateral shapes on the square lattice
and determines the area by counting points, but he counts I + E/2 + C/4, which works for quadrilaterals but is not valid in general.
H. Steinhaus. O mierzeniu pól płaskich. Przegląd Matematyczno‑Fizyczny 2
(1924) 24‑29. Gives a version of
Pick's theorem, but doesn't cite Pick. (My
thanks to A. Mąkowski for an English summary of this.)
H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938, pp. 16-17 &
132. OUP, NY: 1950: pp. 76‑77 & 260 (note 77); 1960: pp. 99‑100 & 324 (note
95); 1969 (1983): pp. 96‑97 &
301 (note 107). In 1938 he simply notes
the theorem and gives one example. In
1950, he outlines Pick's argument.
He refers to Pick's paper, but
in "Ztschr. d. Vereins 'Lotos' in Prag". Steinhaus also cites his own paper, above.
J. F. Reeve. On the volume of lattice polyhedra. Proc. London Math. Soc. 7 (1957) 378‑395. Deals with the failure of the obvious form
of Pick's theorem in 3‑D and finds a valid generalization.
Ivan Niven & H. S.
Zuckerman. Lattice points and polygonal
area. AMM 74 (1967) 1195‑1200. Straightforward proof. Mention failure for tetrahedra.
D. W. De Temple & J. M.
Robertson. The equivalence of Euler's
and Pick's theorems. MTr 67 (1974)
222‑226. ??NYS.
W. W. Funkenbusch. From Euler's formula to Pick's formula using
an edge theorem. AMM 81 (1974) 647‑648. Easy proof though it could be easier.
R. W. Gaskell, M. S. Klamkin
& P. Watson. Triangulations and
Pick's theorem. MM 49 (1976) 35‑37. A bit roundabout.
Richard A. Gibbs. Pick iff Euler. MM 49 (1976) 158. Cites
DeTemple & Robertson and observes that both Pick and Euler can be proven
from a result on triangulations.
John Reay. Areas of hex-lattice polygons, with short
sides. Abstracts Amer. Math. Soc. 8:2
(1987) 174, #832-51-55. Gives a formula
for the area in terms of the boundary and interior points and the
characteristic of the boundary, but it is an open question to determine when
this formula gives the actual area.
6.I. SYLVESTER'S PROBLEM OF COLLINEAR POINTS
If a set of non‑collinear points
in the plane is such that the line through any two points of the set contains a
third point of the set, then the set is infinite.
J. J. Sylvester. Question 11851. The Educational Times 46 (NS, No. 383) (1 Mar 1893) 156.
H. J. Woodall & editorial
comment. Solution to Question
11851. Ibid. (No. 385)
(1 May 1893) 231. A very
spurious solution.
(The above two items appear
together in Math. Quest. with their Sol. Educ. Times 59 (1893) 98‑99.)
E. Melchior. Über Vielseite der projecktiven Ebene. Deutsche Math. 5 (1940) 461‑475. Solution, but in a dual form.
P. Erdös, proposer; R. Steinberg, solver & editorial comment
giving solution of T. Grünwald (later = T. Gallai). Problem 4065. AMM 50
(1943) 65 & 51 (1944) 169‑171.
L. M. Kelly. (Solution.)
In: H. S. M. Coxeter; A problem
of collinear points; AMM 55 (1948) 26‑28. Kelly's solution is on p. 28.
G. A. Dirac. Note 2271:
On a property of circles. MG 36
(No. 315) (Feb 1952) 53‑54.
Replace 'line' by 'circle' in the problem. He shows this is true by inversion. He asks for an independent proof of the result, even for the case
when two, three are replaced by three, four.
D. W. Lang. Note 2577:
The dual of a well‑known theorem.
MG 39 (No. 330) (Dec 1955) 314.
Proves the dual easily.
H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 4.7: Sylvester's problem of collinear points, pp.
65-66. Sketches history and gives
Kelly's proof.
W. O. J. Moser. Sylvester's problem, generalizations and
relatives. In his: Research Problems in Discrete Geometry 1981,
McGill University, Montreal, 1981.
Section 27, pp. 27‑1 -- 27‑14. Survey with 73 references.
(This problem is not in Part 1 of the 1984 ed. nor in the 1986 ed.)
6.J. FOUR BUGS AND OTHER PURSUIT PROBLEMS
The
general problem becomes too technical to remain recreational, so I will not try
to be exhaustive here.
Arthur Bernhart.
Curves
of pursuit. SM 20 (1954) 125‑141.
Curves
of pursuit -- II. SM 23 (1957) 49‑65.
Polygons
of pursuit. SM 24 (1959) 23‑50.
Curves
of general pursuit. SM 24 (1959) 189‑206.
Extensive history and analysis. First article covers one dimensional
pursuit, then two dimensional linear pursuit.
Second article deals with circular pursuit. Third article is the 'four bugs' problem -- analysis of
equilateral triangle, square, scalene triangle, general polygon, Brocard
points, etc. Last article includes such
variants as variable speed, the tractrix, miscellaneous curves, etc.
Mr. Ash, proposer; editorial
note saying there is no solver. Ladies'
Diary, 1748-47 = T. Leybourn, II: 15-17, quest. 310, with
'Solution by ΦIΛΟΠΟΝΟΣ, taken from
Turner's Exercises, where this question was afterwards proposed and answered
...' A fly is constrained to move on
the periphery of a circle. Spider
starts 30o away from the fly, but walks across the circle, always
aiming at the fly. If she catches the
fly 180o from her starting point, find the ratio of their
speeds.
ΦIΛΟΠΟΝΟΣ solves the more
general problem of finding the curve when the spider starts anywhere.
Carlile. Collection.
1793. Prob. CV, p. 62. A dog and a duck are in a circular pond of
radius 40 and they swim at the same speed.
The duck is at the edge and swims around the circumference. The dog starts at the centre and always
swims toward the duck, so the dog and the duck are always on a radius. How far does the dog swim in catching the
duck? He simply gives the result
as 20π. Letting R be the radius of the pond and V be
the common speed, I find the radius of the dog, r, is given by r = R sin Vt/R. Since the angle,
θ, of both the duck and the
dog is given by θ = Vt/R, the polar equation of the dog's path is r = R sin θ and the path is a semicircle whose diameter is the appropriate
radius perpendicular to the radius to the duck's initial position.
Cambridge Math. Tripos examination,
5 Jan 1871, 9 to 12. Problem 16, set by
R. K. Miller. Three bugs in general
position, but with velocities adjusted to make paths similar and keep the
triangle similar to the original.
Lucas. (Problem of three dogs.)
Nouvelle Correspondance Mathématique 3 (1877) 175‑176. ??NYS -- English in Arc., AMM 28 (1921) 184‑185
& Bernhart.
H. Brocard. (Solution of Lucas' problem.) Nouv. Corr. Math. 3 (1877) 280. ??NYS -- English in Bernhart.
Pearson. 1907.
Part II, no. 66: A duck hunt, pp. 66 & 172. Duck swims around edge of pond; spaniel starts for it from the centre at the
same speed.
A. S. Hathaway, proposer and
solver. Problem 2801. AMM 27 (1920) 31 & 28 (1921) 93‑97. Pursuit of a prey moving on a circle. Morley's and other solutions fail to deal
with the case when the velocities are equal.
Hathaway resolves this and shows the prey is then not caught.
F. V. Morley. A curve of pursuit. AMM 28 (1921) 54-61. Graphical solution of Hathaway's problem.
R. C. Archibald [Arc.] & H. P.
Manning. Remarks and historical notes
on problems 19 [1894], 160 [1902], 273 [1909] & 2801 [1920]. AMM 28 (1921) 91-93.
W. W. Rouse Ball. Problems -- Notes: 17: Curves of
pursuit. AMM 28 (1921) 278‑279.
A. H. Wilson. Note 19: A curve of pursuit. AMM 28 (1921) 327.
Editor's note to Prob. 2
(proposed by T. A. Bickerstaff), National Mathematics Magazine (1937/38) 417
cites Morley and Archibald and adds that some authors credit the problem to
Leonardo da Vinci -- e.g. MG (1930-31) 436 -- ??NYS
Nelson F. Beeler & Franklyn
M. Branley. Experiments in Optical
Illusion. Ill. by Fred H. Lyon. Crowell, 1951, An illusion doodle, pp. 68-71,
describes the pattern formed by four bugs starting at the corners of a square,
drawing the lines of sight at (approximately) regular intervals. Putting several of the squares together,
usually with alternating directions of motion, gives a pleasant pattern which
is now fairly common. They call this
'Huddy's Doodle', but give no source.
J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. 'Lion and man',
pp. 135‑136 (114‑117).
The 1986 ed. adds three diagrams and revises the text somewhat. I quote from it. "A lion and a man in a closed circular arena have equal
maximum speeds. What tactics should the
lion employ to be sure of his meal?"
This was "invented by R. Rado in the late thirties" and
"swept the country 25 years later".
[The 1953 ed., says Rado didn't publish it.] The correct solution "was discovered by Professor
A. S. Besicovitch in 1952".
[The 1953 ed. says "This has just been discovered ...; here is the
first (and only) version in print."]
C. C. Puckette. The curve of pursuit. MG 37 (No. 322) (Dec 1953) 256‑260. Gives the history from Bouguer in 1732. Solves a variant of the problem.
R. H. Macmillan. Curves of pursuit. MG 40 (No. 331) (Feb 1956) 1‑4. Fighter pursuing bomber flying in a straight line. Discusses firing lead and acceleration
problems.
Gamow & Stern. 1958.
Homing missiles. Pp. 112‑114.
Howard D. Grossman, proposer; unspecified solver. Problem 66 -- The walk around. In:
L. A. Graham; Ingenious Mathematical Problems and Methods; Dover,
1959, pp. 40 & 203‑205. Four
bugs -- asserts Grossman originated the problem.
I. J. Good. Pursuit curves and mathematical art. MG 43 (No. 343) (Feb 1959) 34‑35. Draws tangent to the pursuit curves in an
equilateral triangle and constructs various patterns with them. Says a similar but much simpler pattern was
given by G. B. Robison; Doodles; AMM 61 (1954) 381-386, but Robison's doodles
are not related to pursuit curves, though they may have inspired Good to use
the pursuit curves.
J. Charles Clapham. Playful mice. RMM 10 (Aug 1962) 6‑7.
Easy derivation of the distance travelled for n bugs at corners of a
regular n‑gon. [I don't see this result in Bernhart.]
C. G. Paradine. Note 3108:
Pursuit curves. MG 48 (No. 366)
(Dec 1964) 437‑439. Says Good
makes an error in Note 3079. He shows
the length of the pursuit curve in the equilateral triangle is ⅔
of the side and describes the curve as an equiangular spiral. Gives a simple proof that the length of the
pursuit curve in the regular n‑gon
is the side divided by (1 ‑ cos
2π/n).
M. S. Klamkin & D. J.
Newman. Cyclic pursuit or "The
three bugs problem". AMM 78 (1971)
631‑639. General treatment. Cites Bernhart's four SM papers and some of
the history therein.
P. K. Arvind. A symmetrical pursuit problem on the sphere
and the hyperbolic plane. MG 78 (No.
481) (Mar 1994) 30-36. Treats the n
bugs problems on the surfaces named.
Barry Lewis. A mathematical pursuit. M500 170 (Oct 1999) 1-8. Starts with equilateral triangular case,
giving QBASIC programs to draw the curves as well as explicit solutions. Then considers regular n-gons.
Then considers simple pursuit, one beast pursuing another while the
other moves along some given path.
Considers the path as a straight line or a circle. For the circle, he asserts that the analytic
solution was not determined until 1926, but gives no reference.
6.K. DUDENEY'S SQUARE TO TRIANGLE DISSECTION
Dudeney. Weekly Dispatch (6 Apr, 20 Apr, 4 May, 1902)
all p. 13.
Dudeney. The haberdasher's puzzle. London Mag. 11 (No. 64) (Nov 1903) 441 &
443. (Issue with solution not found.)
Dudeney. Daily Mail (1 & 8 Feb 1905) both p.
7.
Dudeney. CP.
1907. Prob. 25: The
haberdasher's puzzle, pp. 49‑50 & 178‑180.
Western Puzzle Works, 1926
Catalogue. No. 1712 -- unnamed, but
shows both the square and the triangle.
Apparently a four piece puzzle.
M. Adams. Puzzle Book. 1939. Prob. C.153:
Squaring a triangle, pp. 162 & 189.
Asserts that Dudeney's method works for any triangle, but his example is
close to equilateral and I recall that this has been studied and only certain
shapes will work??
Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge,
1941); revised ed., Educational
Publishers, St. Louis, 1949. Pp.
40-41. Extends to dissecting a
quadrilateral to a specified triangle and gives a number of related problems.
Two
ladders are placed across a street, each reaching from the base of the house on
one side to the house on the other side.
The
simple problem gives the heights
a, b that the ladders reach on the walls. If the height of the crossing is
c, we easily get 1/c = 1/a + 1/b. NOTATION -- this problem will be denoted by (a, b).
The
more common and more complex problem is where the ladders have lengths a
and b, the height of their crossing is
c and one wants the width d of
the street. If the heights of the
ladder ends are x, y,
this situation gives x2
‑ y2 = a2 ‑ b2 and
1/x + 1/y = 1/c which leads to a
quartic and there seems to be no simple solution. NOTATION -- this will be denoted
(a, b, c).
Mahavira. 850.
Chap. VII, v. 180-183, pp. 243-244.
Gives the simple version with a modification -- each ladder reaches from
the top of a pillar beyond the foot of the other pillar. The ladder from the top of pillar Y
(of height y) extends by
m beyond the foot of pillar X
and the ladder from the top of pillar
X (of height x)
reaches n beyond the foot of pillar Y.
The pillars are d apart.
Similar triangles then yield:
(d+m+n)/c = (d+n)/x + (d+m)/y and one can compute the various distances along the ground. He first does problems with m = n = 0,
which are the simple version of the problem, but since d is
given, he also asks for the distances on the ground.
v.
181. (16, 16) with d = 16.
v.
182. (36, 20) with d = 12.
v.
183. x, y, d, m, n =
12, 15, 4, 1, 4.
Bhaskara II. Lilavati.
1150. Chap. VI, v. 160. In Colebrooke, pp. 68‑69. (10, 15).
(= Bijaganita, ibid., chap. IV, v. 127, pp. 205‑206.)
Fibonacci. 1202.
Pp. 397‑398 (S: 543-544) looks like a crossed ladders problem but
is a simple right triangle problem.
Pacioli. Summa.
1494. Part II.
F.
56r, prob. 48. (4, 6).
F.
60r, prob. 64. (10, 15).
Hutton. A Course of Mathematics. 1798?
Prob. VIII, 1833: 430; 1857: 508.
A ladder 40 long in a roadway can reach 33
up one side and, from the same point, can reach 21
up the other side. How wide is
the street? This is actually a simple
right triangle problem.
Victor Katz reports that
Hutton's problem, with values 60; 37,
23 appears in a notebook of Benjamin
Banneker (1731-1806).
Loyd. Problem 48: A jubilee problem.
Tit‑Bits 32 (21 Aug,
11 & 25 Sep 1897) 385, 439 & 475.
Given heights of the ladder ends above ground and the width of the
street, find the height of the intersection.
However one wall is tilted and the drawing has it covered in decoration
so one may interpret the tilt in the wrong way.
Jno. A. Hodge, proposer; G. B. M. Zerr, solver. Problem 131. SSM 8 (1908) 786 & 9 (1909) 174‑175. (100, 80, 10).
W. V. N. Garretson,
proposer; H. S. Uhler, solver. Problem 2836. AMM 27 (1920) & 29 (1922) 181. (40, 25, 15).
C. C. Camp, proposer; W. J. Patterson & O. Dunkel,
solvers. Problem 3173. AMM 33 (1926) 104 & 34 (1927) 50‑51. General solution.
Morris Savage, proposer; W. E. Batzler, solver. Problem 1194. SSM 31 (1931) 1000
& 32 (1932) 212. (100, 80, 10).
S. A. Anderson, proposer; Simon Vatriquant, solver. Problem E210. AMM 43 (1936) 242 & 642‑643. General solution in integers.
C. R. Green, proposer; C. W. Trigg, solver. Problem 1498. SSM 37 (1937) 484 & 860‑861. (40, 30, 15). Trigg cites
Vatriquant for smallest integral case.
A. A. Bennett, proposer; W. E. Buker, solver. Problem E433. AMM 47 (1940) 487 & 48 (1941) 268‑269. General solution in integers using four
parameters.
J. S. Cromelin, proposer; Murray Barbour, solver. Problem E616 -- The three ladders. AMM 51 (1944) 231 & 592. Ladders of length 60 & 77
from one side. A ladder from the
other side crosses them at heights
17 & 19.
How long is the third ladder and how wide is the street?
Geoffrey Mott-Smith. Mathematical Puzzles for Beginners and
Enthusiasts. (Blakiston, 1946); revised ed., Dover, 1954. Prob. 103: The extension ladder, pp. 58-59
& 176‑178. Complex problem
with three ladders.
Arthur Labbe, proposer; various solvers. Problem 25 -- The two ladders.
Sep 1947 [date given in Graham's second book, cited at 1961]. In:
L. A. Graham; Ingenious Mathematical Problems and Methods; Dover, 1959,
pp. 18 & 116‑118. (20, 30,
8).
M. Y. Woodbridge, proposer and
solver. Problem 2116. SSM 48 (1948) 749 & 49 (1949) 244‑245. (60, 40, 15). Asks for a trigonometric solution. Trigg provides a list of early references.
Robert C. Yates. The ladder problem. SSM 51 (1951) 400‑401. Gives a graphical solution using hyperbolas.
G. A. Clarkson. Note 2522:
The ladder problem. MG 39 (No.
328) (May 1955) 147‑148. (20, 30,
10). Let A = Ö(a2 ‑ b2) and set
x = A sec t, y = A tan t. Then
cos t + cot t = A and he gets a trigonometrical solution. Another method leads to factoring the
quartic in terms of a constant k whose square satisfies a cubic.
L. A. Graham. The Surprise Attack in Mathematical
Problems. Dover, 1968. Problem 6: Searchlight on crossed ladders,
pp. 16-18. Says they reposed Labbe's
Sep 1947 problem in Jun 1961. Solution
by William M. Dennis which is the same trigonometric method as Clarkson.
H. E. Tester. Note 3036:
The ladder problem. A solution
in integers. MG 46 (No. 358) (Dec 1962)
313‑314. A four parameter,
incomplete, solution. He finds the
example (119, 70, 30).
A. Sutcliffe. Complete solution of the ladder problem in
integers. MG 47 (No. 360) (May 1963)
133‑136. Three parameter
solution. First few examples are: (119, 70, 30); (116, 100, 35);
(105, 87, 35). Simpler than
Anderson and Bennett/Buker.
Alan Sutcliffe, proposer; Gerald J. Janusz, solver. Problem 5323 -- Integral solutions of the
ladder problem. AMM 72 (1965) 914 &
73 (1966) 1125-1127. Can the
distance f between the tops of the ladders be integral? (80342, 74226, 18837) has
x = 44758,
y = 32526, d =
66720, f = 67832. This is not known to be the smallest
example.
Anon. A window cleaner's problem.
Mathematical Pie 51 (May 1967) 399.
From a point in the road, a ladder can reach 30 ft up on one side
and 40
ft up on the other side. If the
two ladder positions are at right angles, how wide is the road?
J. W. Gubby. Note 60.3:
Two chestnuts (re-roasted). MG
60 (No. 411) (Mar 1976) 64-65. 1. Given
heights of ladders as a, b, what is the height c of their intersection? Solution:
1/c = 1/a + 1/b or c = ab/(a+b). 2.
The usual ladder problem -- he finds a quartic.
J. Jabłkowski. Note 61:11:
The ladder problem solved by construction. MG 61 (No. 416) (Jun 1977) 138.
Gives a 'neusis' construction.
Cites Gubby.
Birtwistle. Calculator Puzzle Book. 1978.
Prob. 83, A second ladder problem, pp. 58-59 & 115-118. (15, 20, 6). Uses xy as a variable to simplify the quartic for
numerical solution and eventually gets
11.61.
See: Gardner, Circus, p. 266 & Schaaf for more references. ??follow up.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. The tangled ladders, pp. 43-44 & 116. (30, 20, 10). Gives answer 12.311857... with no explanation.
A
ladder of length L is placed to just clear a box of width w
and height h at the base of a wall. How high does the ladder reach? Denote this by (w, h, L). Letting x be
the horizontal distance of the foot and
y be the vertical distance of
the top of the ladder, measured from the foot of the wall, we get x2 + y2 = L2 and
(x‑w)(y‑h) = wh,
which gives a quartic in general.
But if w = h, then use of
x + y as a variable reduces the
quartic to a quadratic. In this case,
the idea is old -- see e.g. Simpson.
The
question of determining shortest ladder which can fit over a box of width w
and height h is the same as determining the longest
ladder which will pass from a corridor of width w into another corridor
of width h. See Huntington below and section 6.AG.
Simpson. Algebra.
1745. Section XVIII, prob. XV,
p. 250 (1790: prob. XIX, pp. 272-273).
"The Side of the inscribed Square
BEDF, and the Hypotenuse AC
of a right-angled Triangle
ABC being given; to determine
the other two Sides of the Triangle AB and
BC." Solves "by
considering x + y as one Quantity".
Pearson. 1907.
Part II, no. 102: Clearing the wall, p. 103. For (15, 12, 52), the ladder reaches 48.
D. John Baylis. The box and ladder problem. MTg 54 (1971) 24. (2, 2, 10). Finds the
quartic which he solves by symmetry.
Editorial note in MTg 57 (1971) 13 says several people wrote to say that
use of similar triangles avoids the quartic.
Birtwistle. Math. Puzzles & Perplexities. 1971.
The ladder and the box problem, pp. 44-45. = Birtwistle; Calculator Puzzle Book; 1978; Prob. 53: A ladder
problem, pp. 37 & 96‑98.
(3, 3, 10). Solves by
using x + y - 6 as a variable.
Monte Zerger. The "ladder problem". MM 60:4 (1987) 239‑242. (4, 4, 16).
Gives a trigonometric solution and a solution via two quadratics.
Oliver D. Anderson. Letter.
MM 61:1 (1988) 63. In response
to Zerger's article, he gives a simpler derivation.
Tom Heyes. The old box and ladder problem --
revisited. MiS 19:2 (Mar 1990) 42‑43. Uses a graphic calculator to find roots
graphically and then by iteration.
A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5.
Does usual problem, getting a quartic.
Then finds the shortest ladder.
[This turns out to be the same as the longest ladder one can get around
a corner from corridors of widths
w and h, so this problem is
connected to 6.AG.]
David Singmaster. Integral solutions of the ladder over box
problem. In preparation. Easily constructs all the primitive integral
examples from primitive Pythagorean triples.
E.g. for the case of a square box, i.e.
w = h, if X, Y, Z
is a primitive Pythagorean triple, then the corresponding primitive
solution has w = h = XY, x = X (X + Y), y = Y (X + Y), L = Z (X + Y), and
remarkably, x - h = X2, y - w = Y2.
These involve finding the shortest
distance over the surface of a cube or cylinder. I've just added the cylindrical form -- see Dudeney (1926),
Perelman and Singmaster. The shortest
route from a corner of a cube or cuboid to a diagonally opposite corner must
date back several centuries, but I haven't seen any version before 1937! I don't know if other shapes have been done
-- the regular (and other) polyhedra and the cone could be considered.
Two-dimensional
problems are in 10.U.
Loyd. The Inquirer (May 1900).
Gives the Cyclopedia problem.
??NYS -- stated in a letter from Shortz.
Dudeney. Problem 501 -- The spider and the fly. Weekly Dispatch (14 &
28 Jun 1903) both p. 16. 4
side version.
Dudeney. Breakfast table problems, No. 320 -- The
spider and the fly. Daily Mail (18 &
21 Jan 1905) both p. 7. Same
as the above problem.
Dudeney. Master of the breakfast table problem. Daily Mail (1 & 8 Feb 1905) both p.
7. Interview with Dudeney in which he gives
the 5 side version.
Ball. MRE, 4th ed., 1905, p. 66.
Gives the 5 side version, citing the Daily Mail of
1 Feb 1905. He says he heard
a similar problem in 1903 -- presumably Dudeney's first version. In the 5th ed., 1911, p. 73, he attributes
the problem to Dudeney.
Dudeney. CP.
1907. Prob. 75: The spider and
the fly, pp. 121‑122 & 221‑222. 5 side version with discussion of various generalizations.
Dudeney. The world's best problems. 1908.
Op. cit. in 2. P. 786 gives the
five side version.
Sidney J. Miller. Some novel picture puzzles -- No. 6. Strand Mag. 41 (No. 243) (Mar 1911) 372 &
41 (No. 244) (Apr 1911) 506.
Contest between two snails.
Better method uses four sides, similar to Dudeney's version, but with
different numbers.
Loyd. The electrical problem.
Cyclopedia, 1914, pp. 219 & 368 (= MPSL2, prob. 149, pp. 106
& 169 = SLAHP: Wiring the hall, pp.
72 & 114). Same as Dudeney's first,
four side, version. (In MPSL2, Gardner
says Loyd has simplified Dudeney's 5 side problem. More likely(?) Loyd had only seen Dudeney's earlier 4 side
problem.)
Dudeney. MP.
1926. Prob. 162: The fly and the
honey, pp. 67 & 157. (= 536, prob.
325, pp. 112 & 313.)
Cylindrical problem.
Perelman. FFF.
1934. The way of the fly. 1957: Prob. 68, pp. 111‑112 & 117‑118; 1979: Prob. 72, pp. 136 & 142‑144. MCBF: Prob. 72, pp. 134 & 141-142. Cylindrical form, but with different numbers
and arrangement than Dudeney's MP problem.
Haldeman-Julius. 1937.
No. 34: The louse problem, pp. 6 & 22. Room 40 x 20 x 10 with louse at a corner wanting to go to a
diagonally opposite corner. Problem
sent in by J. R. Reed of Emmett, Idaho.
Answer is 50!
M. Kraitchik. Mathematical Recreations, 1943, op. cit. in
4.A.2, chap. 1, prob. 7, pp. 17‑21.
Room with 8 equal routes from spider to fly. (Not in his Math. des Jeux.)
Sullivan. Unusual.
1943. Prob. 10: Why not
fly? Find shortest route from a corner
of a cube to the diagonally opposite corner.
William R. Ransom. One Hundred Mathematical Curiosities. J. Weston Walch, Portland, Maine, 1955. The spider problem, pp. 144‑146. There are three types of path, covering 3, 4 and 5
sides. He determines their
relative sizes as functions of the room dimensions.
Birtwistle. Math. Puzzles & Perplexities. 1971.
Round
the cone, pp. 144 & 195. What is
the shortest distance from a point
P around a cone and back to P?
Answer is "An ellipse", which doesn't seem to answer the
question. If the cone has height H,
radius R and
P is l from the apex, then the slant height L
is Ö(R2 + H2), the angle of the opened out cone is θ = 2πR/L and the required distance is 2l sin θ/2.
Spider
circuit, pp. 144 & 198. Spider is
at the midpoint of an edge of a cube.
He wants to walk on each of the faces and return. What is his shortest route? Answer is "A regular hexagon. (This may be demonstrated by putting a
rubber band around a cube.)"
David Singmaster. The spider spied her. Problem used as: More than one way to catch a fly, The Weekend Telegraph (2 Apr
1984). Spider inside a glass tube, open
at both ends, goes directly toward a fly on the outside. When are there two equally short paths? Can there be more than two shortest routes?
Yoshiyuki Kotani has posed the
following general and difficult problem.
On an a x b x c cuboid, which two points are furthest apart,
as measured by an ant on the surface?
Dick Hess has done some work on this, but I believe that even the case
of square cross-section is not fully resolved.
6.N. DISSECTION OF A 1 x 1 x 2 BLOCK TO A CUBE
W. F. Cheney, Jr.,
proposer; W. R. Ransom; A. H. Wheeler,
solvers. Problem E4. AMM 39 (1932) 489; 40 (1933) 113-114
& 42 (1934) 509-510. Ransom finds a solution in 8
pieces; Wheeler in 7.
Harry Lindgren. Geometric Dissections. Van Nostrand, Princeton, 1964. Section 24.2, p. 120 gives a variant of
Wheeler's solution.
Michael Goldberg. A duplication of the cube by dissection and
a hinged linkage. MG 50 (No. 373)
(Oct 1966) 304‑305. Shows that a
hinged version exists with 10 pieces. Hanegraaf, below, notes that there are actually 12 pieces here.
Anton Hanegraaf. The Delian Altar Dissection. Polyhedral Dissections, Elst, Netherlands,
1989. Surveys the problem, gives a 6
piece solution and a 7 piece hinged solution.
6.O. PASSING A CUBE THROUGH AN EQUAL OR SMALLER CUBE. PRINCE RUPERT'S PROBLEM
The
projection of a unit cube along a space diagonal is a regular hexagon of
side Ö2/Ö3. The largest square inscribable in this hexagon has edge Ö6 - Ö2 =
1.03527618. By passing the larger cube
on a slant to the space diagonal, one can get the larger cube having edge 3Ö2/4 = 1.06066172.
There
are two early attributions of this.
Wallis attributes it to Prince Rupert, but Hennessy says Philip Ronayne
of Cork invented it. I have discovered
a possible connection. Prince Rupert of
the Rhine (1619-1682), nephew of Charles I, was a major military figure of his
time, becoming commander-in-chief of Charles I's armies in the 1640s. In 1648-1649, he was admiral of the King's fleet
and was blockaded with 16 ships in Kinsale Harbor for 20 months. Kinsale is about 20km south of Cork.
Ronayne
wrote an Algebra, of which only a second edition of 1727 is in the BL. Schrek has investigated the family histories
and says Ronayne lived in the early 18C.
This would seem to make him too young to have met Rupert. Perhaps Rupert invented the problem while in
Kinsale and this was conveyed to Ronayne some years later. Does anyone know the dates of Ronayne or of
the 1st ed (Schrek only located the BL example of the 2nd ed)? I cannot find anything on him in Wallis,
May, Poggendorff, DNB, but Google has turned up a reference to a 1917 history
of the family which Schrek cites, but I have not yet tried to find this.
Hennessy's
article says a little about Daniel Voster and details are in Wallis's . His father, Elias (1682 ‑ >1728)
wrote an Arithmetic, of which Wallis lists 30 editions. The BL lists one as late as 1829. The son, Daniel (1705 ‑ >1760) was
a schoolmaster and instrument maker who edited later versions of his father's
arithmetic. The 1750 History of Cork
quoted by Hennessy says the author had seen the cubes with Daniel. Hennessy conjectures that his example was
made specially, perhaps under the direction of a mathematician. It seems likely that Daniel knew Ronayne and
made this example for him.
John Wallis. Perforatio cubi, alterum ipsi aequalem
recipiens. (De Algebra Tractatus; 1685;
Chap. 109) = Opera Mathematica,
vol. II, Oxford, 1693, pp. 470‑471, ??NYS. Cites Rupert as the source of the equal cube version. (Latin and English in Schrek.) Scriba, below, found an errata slip in
Wallis's copy of his Algebra in the Bodleian.
This corrects the calculations, but was published in the Opera, p. 695.
Ozanam‑Montucla. 1778.
Percer un cube d'une ouverture, par laquelle peut passer un autre cube
égal au premier. Prob. 30 & fig.
53, plate 7, 1778: 319-320; 1803:
315-316; 1814: 268-269. Prob. 29, 1840: 137. Equal cubes with diagonal movement.
J. H. van Swinden. Grondbeginsels der Meetkunde. 2nd ed., Amsterdam, 1816, pp. 512‑513,
??NYS. German edition by C. F. A.
Jacobi, as: Elemente der Geometrie,
Friedrich Frommann, Jena, 1834, pp. 394-395. Cites Rupert and Wallis and gives a simple construction, saying
Nieuwland has found the largest cube which can pass through a cube.
Peter Nieuwland. (Finding of maximum cube which passes
through another). In: van Swinden, op. cit., pp. 608‑610; van Swinden‑Jacobi, op. cit. above,
pp. 542-544, gives Nieuwland's proof.
Cundy and Rollett, p. 158, give
references to Zacharias (see below) and to Cantor, but Cantor only cites
Hennessy.
H. Hennessy. Ronayne's cubes. Phil. Mag. (5) 39 (Jan‑Jun 1895) 183‑187. Quotes, from Gibson's 'History of Cork', a
passage taken from Smith's 'History of Cork', 1st ed., 1750, vol. 1, p. 172,
saying that Philip Ronayne had invented this and that a Daniel Voster had made
an example, which may be the example owned by Hennessy. He gives no reference to Rupert. He finds the dimensions.
F. Koch & I. Reisacher. Die Aufgabe, einen Würfel durch einen andern
durchzuschieben. Archiv Math. Physik
(3) 10 (1906) 335‑336. Brief
solution of Nieuwland's problem.
M. Zacharias. Elementargeometrie und elementare
nicht-Euklidische Geometrie in synthetischer Behandlung. Encyklopädie der Mathematischen
Wissenschaften. Band III, Teil 1,
2te Hälfte. Teubner, Leipzig,
1914-1931. Abt. 28: Maxima und
Minima. Die isoperimetrische
Aufgabe. Pp. 1133-1134. Attributes it to Prince Rupert, following
van Swinden. Cites Wallis &
Ronayne, via Cantor, and Nieuwland, via van Swinden.
U. Graf. Die Durchbohrung eines Würfels mit einem
Würfel. Zeitschrift math. naturwiss.
Unterricht 72 (1941) 117. Nice photos
of a model made at the Technische Hochschule Danzig. Larger and better versions of the same photos can be found in: W.
Lietzmann & U. Graf; Mathematik in Erziehung und Unterricht; Quelle &
Meyer, Leipzig, 1941, vol. 2, plate 3, opp. p. 168, but I can't find any
associated text for it.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. No. 12: Curios [sic] cubes, p. 14. First says it can be done with equal cubes
and then a larger can pass through a smaller.
Claims that the larger cube can be about 1.1, but this is due to
an error -- he thinks the hexagon has the same diameter as the cube itself.
H. D. Grossman, proposer; C. S. Ogilvy & F. Bagemihl,
solvers. Problem E888 -- Passing a cube
through a cube of same size. AMM 56 (1949)
632 ??NYS & 57 (1950) 339. Only considers cubes of the same size, though Bagemihl's solution
permits a slightly larger cube. No
references.
D. J. E. Schrek. Prince Rupert's problem and its extension by
Pieter Nieuwland. SM 16 (1950) 73‑80
& 261‑267. Historical survey,
discussing Rupert, Wallis, Ronayne, van Swinden & Nieuwland. Says Ronayne is early 18C.
M. Gardner. SA (Nov 1966) = Carnival, pp. 41‑54. The largest square inscribable in a cube is
the cross section of the maximal hole through which another cube can pass.
Christoph J. Scriba. Das Problem des Prinzen Ruprecht von der
Pfalz. Praxis der Mathematik 10 (1968)
241-246. ??NYS -- described by Scriba
in an email to HM Mailing List, 20 Aug 1999.
Describes the correction to Wallis's work and considers the problem for
the tetrahedron and octahedron.
Gardner. MM&M.
1956. Chap. 7 & 8:
Geometrical Vanishing -- Parts I & II, pp. 114‑155. Best extensive discussion of the subject and
its history.
Gardner. SA (Jan 1963) c= Magic Numbers, chap.
3. Discusses application to making an
extra bill and Magic Numbers adds citations to several examples of people
trying it and going to jail.
Gardner. Advertising premiums to beguile the mind:
classics by Sam Loyd, master puzzle‑poser. SA (Nov 1971) = Wheels, Chap. 12. Discusses several forms.
S&B, p. 144, shows several
versions.
6.P.1. PARADOXICAL DISSECTIONS OF THE CHESSBOARD BASED ON FIBONACCI NUMBERS
Area
63 version: AWGL, Dexter, Escott,
White, Loyd, Ahrens, Loyd Jr., Ransom.
(W. Leybourn. Pleasure with Profit. 1694.
?? I cannot recall the source of
this reference and think it may be an error.
I have examined the book and find nothing relevant in it.)
Loyd. Cyclopedia, 1914, pp. 288 & 378. 8 x 8 to 5 x 13
and to an area of 63. Asserts Loyd presented the first of these in
1858. Cf Loyd Jr, below.
O. Schlömilch. Ein geometrisches Paradoxon. Z. Math. Phys., 13 (1868) 162. 8 x 8
to 5 x 13. (This article is only signed Schl. Weaver, below, says this is Schlömilch, and
this seems right as he was a co‑editor at the time. Coxeter (SM 19 (1953) 135‑143) says it
is V. Schlegel, apparently confusing it with the article below.) Doesn't give any explanation, leaving it as
a student exercise.
F. J. Riecke. Op. cit. in 4.A.1. Vol. 3, 1873. Art. 16:
Ein geometrisches Paradoxon. Quotes Schlömilch
and explains the paradox.
G. H. Darwin. Messenger of Mathematics 6 (1877) 87. 8 x 8
to 5 x 13 and generalizations.
V. Schlegel. Verallgemeinerung eines geometrischen
Paradoxons. Z. Math. Phys. 24
(1879) 123‑128 & Plate I. 8 x
8 to
5 x 13 and generalizations.
Mittenzwey. 1880.
Prob. 299, pp. 54 & 105;
1895?: 332, pp. 58 & 106-107;
1917: 332, pp. 53 & 101. 8 x
8 to
5 x 13. Clear explanation.
The Boy's Own Paper. No. 109, vol. III (12 Feb 1881) 327. A puzzle.
8 x 8 to 5 x 13
without answer.
Richard A. Proctor. Some puzzles. Knowledge 9 (Aug 1886) 305-306.
"We suppose all the readers ... know this old puzzle." Describes and explains 8 x 8
to 5 x 13. Gives a different method of cutting so that
each rectangle has half the error -- several typographical errors.
Richard A. Proctor. The sixty-four sixty-five puzzle. Knowledge 9 (Oct 1886) 360-361. Corrects the above and explains it in more
detail.
Will Shortz has a puzzle trade
card with the 8 x 8 to 5
x 13 version, c1889.
Ball. MRE, 1st ed., 1892, pp. 34‑36. 8 x 8 to 5 x 13
and generalizations. Cites
Darwin and describes the examples in Ozanam-Hutton (see Ozanam-Montucla in
6.P.2). In the 5th ed., 1911, p. 53, he
changes the Darwin reference to Schlömilch.
In the 7th ed., 1917, he only cites the Ozanam-Hutton examples.
Clark. Mental Nuts. 1897, no.
33; 1904, no. 41; 1916, no. 43. Four peculiar drawings. 8
x 8 to
5 x 13.
Carroll-Collingwood. 1899.
Pp. 316-317 (Collins: 231 and/or 232 (lacking in my copy)) = Carroll-Wakeling II, prob. 7: A
geometrical paradox, pp. 12 & 7.
8 x 8 to 5 x 13.
Carroll may have stated this as early as 1888. Wakeling says the papers among which this was found on Carroll's
death are now in the Parrish Collection at Princeton University and suggests
Schlömilch as the earliest version.
AWGL (Paris). L'Echiquier Fantastique. c1900.
Wooden puzzle of 8 x 8 to 5
x 13 and to area 63.
??NYS -- described in S&B, p. 144.
Walter Dexter. Some postcard puzzles. Boy's Own Paper (14 Dec 1901) 174‑175. 8 x 8
to 5 x 13 and to area
63.
C. A. Laisant. Initiation Mathématique. Georg, Geneva & Hachette, Paris,
1906. Chap. 63: Un paradoxe: 64 = 65,
pp. 150-152.
Wm. F. White. In the mazes of mathematics. A series of perplexing puzzles. III.
Geometric puzzles. The Open
Court 21 (1907) 241‑244.
Shows 8 x 8 to 5
x 13 and a two‑piece 11 x 13
to area 145.
E. B. Escott. Geometric puzzles. The Open Court 21 (1907) 502‑505. Shows 8 x 8 to area
63 and discusses the connection
with Fibonacci numbers.
William F. White. Op. cit. in 5.E. 1908. Geometric puzzles,
pp. 109‑117. Partly based on
above two articles. Gives 8 x 8
to 5 x 13 and to area
63. Gives an extension which
turns 12 x 12 into 8 x 18 and into area 144, but turns 23 x 23
into 16 x 33 and into area 145. Shows a puzzle of
Loyd: three‑piece 8 x 8
into 7 x 9.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. 5 x 5 into four pieces that make a 3 x 8.
M. Adams. Indoor Games. 1912. Is 64
equal to 65? Pp. 345-346 with fig. on p. 344.
Loyd. Cyclopedia. 1914. See entry at 1858.
W. Ahrens. Mathematische Spiele. Teubner, Leipzig. 3rd ed., 1916, pp. 94‑95 & 111‑112. The 4th ed., 1919, and 5th ed., 1927, are
identical with the 3rd ed., but on different pages: pp. 101‑102
& pp. 118‑119. Art. X.
65 = 64 = 63 gives 8 x 8
to 5 x 13 and to area
63. The area 63
case does not appear in the 2nd ed., 1911, which has Art. V. 64 = 65, pp. 107 & 118‑119 and this material is not in the 1st
ed. of 1907.
Tom Tit?? In Knott, 1918, but I can't find it in Tom
Tit. No. 3: The square and the
rectangle: 64 = 65!, pp. 15-16.
Clearly explained.
Hummerston. Fun, Mirth & Mystery. 1924.
A puzzling paradox, pp. 44 & 185.
Usual 8 x 8 to
5 x 13, but he erases
the chessboard lines except for the cells the cuts pass through, so one way has
16 cells, the other way has 17 cells.
Reasonable explanation.
Collins. Book of Puzzles. 1927. A paradoxical
puzzle, pp. 4-5. 8 x 8 to 5
x 13. Shades the unit cells that the
lines pass through and sees that one way has 16 cells, the other way has 17
cells, but gives only a vague explanation.
Loyd Jr. SLAHP.
1928. A paradoxical puzzle, pp.
19‑20 & 90. Gives 8 x 8
to 5 x 13. "I have discovered a companion piece
..." and gives the 8 x 8 to area
63 version. But cf AWGL, Dexter, etc. above.
W. Weaver. Lewis Carroll and a geometrical
paradox. AMM 45 (1938) 234‑236. Describes unpublished work in which Carroll
obtained (in some way) the generalizations of the 8 x 8 to 5 x 13
in about 1890‑1893. Weaver
fills in the elementary missing arguments.
W. R. Ransom, proposer; H. W. Eves, solver. Problem E468. AMM 48 (1941) 266 & 49 (1942) 122‑123. Generalization of the 8 x 8
to area 63 version.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. No. 23: Summat for nowt?, pp. 27-28. 8 x 8
to 5 x 13, clearly drawn.
Warren Weaver. Lewis Carroll: Mathematician. Op. cit. in 1. 1956. Brief mention of 8 x 8
to 5 x 13. John B. Irwin's letter gives generalizations
to other consecutive triples of Fibonacci numbers (though he doesn't call them
that). Weaver's response cites his 1938
article, above.
In
several early examples, the authors appear unaware that area has vanished!
Pacioli. De Viribus.
c1500. Ff. 189v - 191r. Part 2.
LXXIX. Do(cumento). un tetragono saper lo longare con restregnerlo
elargarlo con scortarlo (a tetragon knows lengthening and contraction,
enlarging with shortening ??) = Peirani
250-252. Convert a 4 x 24
rectangle to a 3 x 32 using one cut into two pieces. Pacioli's
description
is cryptic but seems to have two cuts, making d c
three
pieces. There is a diagram at the bottom
of f. 190v, badly k f
e
redrawn
on Peirani 458. Below this is a
inserted note which Peirani
252
simply mentions as difficult to read, but can make sense. The g
points
are as laid out at the right. abcd is the original 4 x 24 h a
o b
rectangle. g is
one unit up from a and
e is one unit down from c.
Cut
from c
to g and from e parallel to the base, meeting cg
at f. Then move cdg to
fkh and move fec
to hag. Careful rereading of Pacioli seems to show
he is using a trick! He cuts from e
to f to g. then turns over the upper piece and slides
it along so that he can continue his cut from
g to h, which is where f
to c is now. This gives three
pieces from a single cut! Pacioli clearly
notes that the area is conserved.
Although
not really in this topic, I have put it here as it seems to be a predecessor of
this topic and of 6.AY.
Sebastiano Serlio. Libro Primo d'Architettura. 1545.
This is the first part of his Architettura, 5 books, 1537-1547, first
published together in 1584. I have seen
the following editions.
With
French translation by Jehan Martin, no publisher shown, Paris, 1545,
f. 22.r. ??NX
1559. F. 15.v.
Francesco
Senese & Zuane(?) Krugher, Venice, 1566, f. 16.r. ??NX
Jacomo
de'Franceschi, Venice, 1619, f. 16.r.
Translated
into Dutch by Pieter Coecke van Aelst as:
Den eerstē vijfsten boeck van architecturē; Amsterdam,
1606. This was translated into English
as: The Five Books of Architecture;
Simon Stafford, London, 1611 = Dover,
1982. The first Booke, f. 12v.
3
x 10 board is cut on a diagonal and
slid to form a 4 x 7 table with
3 x 1 left over, but
he doesn't actually put the two leftover pieces together nor notice the area
change!
Pietro Cataneo. L'Architettura di Pietro Cataneo
Senese. Aldus, Venice, 1567. ??NX.
Libro Settimo.
P.
164, prop. XXVIIII: Come si possa accresciere una stravagante larghezza. Gives a correct version of Serlio's process.
P.
165, prop. XXX: Falsa solutione del Serlio.
Cites p. xxii of Serlio.
Carefully explains the error in Serlio and says his method is
"insolubile, & mal pensata".
Schwenter. 1636.
Part 15, ex. 14, p. 541: Mit einem länglichten schmahlen Brett /
für ein bräites Fenster einen Laden zu
machen. Cites Gualtherus Rivius,
Architectur. Discusses Serlio's
dissection as a way of making a 4 x
7 from a 3 x 10 but doesn't notice
the area change.
Gaspar Schott. Magia Universalis. Joh. Martin Schönwetter, Bamberg, Vol. 3, 1677. Pp. 704-708 describes Serlio's error in
detail, citing Serlio. ??NX of plates.
I have a vague reference to the
1723 ed. of Ozanam, but I have not seen it in the 1725 ed. -- this may be an
error for the 1778 ed. below.
Minguet. 1755.
Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Same as Hooper. Not in 1733 ed.
Vyse. Tutor's Guide. 1771? Prob. 8, 1793: p. 304, 1799: p. 317 &
Key p. 358. Lady has a table 27
square and a board 12 x 48. She cuts the board into two 12 x 24
rectangles and cuts each rectangle along a diagonal. By placing the diagonals of these pieces on
the sides of her table, she makes a table
36 square. Note that
362 = 1296 and 272 + 12 x 48 =
1305. Vyse is clearly unaware
that area has been created. By dividing
all lengths by 3, one gets a version where one unit of area is
lost. Note that 4, 8, 9
is almost a Pythagorean triple.
William Hooper. Rational Recreations. 1774.
Op. cit. in 4.A.1. Vol. 4, pp.
286‑287: Recreation CVI -- The geometric money. 3 x 10 cut into four pieces
which make a 2 x 6 and a
4 x 5. (The diagram is shown in
Gardner, MM&M, pp. 131‑132.)
(I recently saw that an edition erroneously has a 3 x 6
instead of a 2 x 6 rectangle.
This must be the 1st ed. of 1774, as it is correct in my 2nd ed. of 1782.)
Ozanam-Montucla. 1778.
Transposition de laquelle semble résulter que le tout peut être égal à
la partie. Prob. 21 & fig. 127,
plate 16, 1778: 302-303 & 363;
1803: 298-299 & 361; 1814:
256 & 306; 1840: omitted. 3 x 11
to 2 x 7 and
4 x 5. Remarks that M. Ligier
probably made some such mistake in showing
172 = 2 x 122
and this is discussed further on the later page.
E. C. Guyot. Nouvelles Récréations Physiques et
Mathématiques. Nouvelle éd. La Librairie, Rue S. André‑des‑Arcs[sic],
Paris, Year 7 [1799]. Vol. 2, Deuxième
récréation: Or géométrique -- construction, pp. 41‑42 & plate 6, opp.
p. 37. Same as Hooper.
Manuel des Sorciers. 1825.
Pp. 202-203, art. 19. ??NX Same as Hooper.
The Boy's Own Book. The geometrical money. 1828: 413;
1828-2: 419; 1829 (US):
212; 1855: 566‑567; 1868: 669.
Same as Hooper.
Magician's Own Book. 1857.
Deceptive vision, pp. 258-259.
Same as Hooper. = Book of 500
Puzzles, 1859, pp. 72-73.
Illustrated Boy's Own
Treasury. 1860. Optics: Deceptive vision, p. 445. Same as Hooper. Identical to Book of 500 Puzzles.
Wemple & Company (New
York). The Magic Egg Puzzle. ©1880.
S&B, p. 144. Advertising card,
the size of a small postcard, but with ads for Rogers Peet on the back. Starts with 9 eggs. Cut into four rectangles and reassemble to
make 6, 7, 8, 10, 11, 12 eggs.
R. March & Co. (St. James's
Walk, Clerkenwell). 'The Magical Egg
Puzzle', nd [c1890]. (I have a
photocopy.) Four rectangles which
produce 6, 7, ..., 12 eggs.
Identical to the Wemple version, but with Wemple's name removed. I only have a photocopy of the front of this
and I don't know what's on the back. I also have a photocopy of the
instructions.
Loyd. US Patent 563,778 -- Transformation Picture. Applied: 11 Mar 1896; patented: 14 Jul 1896. 1p + 1p diagrams. Simple rotating version using 8 to 7 objects.
Loyd. Get Off the Earth. Puzzle
notices in the Brooklyn Daily Eagle (26 Apr ‑ 3 May 1896), printing
individual Chinamen. Presenting all of
these at an office of the newspaper gets you an example of the puzzle. Loyd ran discussions on it in his Sunday
columns until 3 Jan 1897 and he also sold many versions as advertising
promotions. S&B, p. 144, shows
several versions.
Loyd. Problem 17: Ye castle donjon.
Tit‑Bits 31 (6 & 27 Feb
& 6 &
20 Mar 1897) 343, 401, 419
& 455. = Cyclopedia, 1914, The architect's puzzle,
pp. 241 & 372. 5 x 25 to area 124.
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. Discusses and shows
Get Off the Earth.
Ball. MRE, 4th ed., 1905, pp. 50-51: Turton's seventy-seven
puzzle. Additional section describing
Captain Turton's 7 x 11 to 7
x 11 with one projecting square, using
bevelled cuts. This is dropped from the
7th ed., 1917.
William F. White. 1907 & 1908. See entries in 6.P.1.
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Gives "Get Off
the Earth" on p. 785.
Loyd. Teddy and the Lions.
Gardner, MM&M, p. 123, says he has seen only one example, made as a
promotional item for the Eden Musee in Manhattan. This has a round disc, but two sets of figures -- 7 natives and 7
lions which become 6 natives and 8 lions.
Dudeney. A chessboard fallacy. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909) 676 (= AM, prob. 413,
pp. 141 & 247). (There is a solution
in Strand Mag. 39 (No. 229) (Jan 1910) ??NYS.) 8 x 8 into 3 pieces which
make a 9 x 7.
Fun's Great Baseball
Puzzle. Will Shortz gave this out at
IPP10, 1989, as a colour photocopy, 433 x 280 mm (approx. A3). ©1912 by the Press Publishing Co (The New
York World). I don't know if Fun was
their Sunday colour comic section or what.
One has to cut it diagonally and slide one part along to change from 8
to 9 boys.
Loyd. The gold brick puzzle.
Cyclopedia, 1914, pp. 32 & 342 (= MPSL1, prob. 24, pp. 22 & 129). 24 x 24
to 23 x 25.
Loyd. Cyclopedia. 1914. "Get off the earth", p. 323. Says over 10 million were sold. Offers prizes for best answers received in
1909.
Loyd Jr. SLAHP.
1928. "Get off the
Earth" puzzle, pp. 5‑6. Says
'My "Missing Chinaman Puzzle"' of 1896. Gives a simple and clear explanation.
John Barnard. The Handy Boy's Book. Ward, Lock & Co., London, nd
[c1930?]. Some interesting optical
illusions, pp. 310-311. Shows a card
with 11 matches and a diagonal cut so that sliding it one place makes 10
matches.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. No. 24: A chessboard fallacy, pp. 28-29. 8 x 8
cut with a diagonal of a 8 x
7 region, then pieces slid and a
triangle cut off and moved to the other end to make a 9 x 7. Clear
illustration.
Mel Stover. From 1951, he devised a number of variations
of both Get off the Earth (perhaps the best is his Vanishing Leprechaun) and of
Teddy and the Lions (6 men and 4 glasses of beer become 5 men and 5 glasses). I have examples of some of these from Stover
and I have looked at his notebooks, which are now with Mark Setteducati. See
Gardner, MM&M, pp. 125-128.
Gardner. SA (May 1961) c= NMD, chap. 11.
Mentioned in Workout, chap. 27.
Describes his adaptation of a principle of Paul Curry to produce The
Disappearing Square puzzle, where 16 or 17 pieces seem to make the same
square. The central part of the 17
piece version consists of five equal squares in the form of a Greek cross. The central part of the 16 piece version has
four of the squares in the shape of a square.
This has since been produced in several places.
Ripley's Puzzles and Games. 1966.
P. 60. Asserts that when you cut
a 2½ x 4½ board into six right triangles with legs 1½ and 2½,
then they assemble into an equilateral triangle of edge 5. This has an area loss of about 4%.
John Fisher. John Fisher's Magic Book. Muller, London, 1968.
Financial
Wizardry, pp. 18-19. 7 x 8 region with
£ signs marking the area. A line cuts off a triangle of width 7 and height
2 at the top. The rest of the area is
divided by a vertical into strips of widths 4 and 3, with a small rectangle 3
by 1 cut from the bottom of the width 3 strip.
When the strips are exchanged, one unit of area is lost and one £
sign has vanished.
Try-Angle,
pp. 126-127. This is one of Curry's
triangles -- see Gardner, MM&M, p. 147.
Alco-Frolic!,
pp. 148-149. This is a form of
Stover's 6 & 4 to 5
& 5 version.
D. E. Knuth. Disappearances. In: The Mathematical
Gardner; ed. by David Klarner; Prindle, Weber & Schmidt/Wadsworth,
1981. P. 264. An eight line poem which rearranges to a seven line poem.
Dean Clark. A centennial tribute to Sam Loyd. CMJ 23:5 (Nov 1992) 402‑404. Gives an easy circular version with 11 & 12
astronauts around the earth and a
15 & 16 face version with
three pieces, a bit like the Vanishing Leprechaun.
6.Q. KNOTTING A STRIP TO MAKE A REGULAR PENTAGON
Urbano d'Aviso.
Trattato della Sfera e Pratiche per Uso di Essa. Col modeo di fare la figura celeste, opera cavata
dalli manoscritti del. P. Bonaventura Cavalieri. Rome, 1682. ??NYS cited
by Lucas (1895) and Fourrey.
Dictionary of Representative
Crests. Nihon Seishi Monshō
Sōran (A Comprehensive Survey of Names and Crests in Japan), Special issue of Rekishi Dokuhon (Readings in History), Shin Jinbutsu Oraisha, Tokyo, 1989, pp. 271-484. Photocopies of relevant pages kindly sent by
Takao Hayashi.
Crests
3504 and 3506 clearly show a strip knotted to make a pentagon. 3507 has two such knots and 3508 has five. I don't know the dates, but most of these
crests are several centuries old.
Lucas. RM2, 1883, pp. 202‑203.
Tom Tit.
Vol.
2, 1892. L'Étoile à cinq branches, pp.
153-154. = K, no. 5: The pentagon and
the five pointed star, pp. 20‑21.
He adds that folding over the free end and holding the knot up to the
light shows the pentagram.
Vol.
3, 1893. Construire d'un coup de poing
un hexagone régulier, pp. 159-161.
= K, no. 17: To construct a hexagon by finger pressure, pp. 49‑51. Pressing an appropriate size Möbius strip
flat gives a regular hexagon.
Vol.
3, 1893. Les sept pentagones, pp.
165-166. = K, no. 19: The seven
pentagons, pp. 54‑55. By
tying five pentagons in a strip, one gets a larger pentagon with a pentagonal
hole in the middle.
Somerville Gibney. So simple!
The hexagon, the enlarged ring, and the handcuffs. The Boy's Own Paper 20 (No. 1012) (4 Jun
1898) 573-574. As in Tom Tit, vol. 3,
pp. 159-161.
Lucas. L'Arithmétique Amusante.
1895. Note IV: Section II: Les Jeux de Ruban, Nos. 1 &
2: Le nœud de cravate &
Le nœud marin, pp. 220-222.
Cites d'Aviso and says he does both the pentagonal and hexagonal knots,
but Lucas only shows the pentagonal one.
E. Fourrey. Procédés Originaux de Constructions
Géométriques. Vuibert, Paris, 1924. Pp. 113 & 135‑139. Cites Lucas and cites d'Aviso as Traité de
la Sphère and says he gives the pentagonal and hexagonal knots. Fourrey shows and describes both, also
giving the pictures on his title page.
F. V. Morley. A note on knots. AMM 31 (1924) 237-239.
Cites Knott's translation of Tom Tit.
Says the process generalizes to
(2n+3)‑gons by using
n loops. Gets even-gons by using two strips. Discusses using twisted strips.
Robert C. Yates. Geometrical Tools. (As: Tools; Baton Rouge,
1941); revised ed., Educational
Publishers, St. Louis, 1949. Pp. 64-65
gives square (a bit trivial), pentagon, hexagon, heptagon and octagon. Even case need two strips.
Donovan A. Johnson. Paper Folding for the Mathematics
Class. NCTM, 1957, pp. 16-17: Polygons
constructed by tying paper knots. Shows
how to tie square, pentagon, hexagon, heptagon and octagon.
James K. Brunton. Polygonal knots. MG 45 (No. 354) (Dec 1961) 299‑302. All regular
n‑gons, n > 4, can be obtained, except n = 6
which needs two strips.
Discusses which can be made without central holes.
Marius Cleyet-Michaud. Le Nombre d'Or. Presses Universitaires de France, Paris, 1973. On pp. 47-48, he calls this the 'golden
knot' (Le "nœud doré") and describes how to make it.
General
surveys of such fallacies can be found in the following. See also:
6.P, 10.A.1.
These
fallacies are actually quite profound as the first two point out some major
gaps in Euclid's axioms -- the idea of a point being inside a triangle really
requires notions of order of points on a line and even the idea of continuity,
i.e. the idea of real numbers.
Ball. MRE. 1st ed., 1892, pp.
31‑34, two examples, discussed below.
3rd ed., 1896, pp. 39‑46
= 4th ed., 1905, pp. 41-48,
seven examples. 5th ed., 1911, pp.
44-52 = 11th ed., 1939, pp. 76-84, nine example.
Walther Lietzmann. Wo steckt der Fehler? Teubner, Stuttgart, (1950), 3rd ed.,
1953. (Strens/Guy has 3rd ed.,
1963(?).) (There are 2nd ed, 1952??; 5th ed, 1969; 6th ed,
1972. MG 54 (1970) 182 says the 5th ed
appears to be unchanged from the 3rd ed.)
Chap. B: V, pp. 87-99 has 18 examples.
(An
earlier version of the book, by Lietzmann & Trier, appeared in 1913, with
2nd ed. in 1917. The 3rd ed. of 1923
was divided into two books: Wo Steckt
der Fehler? and Trugschlüsse. There was a 4th ed. in 1937.
The relevant material would be in Trugschlüsse, but I have not seen any
of the relevant books, though E. P. Northrop cites Lietzmann, 1923, three times
-- ??NYS.)
E. P. Northrop. Riddles in Mathematics. 1944.
Chap. 6, 1944: 97-116, 232-236
& 249-250; 1945: 91-109, 215-219
& 230-231; 1961: 98-115, 216-219
& 229. Cites Ball, Lietzmann
(1923), and some other individual items.
V. M. Bradis, V. L. Minkovskii & A. K.
Kharcheva. Lapses in Mathematical
Reasoning. (As: [Oshibki v
Matematicheskikh Rassuzhdeniyakh], 2nd ed, Uchpedgiz, Moscow, 1959.) Translated by J. J. Schorr-Kon, ed. by E. A.
Maxwell. Pergamon & Macmillan, NY,
1963. Chap. IV, pp. 123-176. 20 examples plus six discussions of other
examples.
Edwin Arthur Maxwell. Fallacies in Mathematics. CUP, (1959), 3rd ptg., 1969. Chaps. II-V, pp. 13-36, are on geometric
fallacies.
Ya. S. Dubnov. Mistakes in Geometric Proofs. (2nd ed., Moscow?, 1955). Translated by Alfred K. Henn & Olga A.
Titelbaum. Heath, 1963. Chap 1-2, pp. 5-33. 10 examples.
А. Г.
Конфорович. [A. G.
Konforovich].
(Математичні
Софізми і
Парадокси
[Matematichnī Sofīzmi ī Paradoksi] (In Ukrainian). Радянська
Школа [Radyans'ka Shkola], Kiev, 1983.) Translated into German by Galina &
Holger Stephan as: Konforowitsch, Andrej Grigorjewitsch; Logischen Katastrophen
auf der Spur – Mathematische Sophismen und Paradoxa; Fachbuchverlag, Leipzig,
1990. Chap. 4: Geometrie,
pp. 102-189 has 68 examples, ranging from the type considered here up
through fractals and pathological curves.
S. L. Tabachnikov. Errors in geometrical proofs. Quantum 9:2 (Nov/Dec 1998) 37-39 &
49. Shows: every triangle is isosceles (6.R.1); the sum of the angles of a triangle is 180o without
use of the parallel postulate; a
rectangle inscribed in a square is a square;
certain approaching lines never meet (6.R.3); all circles have the same circumference (cf Aristotle's Wheel
Paradox in 10.A.1); the circumference
of a wheel is twice its radius; the
area of a sphere of radius R is
π2R2.
6.R.1. EVERY TRIANGLE IS ISOSCELES
This is sometimes claimed to
have been in Euclid's lost Pseudaria (Fallacies).
Ball. MRE, 1st ed., 1892, pp. 33‑34. On p. 32, Ball refers to Euclid's lost Fallacies and presents
this fallacy and the one in 6.R.2:
"I do not know whether either of them has been published
previously." In the 3rd ed., 1896,
pp. 42-43, he adds the heading: To
prove that every triangle is isosceles.
In the 5th ed., 1911, p. 45, he adds a note that he believes these two
were first published in his 1st ed. and notes that Carroll was fascinated by
them and they appear in The Lewis Carroll Picture Book (= Carroll-Collingwood)
-- see below.
Mathesis (1893). ??NYS.
[Cited by Fourrey, Curiosities Geometriques, p. 145. Possibly Mathesis (2) 3 (Oct 1893) 224,
cited by Ball in MRE, 3rd ed, 1896, pp. 44-45, cf in Section 6.R.4.]
Carroll-Collingwood. 1899.
Pp. 264-265 (Collins: 190-191).
= Carroll-Wakeling II, prob. 27: Every triangle has a pair of
equal sides!, pp. 43 & 27. Every
triangle is isosceles. Carroll may have
stated this as early as 1888.
Wakeling's solution just suggests making an accurate drawing. Carroll-Gardner, p. 65, mentions this and
says it was not original with Carroll.
Ahrens. Mathematische Spiele. Teubner.
Alle Dreiecke sind gleichschenklige.
2nd ed., 1911, chap. X, art. VI, pp. 108 & 119‑120. 3rd ed., 1916, chap. IX, art. IX, pp. 92-93
& 109-111. 4th ed., 1919 & 5th ed., 1927, chap IX, art. IX, pp. 99‑101 & 116‑118.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. Call Mr. Euclid -- No. 15: To prove all triangles
are equilateral, pp. 16-17. Clear
exposition of the fallacy.
See Read in 6.R.4 for a
different proof of this fallacy.
6.R.2. A RIGHT ANGLE IS OBTUSE
Ball. MRE, 1st ed.,
1892, pp. 32‑33. See 6.R.1. In the 3rd ed., 1896, pp. 40-41, he adds the
heading: To prove that a right angle is
equal to an angle which is greater than a right angle.
Mittenzwey. 1895?.
Prob. 331, pp. 58 & 106;
1917: 331, pp. 53 & 101.
Carroll-Collingwood. 1899.
Pp. 266‑267 (Collins 191-192).
An obtuse angle is sometimes equal to a right angle. Carroll-Gardner, p. 65, mentions this and
says it was not original with Carroll.
H. E. Licks. 1917.
Op. cit. in 5.A. Art. 82, p. 56.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. Call Mr. Euclid -- No. 16: To prove one right
angle greater than another right angle, pp. 18-19. "Here again, if you take the trouble to draw an accurate
diagram, you will find that the "construction" used for the alleged
proof is impossible."
E. A. Maxwell. Note 2121:
That every angle is a right angle.
MG 34 (No. 307) (Feb 1950) 56‑57.
Detailed demonstration of the error.
6.R.3. LINES APPROACHING BUT NOT MEETING
Proclus. 5C.
A Commentary on the First Book of Euclid's Elements. Translated by Glenn R. Morrow. Princeton Univ. Press, 1970. Pp. 289-291. Gives the argument and tries to refute it.
van Etten/Henrion/Mydorge. 1630.
Part 2, prob. 7: Mener une ligne laquelle aura inclination à une autre
ligne, & ne concurrera jamais contre l'Axiome des paralelles, pp. 13‑14.
Schwenter. 1636.
To be added.
Ozanam-Montucla. 1778.
Paradoxe géométrique des lignes ....
Prob. 70 & fig. 116-117, plate 13, 1778: 405-407; 1803: 411-413; 1814: 348-350. Prob. 69,
1840: 180-181. Notes that these
arguments really produce a hyperbola and a conchoid. Hutton adds that a great many other examples might be found.
E. P. Northrop. Riddles in Mathematics. 1944.
1944: 209-211 & 239;
1945: 195‑197 & 222;
1961: 197‑198 & 222.
Gives the 'proof' and its fallacy, with a footnote on p. 253
(1945: 234; 1961: 233) saying the argument "has been attributed to
Proclus."
Jeremy Gray. Ideas of Space. OUP, 1979. Pp. 37-39
discusses Proclus' arguments in the context of attempts to prove the parallel
postulate.
Ball. MRE, 3rd ed, 1896, pp. 44-45.
To prove that, if two opposite sides of a quadrilateral are equal, the
other two sides must be parallel. Cites
Mathesis (2) 3 (Oct 1893) 224 -- ??NYS
Cecil B. Read. Mathematical fallacies &
More mathematical fallacies. SSM
33 (1933) 575‑589 & 977-983.
There are two perpendiculars from a point to a line. Part of a line is equal to the whole
line. Every triangle is isosceles (uses
trigonometry). Angle trisection (uses a
marked straightedge).
P. Halsey. Class Room Note 40: The ambiguous case. MG 43 (No. 345) (Oct 1959) 204‑205. Quadrilateral ABCD with angle A = angle C and AB = CD. Is this a parallelogram?
Hoffmann. 1893.
Chap III, pp. 74‑90, 96-97, 111-124 & 128
= Hoffmann-Hordern, pp. 62‑79 & 86-87 with several
photos. Describes Tangrams and Richter
puzzles at some length. Lots of photos
in Hordern. Photos on pp. 67, 71, 75,
87 show Richter's: Anchor (1890‑1900, = Tangram), Tormentor (1898),
Pythagoras (1892), Cross Puzzle (1892), Circular Puzzle (1891), Star Puzzle
(1899), Caricature (1890-1900, = Tangram) and four non-Richter Tangrams in
Tunbridge ware, ivory, mother-of-pearl and tortoise shell. Hordern Collection, pp. 45-57 & 60,
(photos on pp. 46, 49, 50, 52, 54, 56, 60) shows different Richter versions of
Tormentor (1880-1900), Pythagoras (1880-1900), Circular Puzzle (1880-1900),
Star Puzzle (1880‑1900) and has a wood non-Richter version instead of the
ivory version in the last photo.
Ronald C. Read. Tangrams -- 330 Puzzles. Dover, 1965. The Introduction, pp. 1-6, is a sketch of the history. Will Shortz says this is the first serious
attempt to counteract the mythology created by Loyd and passed on by
Dudeney. Read cannot get back before
the early 1800s and notes that most of the Loyd myth is historically
unreasonable. However, Read does not
pursue the early 1800s history in depth and I consider van der Waals to be the
first really serious attempt at a history of the subject.
Peter van Note. Introduction. IN: Sam Loyd; The Eighth Book of Tan; (Loyd & Co., 1903);
Dover, 1968, pp. v-viii. Brief
debunking of the Loyd myth.
Jan van der Waals. History
& Bibliography. In:
Joost Elffers; Tangram; (1973), Penguin, 1976. Pp. 9‑27 & 29‑31. Says the Chinese term "ch'i ch'ae" dates from the Chu
era (‑740/‑330), but the earliest known Chinese book is 1813. The History reproduces many pages from early
works. The Bibliography cites 8
versions of 4 Chinese books (with locations!) from 1813 to 1826 and 18 Western
books from 1805 to c1850. The 1805, and
several other references, now seem to be errors.
S&B. 1986.
Pp. 22‑33 discusses loculus of Archimedes, Chie no Ita, Tangrams
and Richter puzzles.
Alberto Milano. Due giochi di società dell'inizio
dell'800. Rassegna di Studi e di
Notizie 23 (1999) 131-177. [This is a
publication by four museums in the Castello Sforzesco, Milan: Raccolta delle Stampe Achille Bertarelli;
Archivio Fotografico; Raccolte d'Arte Applicata; Museo degli Strumenti
Musicali. Photocopy from Jerry Slocum.] This surveys early books on tangrams, some
related puzzles and the game of bell and hammer, with many reproductions of TPs
and problems.
Jerry Slocum. The Tangram Book. (With Jack Botermans, Dieter Gebhardt, Monica Ma, Xiaohe Ma,
Harold Raizer, Dic Sonneveld and Carla van Spluntern.) ©2001 (but the first publisher collapsed),
Sterling, 2003. This is the long
awaited definitive history of the subject!
It will take me sometime to digest and summarize this, but a brief
inspection shows that much of the material below needs revision!
Recent
research by Jerry Slocum, backed up by The Admired Chinese Puzzle, indicates
that the introduction of tangrams into Europe was done by a person or persons
in Lord Amherst's 1815-1817 embassy to China, which visited Napoleon on St.
Helena on its return voyage. If so, then
the conjectural dating of several items below needs to be amended. I have amended my discussion accordingly and
marked such dates with ??. Although
watermarking of paper with the correct date was a legal requirement at the
time, paper might have been stored for some time before it was printed on, so
watermark dates only give a lower bound for the date of printing. I have seen several further items dated
1817, but it is conceivable that some material may have been sent back to
Europe or the US a few years earlier -- cf Lee.
On
2 Nov 2003, I did the following brief summary of Slocum's work in a letter to
an editor. I've made a few corrections
and added a citation to the following literature.
Tangrams. The history of this has now been definitely
established in Jerry Slocum's new book: The Tangram Book; ©2001 (but the first publisher collapsed),
Sterling, 2003. This history has been
extremely difficult to unravel because Sam Loyd deliberately obfuscated it in 1903,
claiming the puzzle went back to 2000 BC, because the only previous attempt at
a history had many errors, and because much of the material doesn't survive, or
only a few examples survive. The
history covers a wide range in both time and location, as evidenced by the
presence of seven co-authors from several countries.
Briefly,
the puzzle, in the standard form, dates from about 1800, in China. It is attributed to Yang-cho-chü-shih, but
this is a pseudonym, meaning 'dim-witted recluse', and no copies of his work
are known. The oldest known example of
the game is one dated 1802 in a museum near Philadelphia -- see Lee,
below. The oldest known book on the
puzzle had a preface by Sang-hsia-ko [guest under the mulberry tree] dated June
1813 and a postscript by Pi-wu-chü-shih dated July 1813. This is only known from a Japanese facsimile
of it made in 1839. This book was
republished, with a book of solutions, in two editions in 1815 -- one with
about four problems per page, the other with about eleven. The latter version was the ancestor of many
19C books, both in China and the west.
Another 2 volume version appeared later in 1815. Sang-hsia-ko explicitly says "The
origin of the Tangram lies within the Pythagorean theorem".
In
1816, several ships brought copies of the eleven problems per page books to the
US, England and Europe. The first
western publication of the puzzle is in early 1817 when J. Leuchars of 47
Piccadilly registered a copyright and advertised sets for sale. But the craze was really set off by the
publication of The Fashionable Chinese Puzzle and its Key by John
and Edward Wallis and John Wallis Jr in March 1817. This included a poem with a note that the game was "the
favourite amusement of Ex-Emperor Napoleon". This went through many printings, with some (possibly the first)
versions having nicely coloured illustrations.
By the end of the year, there were many other books, including examples
in France, Italy and the USA.
Dic
Sonneveld, one of the co-authors of Slocum's book, managed to locate the
tangram and books that had belonged to Napoleon in the Château de Malmaison,
outside Paris, but there is no evidence that Napoleon spent much time playing
with it. St. Helena was a regular stop
for ships in the China trade. Napoleon
is recorded as having bought a chess set from one ship and several notables are
recorded as having presented Napoleon with gifts of Chinese objects. A diplomatic letter of Jan 1817 records
sending an example of the game from St. Helena to Prince Metternich, but this
example has not been traced.
The
first American book was Chinese Philosophical and Mathematical Trangram
by James Coxe, appearing in Philadelphia in August 1817. The word 'trangram' meaning 'an odd,
intricately contrived thing' according to Johnson's Dictionary, was
essentially obsolete by 1817, but was still in some use in the US. The earliest known use of the word 'tangram'
is in Thomas Hill's Geometrical Puzzles for the Young, Boston,
1848. One suspects that he was
influenced by Coxe's book, but he may have known that 'T'ang' is the Cantonese
word for 'Chinese'. Hill later became
President of Harvard University and was an active promoter and inventor of
games for classroom use. In 1864, the
word was in Webster's Dictionary.
However,
the above is the story of the seven-piece tangram that we know today. There is a long background to this, dating
back to the 3rd century BC, when Archimedes wrote a letter to Eratosthenes
describing a fourteen piece puzzle, known as the Stomachion or Loculus of
Archimedes. The few surviving texts are
not very clear and there are two interpretations -- in one the standard
arrangement of the pieces is a square and in the other it is a rectangle twice
as wide as high. There are six (at
least) references to the puzzle in the classical world, the last being in the
6th century. The puzzle was used to
make a monstrous elephant, a brutal boar, a ship, a sword, etc., etc. The puzzle then disappears, and no form of
it appears in the Arabic world, which has always surprised me, given the Arabic
interest in patterns.
Further,
several eastern predecessors of the tangrams are known. The earliest is a Japanese version of 1742
by Ganriken (or Granreiken) which has seven pieces, attributed (as were many
things) to Sei Shonagon, a 10th century courtesan famous for her
ingenuity. By the end of the 18th century,
three other dissection/arrangement puzzles appeared in Japan, with 15, 19 and
19 pieces, including some semi-circles.
An 1804 print by Utamaro shows courtesans playing with some version of
the puzzle -- only two copies of this print have been located.
But
the basic puzzle idea has its roots in Chinese approaches to the Theorem of
Pythagoras and similar geometric proofs by dissection and rearrangement which
date back to the 3rd century (and perhaps earlier). But the tangram did not develop directly from these ideas. From the 12th century, there was a Chinese
tradition of making "Banquet Tables" in the form of several pieces
that could be arranged in several ways.
The first known Chinese book on furniture, by Huang Po-ssu in 1194,
describes a Banquet Table formed of seven rectangular pieces: two long, two
medium and three short. In 1617, Ko
Shan described 'Butterfly Wing" tables with 13 pieces, including isosceles
right triangles, right trapeziums and isosceles trapeziums. In 1856, a Chinese scholar noted the
resemblance of these tables with the tangram and a modern Chinese historian of
mathematics has observed that half of the butterfly arrangement can be easily
transformed into the tangrams. No
examples of these tables have survived, but tables (and serving dishes) in the
tangram pattern exist and are probably still being made in China.
Kanchusen. Wakoku Chiekurabe. 1727. Pp. 9 & 28-29:
a simple dissection puzzle with 8 pieces.
The problem appears to consist of a mitre comprising ¾ of
a unit square; 4 isosceles right
triangles of hypotenuse 1 and 3 isosceles right triangles of side ½,
but the solution shows that all the triangles are the same size, say
having hypotenuse 1, and the mitre shape is actually formed from
a rectangle of size 1 x Ö2.
"Ganriken" [pseud.,
possibly of Fan Chu Sen]. Sei
Shōnagon Chie-no-Ita (The Ingenious Pieces by Sei Shōnagon.) (In
Japanese). Kyoto Shobo, Aug 1742, 18pp,
42 problems and solutions. Reproduced
in a booklet, ed. by Kazuo Hanasaki, Tokyo, 1984, as pp. 19‑36. Also reproduced in a booklet, transcribed
into modern Japanese, with English pattern names and an English abstract, by
Shigeo Takagi, 1989. This uses a set of
seven pieces different than the Tangram.
S&B, p. 22, shows these pieces.
Sei Shōnagon (c965-c1010) was a famous courtier, author of The
Pillow Book and renowned for her intelligence.
The Introduction is signed Ganriken.
S&B say this is probably Fan Chu Sen, but Takagi says the author's
real name is unknown.
Utamaro. Interior of an Edo house, from the picture‑book: The Edo Sparrows (or Chattering Guide),
1786. Reproduced in B&W in: J. Hillier; Utamaro -- Colour Prints
and Paintings; Phaidon Press, Oxford, (1961), 2nd ed., 1979, p. 27, fig.
15. I found this while hunting for the
next item. This shows two women
contemplating some pieces but it is hard to tell if it is a tangram‑type
puzzle, or if perhaps they are cakes.
Hiroko and Mike Dean tell me that they are indeed cooking cakes.
Utamaro. Woodcut.
1792. Shows two courtesans
working on a tangram puzzle. Van der
Waals dated this as 1780, but Slocum has finally located it, though he has only
been able to find two copies of it! The
courtesans are clearly doing a tangram-like puzzle with 12(?) pieces -- the
pieces are a bit piled up and one must note that one of the courtesans is
holding a piece. They are looking at a
sheet with 10 problem figures on it.
Early 19C books from China --
??NYS -- cited by Needham, p. 111.
Jean Gordon Lee. Philadelphians and the China Trade 1784-1844.
Philadelphia Museum of Art, 1984, pp. 122-124. (Photocopy from Jerry Slocum.)
P. 124, item 102, is an ivory tangram in a cardboard box, inscribed on
the bottom of the box: F. Waln April 4th 1802. Robert Waln was a noted trader with China
and this may have been a present for his third son Francis (1799‑1822). This item is in the Ryerss Museum, a city
museum in Philadelphia in the country house called Burholme which was built by
one of Robert Waln's sons-in-law.
A New Invented Chinese Puzzle,
Consisting of Seven Pieces of Ivory or Wood, Viz. 5 Triangles, 1 Rhomboid,
& 1 Square, which may be so placed as to form the Figures represented in
the plate. Paine & Simpson,
Boro'. Undated, but the paper is
watermarked 1806. This consists of two
'volumes' of 8 pages each, comprising 159 problems with no solutions. At the end are bound in a few more pages
with additional problems drawn in -- these are direct copies of plates 21, 26,
22, 24, and 28 (with two repeats from plate 22) of The New and Fashionable
Chinese Puzzle, 1817. Bound in plain
covers. This is in Edward Hordern's
collection and he provided a photocopy.
Dalgety also has a copy.
Ch'i Ch'iao t'u ho‑pi (=
Qiqiao tu hebi) (Harmoniously combined book of tangram problems OR Seven clever
pieces). 1813. (Bibliothek Leiden 6891, with an 1815
edition at British Library 15257 d 13.)
van der Waals says it has 323 examples.
The 1813 seems to be the earliest Chinese tangram book of problems, with
the 1815 being the solutions. Slocum
says there was a solution book in 1815 and that the problem book had a preface
by Sang‑hsia K'o (= Sang-xia-ke), which was repeated in the solution book with the same date. Milano mentions this, citing Read and van
der Waals/Elffers, and says an example is on the BL. A version of this appears to have been the book given to Napoleon
and to have started the tangram craze in Europe. I have now received a photocopy from Peter Rasmussen & Wei
Zhang which is copied from van der Waals' copy from BL 15257 d 13. It has a cover, 6 preliminary pages and 28
plates with 318 problems. The pages are
larger than the photocopies of 1813/1815 versions in the BL that Slocum gave
me, which have 334 problems on 86 pages, but I see these are from 15257 d 5 and
14. I have a version of the smaller
page format from c1820s which has 334 problems on 84pp, apparently lacking its
first sheet. The problems are not
numbered, but given Chinese names. They
are identical to those appearing in Wallis's Fashionable Chinese Puzzle, below,
except the pages are in different order, two pages are inverted, Wallis
replaces Chinese names by western numbers and draws the figures a bit more
accurately. Wallis skips one number and
adds four new problems to get 323 problems - van der Waals seems to have taken
323 from Wallis.
Shichi‑kou‑zu
Gappeki [The Collection of Seven‑Piece Clever Figures]. Hobunkoku Publishing, Tokyo, 1881. This is a Japanese translation of an 1813
Chinese book "recognized as the earliest of existing Tangram book",
apparently the previous item. [The book
says 1803, but Jerry Slocum reports this is an error for 1813!] Reprinted, with English annotations by Y.
Katagiri, from N. Takashima's copy, 1989.
129 problems (but he counts 128 because he omits one after no. 124), all
included in my version of the previous item, no solutions.
Anonymous. A Grand Eastern Puzzle. C. Davenporte & Co. Registered on 24 Feb 1817, hence the second
oldest English (and European?) tangram book [Slocum, p. 71.] It is identical to Ch'i Ch'iao t'u ho‑pi,
1815, above, except that plates 25 and 27 have been interchanged. It appears to be made by using Chinese pages
and putting a board cover on it. On the
front cover is the only English text:
A
Grand
Eastern Puzzle
----------
THE
following Chineze Puzzle is recommended
to
the Nobility, Gentry, and others, being superior to
any
hitherto invented for the Amusement of the Juvenile
World,
to whom it will afford unceasing recreation and
information;
being formed on Geometrical principles, it
may
not be considered as trifling to those of mature
years,
exciting interest, because difficult and instructive,
imperceptibly
leading the mind on to invention and per-
severence.
-- The Puzzle consists of five triangles, a
square,
and a rhomboid, which may be placed in upwards
of
THREE HUNDRED and THIRTY Characters, greatly re-
sembling
MEN, BEASTS, BIRDS, BOATS, BOTTLES, GLASS-
ES,
URNS, &c. The whole being the
unwearied exertion
of
many years study and application of one of the Lite-
rati
of China, and is now offered to the Public for their
patronage
and support.
ENTERED
AT STATIONERS HALL
----
Published
and sold by
C.
DAVENPORTE and Co.
No.
20, Grafton Street, East Euston Square.
The Fashionable Chinese
Puzzle. Published by J. & E.
Wallis, 42, Skinner Street and J. Wallis Junr, Marine Library,
Sidmouth, nd [Mar 1817]. Photocopy from
Jerry Slocum. This has an illustrated
cover, apparently a slip pasted onto the physical cover. This shows a Chinese gentleman holding a
scroll with the title. There is a
pagoda in the background, a bird hovering over the scroll and a small person in
the foreground examining the scroll.
Slocum's copy has paper watermarked 1816.
PLUS
A Key to the New and Fashionable
Chinese Puzzle, Published by J. and E. Wallis, 42, Skinner Street, London,
Wherein is explained the method of forming every Figure contained in That
Pleasing Amusement. Nd [Mar 1817]. Photocopy from the Bodleian Library, Oxford,
catalogue number Jessel e.1176. TP
seems to made by pasting in the cover slip and has been bound in as a left hand
page. ALSO a photocopy from Jerry
Slocum. In the latter copy, the
apparent TP appears to be a paste down on the cover. The latter copy does not have the Stanzas mentioned below. Slocum's copy has paper watermarked 1815; I
didn't check this at the Bodleian.
NOTE. This is quite a different book than The New
and Fashionable Chinese Puzzle published by Goodrich in New York, 1817.
Bound
in at the beginning of the Fashionable Chinese Puzzle and the Bodleian copy of
the Key is: Stanzas, Addressed to
Messrs. Wallis, on the Ingenious Chinese Puzzle, Sold by them at the Juvenile
Repository, 42, Skinner Street. In the
Key, this is on different paper than the rest of the booklet. The Stanzas has a footnote referring to the
ex-Emperor Napoleon as being in a debilitated state. (Napoleon died in 1821, which probably led to the Bodleian
catalogue's date of c1820 for the entire booklet - but see below. Then follow 28 plates with 323 numbered
figures (but number 204 is skipped), solved in the Key. In the Bodleian copy of the Key, these are
printed on stiff paper, on one side of each sheet, but arranged as facing
pairs, like Chinese booklets.
[Philip
A. H. Brown; London Publishers and Printers
c. 1800-1870; British Library, 1982, p. 212] says the Wallis firm is
only known to have published under the imprint J. & E. Wallis during 1813
and Ruth Wallis showed me another source giving 1813?-1814. This led me to believe that the booklets
originally appeared in 1813 or 1814, but that later issues or some owner
inserted the c1820 sheet of Stanzas, which was later bound in and led the
Bodleian to date the whole booklet as c1820.
Ruth Wallis showed me a source that states that John Wallis (Jun.) set
up independently of his father at 186 Strand in 1806 and later moved to
Sidmouth. Finding when he moved to
Sidmouth might help date this publication more precisely, but it may be a later
reissue. However, Slocum has now found
the book advertised in the London Times in Mar 1817 and says this is the
earliest Western publication of tangrams, based on the 1813/1815 Chinese
work. Wallis also produced a second
book of problems of his own invention and some copies seem to be coloured.
In
AM, p. 43, Dudeney says he acquired the copy of The Fashionable Chinese Puzzle
which had belonged to Lewis Carroll.
He says it was "Published by J. and E. Wallis, 42 Skinner Street,
and J. Wallis, Jun., Marine Library, Sidmouth" and quotes the Napoleon
footnote, so this copy had the Stanzas included. This copy is not in the Strens Collection at Calgary which has
some of Dudeney's papers.
Van
der Waals cites two other works titled
The Fashionable Chinese Puzzle.
An 1818 edition from A. T. Goodridge [sic], NY, is in the American
Antiquarian Society Library (see below) and another, with no details given, is
in the New York Public Library. Could
the latter be the Carroll/Dudeney copy?
Toole
Stott 823 is a copy with the same title and imprint as the Carroll/Dudeney
copy, but he dates it c1840. This
version is in two parts. Part I has 1
leaf text + 26 col. plates -- it seems clear that col. means coloured, a
feature that is not mentioned in any other description of this book -- perhaps
these were hand-coloured by an owner.
Unfortunately, he doesn't give the number of puzzles. I wonder if the last two plates are missing
from this?? Part II has 1 leaf text +
32 col. plates, giving 252 additional figures.
The only copy cited was in the library of J. B. Findlay -- I have
recently bought a copy of the Findlay sale catalogue, ??NYR.
Toole
Stott 1309 is listed with the title: Stanzas, .... J. & F. [sic] Wallis ... and Marine Library, Sidmouth, nd
[c1815]. This has 1 leaf text and 28
plates of puzzles, so it appears that the Stanzas have been bound in and the
original cover title slip is lost or was not recognised by Toole Stott. The date of c1815 is clearly derived from
the Napoleon footnote but 1817 would have been more reasonable, though this may
be a later reissue. Again only one copy
is cited, in the library of Leslie Robert Cole.
Plates
1-28 are identical to plates 1-28 of The Admired Chinese Puzzle, but in
different order. The presence of the
Chinese text in The Admired Chinese Puzzle made me think the Wallis version was
later than it.
Comparison
of the Bodleian booklet with the first 27 plates of Giuoco Cinese, 1818?,
reveals strong similarities. 5 plates
are essentially identical, 17 plates are identical except for one, two or three
changes and 3 plates are about 50% identical.
I find that 264 of the 322 figures in the Wallis booklet occur in Giuoco Cinese, which is about 82%.
However, even when the plates are essentially identical, there are often
small changes in the drawings or the layout.
Some
of the plates were copied by hand into the Hordern Collection's copy of A New Invented Chinese Puzzle, c1806??.
The Admired Chinese Puzzle A New & Correct Edition From the Genuine
Chinese Copy. C. Taylor, Chester,
nd [1817]. Paper is clearly watermarked
1812, but the Prologue refers to the book being brought from China by someone
in Lord Amherst's embassy to China, which took place in 1815-1817 and which visited
Napoleon on St. Helena on its return.
Slocum dates this to after 17 Aug 1817, when Amherst's mission returned
to England and this seems to be the second western book on tangrams. Not in Christopher, Hall, Heyl or Toole
Stott -- Slocum says there is only one copy known in England! It originally had a cover with an
illustration of two Chinese, titled The
Chinese Puzzle, and one of the men
holds a scroll saying To amuse and
instruct. The Chinese text gives the
title Ch'i ch'iao t'u ho pi
(Harmoniously combined book of tangram problems). I have a photocopy of the cover from Slocum. Prologue facing TP; TP; two pp in Chinese,
printed upside down, showing the pieces;
32pp of plates numbered at the upper left (sometimes with reversed
numbers), with problems labelled in Chinese, but most of the characters are
upside down! The plates are printed
with two facing plates alternating with two facing blank pages. Plate 1 has 12 problems, with solution lines
lightly indicated. Plates 2 - 28
contain 310 problems. Plates 29-32
contain 18 additional "caricature Designs" probably intended to be
artistic versions of some of the abstract tangram figures. The Prologue shows faint guide lines for the
lettering, but these appear to be printed, so perhaps it was a quickly done
copperplate. The text of the Prologue
is as follows.
This
ingenious geometrical Puzzle was introduced into this Kingdom from China.
The
following sheets are a correct Copy from the Chinese Publication, brought to
England by a Gentleman of high Rank in the suit [sic] of Lord Amherst's late
Embassy. To which are added caricature
Designs as an illustration, every figure being emblematical of some Being or
Article known to the Chinese.
The
plates are identical to the plates in The Fashionable Chinese Puzzle above, but
in different order and plate 4 is inverted and this version is clearly upside
down.
Sy Hall. A New Chinese Puzzle, The Above Consists of Seven Pieces of Ivory
or Wood, viz. 5 Triangles, 1 Rhomboid, and 1 Square, which will form the 292
Characters, contained in this Book; Observing the Seven pieces must be used to
form each Character. NB. This Edition has been corrected in all its
angles, with great care and attention.
Engraved by Sy Hall, 14 Bury Street, Bloomsbury. 31 plates with 292 problems. Slocum, the Hordern Collection and BL have
copies. I have a photocopy from a
version from Slocum which has no date but is watermarked 1815. Slocum's recent book [The Tangram Book, pp.
74-75] shows a version of the book with the publisher's name as James Izzard
and a date of 1817. Sy
probably is an abbreviation of Sydney (or possibly Stanley?).
(The
BL copy is watermarked IVY MILL
1815 and is bound with a large folding
Plate 2 by Hall, which has 83 tinted examples with solution lines drawn in (by
hand??), possibly one of four sheets giving all the problems in the book. However there is no relationship between the
Plate and the book -- problems are randomly placed and often drawn in different
orientation. I have a photocopy of the
plate on two A3 sheets and a copy of a different plate with 72 problems,
watermarked J. Green 1816.)
A New Chinese Puzzle. Third Edition: Universally allowed to be the
most correct that has been published.
1817. Dalgety has a copy.
A New Chinese Puzzle Consisting
of Seven Pieces of Ivory or Wood, The Whole of which must be used, and will
form each of the CHARACTERS. J.
Buckland, 23 Brook Street, Holborn, London.
Paper watermarked 1816. (Dalgety
has a copy, ??NYS.)
Miss D. Lowry. A Key to the Only Correct Chinese Puzzle
Which has Yet Been Published, with above a Hundred New Figures. No. 1.
Drawn and engraved by Miss Lowry.
Printed by J. Barfield, London, 1817.
The initial D. is given on the next page. Edward Hordern's collection has a copy.
W. Williams. New Mathematical Demonstrations of Euclid,
rendered clear and familiar to the minds of youth, with no other mathematical
instruments than the triangular pieces commonly called the Chinese Puzzle. Invented by Mr. W. Williams, High Beech Collegiate
School, Essex. Published by the author,
London, 1817. [Seen at BL.]
Enigmes Chinoises. Grossin, Paris, 1817. ??NYS -- described and partly reproduced in
Milano. Frontispiece facing the TP
shows an oriental holding a banner which has the pieces and a few problems on
it. This is a small book, with five or
six figures per page. The figures seem
to be copied from the Fashionable Chinese Puzzle, but some figures are not in
that work. Milano says this is cited as
the first French usage of the term 'tangram', but this does not appear in
Milano's photos and it is generally considered that Loyd introduced the word in
the 1850s. Milano's phrasing might be
interpreted as saying this is the first French work on tangrams.
Chinesische-Raethsel. Produced by Daniel Sprenger with designs by
Matthaeus Loder, Vienna, c1818. ??NYS
-- mentioned by Milano.
Chinesisches Rätsel. Enigmes chinoises. Heinrich Friedrich Muller (or Mueller), Vienna, c1810??. ??NYS (van der Waals). This is probably a German edition of the
above and should be dated 1817 or 1818.
However, Milano mentions a box in the Historisches Museum der Stadt
Wien, labelled Grosse Chinesische
Raethsel, produced by Mueller and dated
1815-1820.
Passe-temps Mathématique, ou
Récréation à l'ile Sainte-Hélène. Ce
jeu qui occupé à qu'on prétend, les loisirs du fameux exilé à St.-Hélène. Briquet, Geneva, 1817. 21pp.
[Copy advertised by Interlibrum, Vaduz, in 1990.]
The New and Fashionable Chinese
Puzzle. A. T. Goodrich & Co., New
York, 1817. TP, 1p of Stanzas (seems
like there should be a second page??), 32pp with 346 problems. Slocum has a copy.
[Key] to the Chinese
Philosophical Amusements. A. T.
Goodrich & Co., New York, 1817. TP,
2pp of stanzas (the second page has the Napoleon footnote and a comment which
indicates it is identical to the material in the problem book), Index to the
Key to the Chinese Puzzle, 80pp of solutions as black shapes with white
spacing. Slocum has a copy.
NOTE. This is quite a different book than The
Fashionable Chinese Puzzle published in London by Wallis in 1817.
Slocum
writes: "Although the Goodrich problem book has the same title as the
British book by Wallis and Goodrich has the "Stanzas" poem (except
for the first 2 paragraphs which he deleted) the problem books have completely
different layouts and Goodrich's solution book largely copies Chinese
books."
Il Nuovo e Dilettevole Giuoco
Chinese. Bardi, Florence, 1817. ??NYS -- mentioned by Milano.
Buonapartes Geliefkoosste
Vermaack op St. Helena, op Chineesch Raadsel.
1er Rotterdam by J. Harcke.
Prijs 1 - 4 ??. 2e
Druck te(?) Rotterdam. Ter
Steendrukkery van F. Scheffers & Co.
Nanco Bordewijk has recently acquired this and Slocum has said it is a
translation of one of the English items in c1818. I have just a copy of the cover, and it uses many fancy letters
which I don't guarantee to have read correctly.
Recueil des plus jolis Jeux de
Sociéte, dans lequel on trouve les
gravures d'un grand nombre d'énigmes chinoises, et l'explication de ce nouveau
jeu. Chez Audot, Librairie, Paris,
1818. Pp. 158-162: Le jeu des énigmes
chinoises. This is a short
introduction, saying that the English merchants in Japan have sent it back to
their compatriots and it has come from England to France. This is followed by 11 plates. The first three are numbered. The first shows the pieces formed into a
rectangle. The others have 99 problems,
with 7 shown solved (all six of those on plate 2 and one (the square) on the
10th plate.)
Das grosse chinesische
Rätselspiel für die elegante Welt.
Magazin für Industrie (Leipzig) (1818).
??NYS (van der Waals). Jerry
Slocum informs me that 'Magazin' here denotes a store, not a periodical, and
that this is actually a game version with a packet of 50 cards of problems,
occurring in several languages, from 1818.
I have acquired a set of the cards which lacks one card (no. 17), in a
card box with labels in French and Dutch pasted on. One side has: Nouvelles /
ENIGMES / Chinoises / en Figures et en Paysages with a dancing Chinaman below.
The other side has: Chineesch /
Raadselspel, / voor / de Geleerde Waereld / in / 50 Beelaachlige /
Figuren. with two birds below. Both labels are printed in red, with the
dancing Chinaman having some black lines.
The cards are
82 x 55 mm and are beautifully printed with coloured
pictures of architectonic, anthropomorphic and zoomorphic designs in
appropriate backgrounds. The first card
has four shapes, three of which show the solution with dotted lines. All other cards have just one problem
shape. The reverses have a simple
design. Slocum says the only complete
set he has seen is in the British Library.
I have scanned the cards and the labels.
Gioco cinese chiamato il
rompicapo. Milan, 1818. ??NYS (van der Waals). Fratelli Bettali, Milan, nd, of which
Dalgety has a copy.
Al Gioco Cinese Chiamato Il
Rompicapo Appendice di Figure Rappresentanti ... Preceduta da un Discorso sul
Rompicapo e sulla Cina intitolato Passatempo Preliminare scritto dall'Autore
Firenze All'Insegna dell'Ancora 1818. 64pp
+ covers. The cover or TP has an almond
shape with the seven shapes inside. Pp.
3-43 are text -- the Passatempo Preliminare and an errata page. 12 plates.
The first is headed Alfabeto in fancy Gothic. Plates 1-3 give the alphabet (J and W are omitted). Plate 4 has the positive digits. Plates 5-12 have facing pages giving the
names of the figures (rather orientalized) and contain 100 problems. Hence a total of 133 problems, no solutions. The Hordern collection has a copy and I have
a photocopy from it. This has some
similarities to Giuoco Cinese.
Described and partly reproduced in Milano.
Al Gioco Cinese chiamato il
Rompicapo Appendice. Pietro &
Giuseppe Vallardi, Milan, 1818.
Possibly another printing of the item above. ??NYS -- described in Milano, who reproduces plates 1 & 2, which
are identical to the above item, but with a simpler heading. Milano says the plates are identical to
those in the above item.
Nuove e Dilettevole Giuoco
Chinese. Milano presso li Frat.
Bettalli Cont. del Cappello N. 4031.
Dalgety has a copy. It is
described and two pages are reproduced in Milano from an example in the
Raccolta Bertarelli. Milano dates it as
1818. Cover illustration is the same as
The Fashionable Chinese Puzzle, with the text changed. But it is followed by some more text: Questa ingegnosa invenzione è fondata sopra
principi Geometrici, e consiste in 7 pezzi cioè 5. triangoli, un quadrato ed un
paralellogrammo i quali possono essere combinati in modo da formare piu di 300
figure curiose. The second photo shows
a double page identical to pp. 3-4 of The Fashionable Chinese Puzzle, except
that the page number on p. 4 was omitted in printing and has been written
in. (Quaritch's catalogue 646 (1947)
item 698 lists this as Nuovo e
dilettevole Giuoco Chinese, from Milan,
[1820?])
Nuove e Dilettevole Giuoco
Chinese. Bologna Stamperia in pietra di
Bertinazzi e Compag. ??NYS -- described
and partly reproduced in Milano from an example in the Raccolta Bertarelli. Identical to the above item except that it
is produced lithographically, the text under the cover illustration has been
redrawn, the page borders, the page numbers and the figure numbers are a little
different. Milano's note 5 says the
dating of this is very controversial.
Apparently the publisher changed name in 1813, and one author claims the
book must be 1810. Milano opts for
1813? but feels this is not consistent with the above item. From Slocum's work and the examples above,
it seems clear it must be 1818?
Supplemento al nuovo giuoco
cinese. Fratelli Bettalli, Milan,
1818. ??NYS -- described in Milano, who
says it has six plates and the same letters and digits as Al Gioco Cinese
Chiamato Il Rompicapo Appendice.
Giuoco Cinese Ossia
Raccolta di 364. Figure Geometrica [last letter is blurred] formate con
un Quadrato diviso in 7. pezzi, colli quali si ponno formare infinite Figure
diversi, come Vuomini[sic], Bestie, Ucelli[sic], Case, Cocchi, Barche, Urne,
Vasi, ed altre suppelletili domestiche: Aggiuntovi l'Alfabeto, e li Numeri
Arabi, ed altre nuove Figure. Agapito Franzetti
alle Convertite, Rome, nd [but 1818 is written in by hand]. Copy at the Warburg Institute, shelf mark
FMH 4050. TP & 30 plates. It has alternate openings blank, apparently
to allow you to draw in your solutions, as an owner has done in a few cases. The first plate shows the solutions with
dotted lines, otherwise there are no solutions. There is no other text than on the TP, except for a florid
heading Alfabeto on plate XXVIII. The diagrams have no numbers or names. The upper part of the TP is a plate of three men, intended to be
Orientals, in a tent? The one on the
left is standing and cutting a card marked with the pieces. The man on the right is sitting at a low
table and playing with the pieces. He
is seated on a box labelled ROMPI CAPO. A third man is seated behind the table and
watching the other seated man. On the
ground are a ruler, dividers and right angle.
The Warburg does not know who put the date 1818 in the book, but the
book has a purchase note showing it was bought in 1913. James Dalgety has the only other copy
known. Sotheby's told him that
Franzetti was most active about 1790, but Slocum finds Sotheby's is no longer
very definite about this. I thought it
possible that a page was missing at the beginning which gave a different form
of the title, but Dalgety's copy is identical to this one. Mario Velucchi says it is not listed in a
catalogue of Italian books published in 1800-1900. The letters and numbers are quite different to those shown in
Elffers and the other early works that I have seen, but there are great
similarities to The New and Fashionable Chinese Puzzle, 1817 (check which??),
and some similarities to Al Gioco Cinese above. I haven't counted the figures to verify the 364. Mentioned in Milano, based on the copy I
sent to Dario Uri.
Jeu du Casse Tete Russe. 1817?
??NYS -- described and partly reproduced in Milano from an example in
the Raccolta Bertarelli but which has only four cards. Here the figures are given anthropomorphic
or architectonic shapes. There are four
cards on one coloured sheet and each card has a circle of three figures at the
top with three more figures along the bottom.
Each card has the name of the game at the top of the circle and
"les secrets des Chinois dévoliés" and "casse tête russe"
inside and outside the bottom of the circle.
The figures are quite different than in the following item.
Nuovo Giuoco Russo. Milano presso li Frat. Bettalli Cont. del
Cappello. [Frat. is an abbreviation of
Fratelli (Brothers) and Cont. is an abbreviation of Contrade (road).] Box, without pieces, but with 16 cards of
problems (one being examples) and instruction sheet (or leaflet). ??NYS - described by Milano with reproductions
of the box cover and four of the cards.
This example is in the Raccolta Bertarelli. Box shows a Turkish(?) man handing a box to another. On the first card is given the title and
publisher in French: Le Casse-Tête Russe
Milan, chez les Fr. Bettalli, Rue du Chapeau. The instruction sheet says that the Giuoco Chinese has had such
success in the principal cities of Europe that a Parisian publisher has
conceived another game called the Casse Tête Russe and that the Brothers
Bettalli have hurried to produce it.
Each card has four problems where the figures are greatly elaborated
into architectonic forms, very like those in Metamorfosi, below. Undated, but Milano first gives 1815‑1820,
and feels this is closely related to Metamorfosi and similar items, so he
concludes that it is 1818 or 1819, and this seems to be as correct as present
knowledge permits. The figures are
quite different than in the French version above.
Metamorfosi del Giuoco detto
l'Enimma Chinese. Firenze 1818
Presso Gius. Landi Libraio sul Canto di Via de Servi. Frontispiece shows an angel drawing a pattern
on a board which has the seven pieces at the top. The board leans against a plinth with the solution for making a
square shown on it. Under the drawing
is A. G. inv. Milano reproduces this
plate. One page of introduction,
headed Idea della Metamorfosi Imaginata dell'Enimma Chinese. 100 shapes, some solved, then with elegant
architectonic drawings in the same shapes, signed Gherardesce inv: et inc: Milano identifies the artist as Alessandro
Gherardesca (1779-1852), a Pisan architect.
See S&B, pp. 24‑25.
Grand Jeu du Casse Tête Français
en X. Pieces. ??NYS -- described and
partly reproduced in Milano, who says it comes from Paris and dates it
1818? The figures are anthropomorphic
and are most similar to those in Jeu du Casse Tete Russe.
Grande Giuocho del Rompicapo
Francese. Milano presso Pietro e
Giuseppe Vallardi Contrada di S. Margherita No 401(? my copy is
small and faint). ??NYS -- described
and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in the previous item,
but the figures have been redrawn rather than copied exactly.
Ch'i Ch'iao pan. c1820.
(Bibliothek Leiden 6891; Antiquariat Israel, Amsterdam.) ??NYS (van der Waals).
Le Veritable Casse‑tete,
ou Enigmes chinoises. Canu Graveur,
Paris, c1820. BL. ??NYS (van der Waals).
L'unico vero Enimma Chinese Tradotto dall'originale, pubblicato a
Londra, da J. Barfield. Florence,
[1820?]. [Listed in Quaritch's catalogue
646 (1947) item 699.)
A tangram appears in Pirnaisches
Wochenblatt of 16 Dec 1820. ??NYS --
described in Slocum, p. 60.
Ch'i Ch'iao ch'u pien ho‑pi. After 1820.
(Bibliothek Leiden 6891.) ??NYS
(van der Waals). 476 examples.
Nouveau Casse‑Tête
Français. c1820 (according to van der
Waals). Reproduced in van der Waals,
but it's not clear how the pages are assembled. Milano dates it a c1815 and indicates it is 16 cards, but van der
Waals looks like it may have been a booklet of 16 pp with TP, example page and
end page. The 16 pp have 80
problems.
Jerry
Slocum has sent 2 large pages with 58 figurative shapes which are clearly the
same pictures. The instructions are
essentially the same, but are followed by rules for a Jeu de Patience on the
second page and there is a 6 x 6 table of words on the first page headed
"Morales trouvées dans les ruines de la célébres Ville de Persépolis
..." which one has to assemble into moral proverbs. It looks like these are copies of folding
plates in some book of games.
Chinese Puzzle Georgina.
A. & S. Josh Myers, & Co 144, Leadenhall Street, London. Ganton Litho. 81 examples on 8 plates with elegant TP. Pages are one-sided sheets, sewn in the
middle, but some are upside down. Seen
at BL (1578/4938).
Bestelmeier, 1823. Item 1278: Chinese Squares. It is not in the 1812 catalogue.
Slocum. Compendium.
Shows the above Bestelmeier entry.
Anonymous. Ch'i ch'iao t'u ho pi (Harmoniously combined
book of Tangram problems) and Ch'i ch'iao t'u chieh (Tangram solutions). Two volumes of tangrams and solutions with
no title page, Chinese labels of the puzzles, in Chinese format (i.e. printed
as long sheets on thin paper, accordion folded and stitched with ribbon. Nd [c1820s??], stiff card covers with
flyleaves of a different paper, undoubtedly added later. 84 pages in each volume, containing 334 problems
and solutions. With ownership stamp of
a cartouche enclosing EWSHING, probably a Mr. E. W. Shing. Slocum says this is a c1820s reprint of the
earliest Chinese tangram book which appeared in 1813 & 1815. This version omits the TP and opening text. I have a photocopy of the opening material
from Slocum. The original problem book
had a preface by Sang‑hsia
K'o, which was repeated in the solution
book with the same date. Includes all
the problems of Shichi-kou-zu Gappeki, qv.
New Series of Ch'i ch'iau
puzzles. Printed by Lou Chen‑wan,
Ch'uen Liang, January 1826. ??NYS. (Copy at Dept. of Oriental Studies, Durham
Univ., cited in R. C. Bell; Tangram Teasers.)
Neues chinesisches Rätselspiel
für Kinder, in 24 bildlichen und alphabetischen Darstellungen. Friese, Pirna. Van der Waals, copying Santi, gives c1805, but Slocum, p. 60,
reports that it first appears in Pirnaisches Wochenblatt of 19 Dec 1829, though
there is another tangram in the issue of 16 Dec 1820. ??NYS.
Child. Girl's Own Book. 1833:
85; 1839: 72; 1842: 156. "Chinese
Puzzles -- These consist of pieces of wood in the form of squares, triangles,
&c. The object is to arrange them
so as to form various mathematical figures."
Anon. Edo Chiekata (How to Learn It??) (In Japanese). Jan 1837, 19pp, 306 problems. (Unclear if this uses the Tangram
pieces.) Reprinted in the same booklet
as Sei Shōnagon, on pp. 37‑55.
A Grand Eastern Puzzle. C. Davenport & Co., London. Nd.
??NYS (van der Waals). (Dalgety
has a copy and gives C. Davenporte (??SP) and Co., No. 20, Grafton Street, East
Euston Square. Chinese pages dated 1813
in European binding with label bearing the above information.)
Augustus De Morgan. On the foundations of algebra, No. 1. Transactions of the Cambridge Philosophical
Society 7 (1842) 287-300. ??NX. On pp. 289, he says "the well-known toy
called the Chinese Puzzle, in which a prescribed number of forms are given, and
a large number of different arrangements, of which the outlines only are drawn,
are to be produced."
Crambrook. 1843.
P. 4, no. 4: Chinese Puzzle.
Chinese Books, thirteen numbers.
Though not illustrated, this seems likely to be the Tangrams -- ??
Boy's Own Book. 1843 (Paris): 439.
No.
19: The Chinese Puzzle. Instructions
give five shapes and say to make one copy of some and two copies of the
others. As written, this has two medium
sized triangles instead of two large ones, though it is intended to be the
tangrams. 11 problem shapes given, no answers. Most of the shapes occur in earlier tangram collections,
particularly in A New Invented Chinese
Puzzle. "The puzzle may be
purchased, ..., at Mr. Wallis's, Skinner street, Snow hill, where numerous
books, containing figures for this ingenious toy may also be
obtained." = Boy's Treasury, 1844,
pp. 426-427, no. 16. It is also
reproduced, complete with the error, but without the reference to Wallis, as: de Savigny, 1846, pp. 355-356, no. 14: Le
casse-tête chinois; Magician's Own
Book, 1857, prob. 49, pp. 289-290;
Landells, Boy's Own Toy-Maker, 1858, pp. 139-140; Book of 500 Puzzles, 1859, pp. 103‑104; Boy's Own Conjuring Book, 1860, pp.
251-252; Wehman, New Book of 200
Puzzles, 1908, pp. 34‑35.
No.
20: The Circassian puzzle. "This
is decidedly the most interesting puzzle ever invented; it is on the same
principle, but composed of many more pieces than the Chinese puzzle, and may
consequently be arranged in more intricate figures. ..." No pieces or
problems are shown. In the next
problem, it says: "This and the Circassian puzzle are published by Mr.
Wallis, Skinner-street, Snow-hill."
= Boy's Treasury, 1844, p. 427, no. 17. = de Savigny, 1846, p. 356, no. 15: Le problème circassien, but
the next problem omits the reference to Wallis.
Although
I haven't recorded a Circassian puzzle yet -- cf in 6.S.2 -- I have just seen
that the puzzle succeeding The Chinese Puzzle in Wehman, New Book of 200
Puzzles, 1908, pp. 35-36, is called The Puzzle of Fourteen which might be the
Circassian puzzle. Taking a convenient
size, this has two equilateral triangles of edge 1 and four each of the
following: a 30o-60o-90o triangle with edges 2, 1, Ö3; a
parallelogram with angles 60o
and 120o with edges 1 and
2; a trapezium with base angles 60o and 60o, with lower and upper base edges 2 and 1,
height Ö3/4 and slant edges 1/2 and Ö3/2.
All 14 pieces make a rectangle 2Ö3 by 4.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob.
584-11, pp. 288 & 405: Chinesisches Verwandlungsspiel. Make a square with the tangram pieces. Shows just five of the pieces, but correctly
states which two to make two copies of.
Prob.
584-16, pp. 289 & 406. Make an
isosceles right triangle with the tangram pieces.
Prob.
584-18/25, pp. 289-291 & 407: Hieroglyphenspiele. Form various figures from various sets of pieces, mostly
tangrams, but the given shapes have bits of writing on them so the assembled
figure gives a word. Only one of the
shapes is as in Boy's Own Book.
Prob.
588, pp. 298 & 410: Etliche Knackmandeln.
Another tangram problem like the preceding, not equal to any in Boy's
Own Book.
Adams & Co., Boston. Advertisement in The Holiday Journal of
Parlor Plays and Pastimes, Fall 1868.
Details?? -- photocopy sent by Slocum.
P. 6: Chinese Puzzle. The
celebrated Puzzle with which a hundred or more symmetrical forms can be made,
with book showing the designs. Though
not illustrated, this seems likely to be the Tangrams -- ??
Mittenzwey. 1880.
Prob. 243-252, pp. 45 & 95-96;
1895?: 272-281, pp. 49 & 97-98;
1917: 272-282, pp. 45 & 92-93.
Make a funnel, kitchen knife, hammer, hat with brim being horizontal or
hanging down or turned up, church, saw, dovecote, hatchet, square, two equal
squares.
J. Murray (editor of the
OED). Two letters to H. E. Dudeney (9
Jun 1910 & 1 Oct 1910). The first inquires about the word 'tangram', following on
Dudeney's mention of it in his "World's best puzzles" (op. cit. in
2). The second says that 'tan' has no
Chinese origin; is apparently mid 19C,
probably of American origin; and the
word 'tangram' first appears in Webster's Dictionary of 1864. Dudeney, AM, 1917, p. 44, excerpts these
letters.
F. T. Wang & C.‑S.
Hsiung. A theorem on the tangram. AMM 49 (1942) 596‑599. They determine the 20 convex regions which
16 isosceles right triangles can form and hence the 13 ones which the Tangram
pieces can form.
Mitsumasa Anno. Anno's Math Games. (Translation of: Hajimete deau sugaku no ehon; Fufkuinkan Shoten,
Tokyo, 1982.) Philomel Books, NY,
1987. Pp. 38-43 & 95-96 show a
simplified 5-piece tangram-like puzzle
which I have not seen before. The
pieces are: a square of side 1; three isosceles right triangles of side 1; a
right trapezium with bases 1 and
2, altitude 1
and slant side Ö2. The trapezium can be viewed as putting
together the square with a triangle. 19
problems are set, with solutions at the back.
James Dalgety. Latest news on oldest puzzles. Lecture to Second Meeting on the History of
Recreational Mathematics, 1 Jun 1996.
10pp. In 1998, he extracted the
two sections on tangrams and added a list of tangram books in his collection
as: The origins of Tangram; © 1996/98; 10pp.
(He lists about 30 books, eight up to 1850.) In 1993, he was buying tangrams in Hong Kong and asked what they
called it. He thought they said
'tangram' but a slower repetition came out 'ta hau ban' and they wrote down the
characters and said it translates as 'seven lucky tiles'. He has since found the characters in 19C
Chinese tangram books. It is quite
possible that Sam Loyd (qv under Murray, above) was told this name and wrote
down 'tangram', perhaps adjusted a bit after thinking up Tan as the inventor.
At the International Congress on
Mathematical Education, Seville, 1996, the Mathematical Association gave
out The 3, 4, 5 Tangram, a cut card tangram, but in a 6 x 8
rectangular shape, so that the medium sized triangle was a 3-4-5 triangle. I modified this in Nov 1999, by stretching along a diagonal to
form a rhombus with angles double the angles of a 3-4-5 triangle, so that four
of the triangles are similar to 3-4-5 triangles. Making the small triangles be actually 3-4-5, all edges are
integral. I made up 35 problems with
these pieces. I later saw that Hans
Wiezorke has mentioned this dissection in CFF, but with no problems. I distributed this as my present at G4G4,
2000.
See S&B 22. I recall there is some dispute as to whether
the basic diagram should be a square or a double square.
E. J. Dijksterhuis. Archimedes.
Munksgaard, Copenhagen, 1956;
reprinted by Princeton Univ. Press, 1987. Pp. 408‑412 is the best discussion of this topic and
supplies most of the classical references.
Archimedes. Letter to Eratosthenes, c-250?. Greek palimpsest, c975, on MS no. 355, from
the Cloister of Saint Sabba (= Mar Saba), Jerusalem, then at Metochion of the
Holy Sepulchre, Constantinopole. [This
MS disappeared in the confusion in Asia Minor in the 1920s but reappeared in
1998 when it was auctioned by Christie's in New York for c2M$. Hopefully, modern technology will allow a
facsimile and an improved transcription in the near future.] Described by J. L. Heiberg (& H. G.
Zeuthen); Eine neue Schrift des Archimedes; Bibliotheca Math. (3) 7 (1906‑1907)
321‑322. Heiberg describes the
MS, but only mentions the loculus. The
text is in Heiberg's edition of Archimedes; Opera; 2nd ed., Teubner, Leipzig,
1913, vol. II, pp. 415‑424, where it has been restored using the Suter
MSS below. Heath only mentions the
problem in passing. Heiberg quotes
Marius Victorinus, Atilius Fortunatianus and cites Ausonius and Ennodius.
H. Suter. Der Loculus Archimedius oder das
Syntemachion des Archimedes. Zeitschr.
für Math. u. Phys. 44 (1899) Supplement
= AGM 9 (1899) 491‑499.
This is a collation from two 17C Arabic MSS which describe the
construction of the loculus. They are
different than the above MS. The German
is included in Archimedes Opera II, 2nd ed., 1913, pp. 420‑424.
Dijksterhuis discusses both of
the above and says that they are insufficient to determine what was
intended. The Greek seems to indicate
that Archimedes was studying the mathematics of a known puzzle. The Arabic shows the construction by cutting
a square, but the rest of the text doesn't say much.
Lucretius. De Rerum Natura. c‑70. ii, 778‑783. Quoted and discussed in H. J. Rose;
Lucretius ii. 778‑83; Classical Review (NS) 6 (1956) 6‑7. Brief reference to assembling pieces into a
square or rectangle.
Decimus Magnus Ausonius. c370.
Works. Edited & translated
by H. G. Evelyn White. Loeb Classical
Library, ??date. Vol. I, Book XVII:
Cento Nuptialis (A Nuptial Cento), pp. 370-393 (particularly the Preface,
pp. 374-375) and Appendix, pp. 395-397.
Refers to 14 little pieces of bone which form a monstrous elephant, a
brutal boar, etc. The Appendix gives
the construction from the Arabic version, via Heiberg, and forms the monstrous
elephant.
Marius Victorinus. 4C.
VI, p. 100 in the edition of Keil, ??NYS. Given in Archimedes Opera II, 2nd ed., 1913, p. 417. Calls it 'loculus Archimedes' and says it
had 14 pieces which make a ship, sword, etc.
Ennodius. Carmina: De ostomachio eburneo. c500.
In: Magni Felicis Ennodii Opera;
ed. by F. Vogel, p. 340. In: Monumenta Germaniae Historica, VII (1885)
249. ??NYS. Refers to ivory pieces to be assembled.
Atilius Fortunatianus. 6C.
??NYS Given in Archimedes Opera
II, p. 417. Same comment as for Marius
Victorinus.
E. Fourrey. Curiositiés Géométriques. (1st ed., Vuibert & Nony, Paris,
1907); 4th ed., Vuibert, Paris,
1938. Pp. 106‑109. Cites Suter, Ausonius, Marius Victorinus,
Atilius Fortunatianus.
Collins. Book of Puzzles. 1927. The loculus of
Archimedes, pp. 7-11. Pieces made from
a double square.
See Hoffmann
& S&B, cited at the beginning of 6.S, for general surveys.
See
Bailey in 6.AS.1 for an 1858 puzzle with 10 pieces and The Sociable and Book of
500 Puzzles, prob. 10, in 6.AS.1 for an 11 piece puzzle.
There
are many versions of this idea available and some are occasionally given in
JRM.
The Richter Anchor Stone puzzles and
building blocks were inspired by Friedrich Froebel (or Fröbel) (1782‑1852),
the educational innovator. He was the
inventor of Kindergartens, advocated children's play blocks and inspired both
the Richter Anchor Stone Puzzles and Milton Bradley. The stone material was invented by Otto Lilienthal (1848‑1896)
(possibly with his brother Gustav) better known as an aviation pioneer -- they
sold the patent and their machines to F. Adolph Richter for 1000 marks. The material might better be described as a
kind of fine brick which could be precisely moulded. Richter improved the stone and began production at Rudolstadt,
Thüringen, in 1882; the plant closed in 1964.
Anchor was the company's trademark.
He made at least 36 puzzles and perhaps a dozen sets of building blocks
which were popular with children, architects, engineers, etc. The Deutsches Museum in Munich has a whole
room devoted to various types of building blocks and materials, including the
Anchor blocks. The Speelgoed Museum 'Op
Stelten' (Sint Vincentiusstraat 86, NL-4902 (or 4901) GL Oosterhout,
Noord-Brabant, The Netherlands; tel: 0262 452 825; fax: 0262 452
413) has a room of Richter blocks and some puzzles. There was an Anker Museum in the Netherlands (Stichting Ankerhaus
(= Anker Museum); Opaalstraat 2‑4 (or Postf. 1061), NL-2400 BB Alphen aan
den Rijn, The Netherlands; tel: 01720‑41188) which produced
replacement parts for Anker stone puzzles.
Modern facsimiles of the building sets are being produced at Rudolstadt.
In 1996 I noticed the ceiling of
the room to the south of the Salon of the Ambassadors in the Alcazar of
Seville. This 15C? ceiling was built by
workmen influenced by the Moorish tradition and has 112 square wooden panels in
a wide variety of rectilineal patterns.
One panel has some diagonal lines and looks like it could be used as a
10 piece tangram-like puzzle. Consider
a 4 x 4 square. Draw both
diagonal lines, then at two adjacent corners, draw two lines making a unit
square at these corners. At the other
two corners draw one of these two lines, namely the one perpendicular to their
common side. This gives six isosceles
right triangles of edge 1; two pentagons with three right angles and
sides 1, 2, 1, Ö2, Ö2; two quadrilaterals with two right angles and
sides 2, 1, Ö2, 2Ö2. Since geometric patterns and panelling are
common features of Arabic art, I wonder if there are any instances of such
patterns being used for a tangram-like puzzle?
Grand Jeu du Casse Tête Français
en X. Pieces. ??NYS -- described and
partly reproduced in Milano, who says it comes from Paris and dates it
1818? The figures are anthropomorphic
and are most similar to those in Jeu du Casse Tete Russe.
Grande Giuocho del Rompicapo
Francese. Milano presso Pietro e
Giuseppe Vallardi Contrada di S. Margherita No 401(? my copy is
small and faint). ??NYS -- described
and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in the previous item,
but the figures have been redrawn rather than copied exactly.
Allizeau. Les Métamorphoses ou Amusemens
Géometriques Dédiée aux Amateurs Par Allizeau. A Paris chex Allizeau
Quai Malaquais, No 15.
??NYS -- described and partly reproduced in Milano. This uses 15 pieces and the problems tend to
be architectural forms, like towers.
Jackson. Rational Amusement. 1821.
Geometrical Puzzles, nos. 20-27, pp. 27-29 & 88-89 & plate II,
figs. 15-22. This is a set of 20 pieces
of 8 shapes used to make a square, a right triangle, three squares, etc.
Crambrook. 1843.
P. 4, no. 1: Pythagorean Puzzle, with Book. Though not illustrated, this is probably(??) the puzzle described
in Hoffmann, below, which was a Richter Anchor puzzle No. 12 of the same name
and is still occasionally seen. See
S&B 28.
Edward Hordern's collection has
a Circassian Puzzle, c1870, with many pieces, but I didn't record the shapes --
cf Boy's Own Book, 1843 (Paris), in section 6.S.
Mittenzwey. 1880.
Prob.
177-179, pp. 34 & 86; 1895?:
202-204, pp. 38-39 & 88; 1917:
202-204, pp. 35 & 84-85. Consider
the ten piece version of dissecting 5 squares to one (6.AS.1). Use the pieces to make:
a squat octagon, a house gable-end, a
church (no solution), etc.;
two dissimilar rectangles;
three dissimilar parallelograms, two
dissimilar trapezoids. Solution says
one can make many other shapes with these pieces, e.g. a trapezoid with
parallel sides in the proportion 9 :
11.
Prob.
181-184, pp. 34-35 & 87-88; 1895?:
206-209, pp. 39 & 89-90; 1917:
206-209, pp. 36 & 85-86. Take six
equilateral triangles of edge 2. Cut an
equilateral triangle of edge 1 from the corner of each of them, giving 12
pieces. Make a hexagon in eight
different ways [there are many more -- how many??] and three tangram-like
shapes.
Prob.
195-196, pp. 36 & 89; 1895?:
220-221, pp. 41 & 91; 1917:
220-221, pp. 37 & 87. Use four
isosceles right triangles, say of leg 1, to make a square, a 1 x 4
rectangle and an isosceles right triangle.
Nicholas Mason. US Patent 232,140 - Geometrical
Puzzle-Block. Applied: 13 May 1880;
patented 14 Sep 1880. 1p plus 2pp
diagrams. Five squares, six units
square, each cut into four pieces in the same way. Start at the midpoint of a side and cut to an opposite
corner. (This is the same cut used to
produce the ten piece 'Five Squares to One' puzzle.) Cut again in the triangle just formed, from the same midpoint to
a point one unit from the right angle corner of the piece just made. This gives a right triangle of sides 3, 1, Ö10 and a triangle of sides 5, Ö10, 3Ö45. Cut again from the same midpoint across the
trapezoidal piece made by the first cut, to a point five units from the corner
previously cut to. This gives a
triangle of sides 5, 3Ö5, 2Ö10 and a right trapezoid with sides 2,Ö10, 1, 6, 3. This was produced as Hill's American
Geometrical Prize Puzzle in England ("Price, One Shilling.") in
1882. Harold Raizer produced a
facsimile version, with facsimile box label and instructions for IPP22. The instructions have 20 problems to solve
and the solutions have to be submitted by 1 May 1882.
Hoffmann. 1893.
Chap. III, no. 3: The Pythagoras Puzzle, pp. 83-85 & 117-118
= Hoffmann‑Hordern, pp. 69-72.
This has 7 pieces and is quite like the Tangram -- see comment under
Crambrook. Photo on p. 71, with
different version in Hordern Collection, p. 50.
C. Dudley Langford. Note 1538:
Tangrams and incommensurables.
MG 25 (No. 266) (Oct 1941) 233‑235. Gives alternate dissections of the square and some hexagonal
dissections.
C. Dudley Langford. Note 2861:
A curious dissection of the square.
MG 43 (No. 345) (Oct 1959) 198.
There are 5 triangles whose angles are multiples of π/8 = 22½o. He uses these to make a square.
See items at the end of 6.S.
6.T. NO THREE IN A LINE PROBLEM
See
also section 6.AO.2.
Loyd. Problem 14: A crow puzzle.
Tit‑Bits 31 (16 Jan
& 6 Feb 1897) 287 &
343. = Cyclopedia, 1914,
Crows in the corn, pp. 110 & 353.
= MPSL1, prob. 114, pp. 113 & 163‑164. 8 queens with no two attacking and no three
in any line.
Dudeney. The Tribune (7 Nov 1906) 1. ??NX.
= AM, prob. 317, pp. 94 & 222.
Asks for a solution with two men in the centre 2 x 2 square.
Loyd. Sam Loyd's Puzzle Magazine, January 1908. ??NYS.
(Given in A. C. White; Sam Loyd and His Chess Problems; 1913, op.
cit. in 1; p. 100, where it is described as the only solution with 2 pieces in
the 4 central squares.)
Ahrens, MUS I 227, 1910, says he
first had this in a letter from E. B. Escott dated 1 Apr 1909.
(W. Moser, below, refers this to the 1st ed., 1900, but this must be due
to his not having seen it.)
C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV: No. 2: Another draught
puzzle, pp. 515 & 520. The problem
says "no three men shall be in a line, either horizontally or
perpendicularly". The solution
says "no three are in a line in any direction" and the diagram shows
this is indeed true.
Loyd. Picket posts. Cyclopedia,
1914, pp. 105 & 352. = MPSL2, prob.
48, pp. 34 & 136. 2 pieces
initially placed in the 4 central squares.
Blyth. Match-Stick Magic.
1921. Matchstick board game, p.
73. 6 x 6 version phrased as putting "only two in any one line:
horizontal, perpendicular, or diagonal."
However, his symmetric solution has three in a row on lines of slope 2.
King. Best 100. 1927. No. 69, pp. 28 & 55. Problem on the 6 x 6 board -- gives a
symmetric solution. Says "there
are two coins on every row" including "diagonally across it",
but he has three in a row on lines of slope 2.
Loyd Jr. SLAHP.
1928. Checkers in rows, pp. 40
& 98. Different solution than in
Cyclopedia.
M. Adams. Puzzle Book. 1939. Prob. C.83: Stars
in their courses, pp. 144 & 181.
Same solution as King, but he says "two stars in each vertical row,
two in each horizontal row, and two in each of the the two diagonals .... There must not be more than two stars in the
same straight line", but he has three in a row on lines of slope 2.
W. O. J. Moser & J.
Pach. No‑three‑in‑line
problem. In: 100 Research Problems in Discrete Geometry 1986; McGill Univ.,
1986. Problem 23, pp. 23.1 -- 23.4. Survey with 25 references. Solutions are known on the n x n
board for n £
16 and for even n £ 26. Solutions with the symmetries of the square are only known
for n = 2, 4, 10.
R. Penrose. The role of aesthetics in pure and applied
mathematical research. Bull. Inst.
Math. Appl. 10 (1974) 266‑272.
M. Gardner. SA (Jan 1977). Extensively rewritten as Penrose Tiles, Chaps. 1 & 2.
R. Penrose. Pentaplexity. Eureka 39 (1978) 16‑22.
= Math. Intell. 2 (1979) 32‑37.
D. Shechtman, I. Blech, D.
Gratias & J. W. Cohn. Metallic
phase with long‑range orientational order and no translational
symmetry. Physical Rev. Letters 53:20
(12 Nov 1984) 1951‑1953.
Describes discovery of 'quasicrystals' having the symmetry of a Penrose‑like
tiling with icosahedra.
David R. Nelson. Quasicrystals. SA 255:2 (Aug 1986) 32‑41 & 112. Exposits the discovery of
quasicrystals. First form is now called
'Shechtmanite'.
Kimberly-Clark Corporation has
taken out two patents on the use of the Penrose pattern for quilted toilet
paper as the non-repetition prevents the tissue from 'nesting' on the
roll. In Apr 1997, Penrose issued a
writ against Kimberly Clark Ltd. asserting his copyright on the pattern and
demanding damages, etc.
John Kay. Top prof goes potty at loo roll
'rip-off'. The Sun (11 Apr 1997) 7.
Patrick McGowan. It could end in tears as maths boffin sues
Kleenex over design. The Evening
Standard (11 Apr 1997) 5.
Kleenex art that ended in
tears. The Independent (12 Apr 1997) 2.
For a knight on the tiles. Independent on Sunday (13 Apr 1997) 24. Says they exclusively reported Penrose's
discovery of the toilet paper on sale in Dec 1996.
D. Trull. Toilet paper plagiarism. Parascope, 1997 -- available on
www.noveltynet.org/content/paranormal/www.parascope.com/arti...
6.U.2. PACKING BRICKS IN BOXES
In
two dimensions, it is not hard to show that a x b packs A x B
if and only if
a divides either A or
B; b
divides either A or B; A and B
are both linear combinations of
a and b. E.g. 2 x 3
bricks pack a 5 x 6 box.
See
also 6.G.1.
Anon. Prob. 52. Hobbies 30 (No. 767) (25 Jun 1910) 268 & 283 &
(No. 770) (16 Jul 1910) 328. Use
at least one of each of 5 x 7, 5 x 10,
6 x 10 to make the smallest
possible square. Solution says to use 4, 4, 1,
but doesn't show how. There are
lots of ways to make the assembly.
Manuel H. Greenblatt ( -1972, see JRM 6:1 (Winter 1973) 69). Mathematical Entertainments. Crowell, NY, 1965. Construction of a cube, pp. 80‑81. Can
1 x 2 x 4 fill 6 x 6 x 6?
He asserts this was invented by R. Milburn of Tufts Univ.
N. G. de Bruijn. Filling boxes with bricks. AMM 76 (1969) 37‑40. If a1
x ... x an fills A1 x ... x An and
b divides k of
the ai, then
b divides at least k of
the Ai. He previously presented the results, in
Hungarian, as problems in Mat. Lapok 12, pp. 110‑112, prob. 109 and
13, pp. 314‑317, prob. 119.
??NYS.
D. A. Klarner. Brick‑packing puzzles. JRM 6 (1973) 112‑117. General survey. In this he mentions a result that I gave him -- that 2 x 3 x 7
fills a 8 x 11 x 21, but that the box cannot be divided into two
packable boxes. However, I gave him the
case 1 x 3 x 4 in 5
x 5 x 12 which is the smallest example
of this type. Tom Lensch makes fine
examples of these packing puzzles.
T. H. Foregger, proposer; Michael Mather, solver. Problem E2524 -- A brick packing
problem. AMM 82:3 (Mar 1975) 300
& 83:9 (Nov 1976) 741-742. Pack
41 1 x 2 x 4 bricks in a 7 x 7 x 7 box.
One cannot get 42 such bricks into the box.
6.V. SILHOUETTE AND VIEWING PUZZLES
Viewing
problems must be common among draughtsmen and engineers, but I haven't seen
many examples. I'd be pleased to see
further examples.
2 silhouettes.
Circle &
triangle -- van Etten,
Ozanam, Guyot, Magician's Own Book (UK version)
Circle &
square -- van Etten
Circle &
rhombus -- van Etten,
Ozanam
Rectangle
with inner rectangle & rectangle with notch --
Diagram Group.
3 silhouettes.
Circle, circle,
circle -- Madachy
Circle, cross,
square -- Shortz collection (c1884), Wyatt,
Perelman
Circle, oval,
rectangle -- van Etten,
Ozanam, Guyot,
Magician's Own Book (UK version)
Circle, oval,
square -- van Etten,
Tradescant, Ozanam, Ozanam‑Montucla, Badcock,
Jackson, Rational
Recreations,
Endless Amusement II,
Young Man's Book
Circle, rhombus,
rectangle -- Ozanam,
Alberti
Circle, square,
triangle -- Catel,
Bestelmeier, Jackson, Boy's Own Book, Crambrook,
Family Friend, Magician's Own
Book,
Book of 500 Puzzles,
Boy's Own Conjuring Book,
Illustrated Boy's Own Treasury, Riecke, Elliott, Mittenzwey,
Tom Tit, Handy Book, Hoffmann,
Williams, Wyatt, Perelman,
Madachy. But see Note below.
Square, tee,
triangle -- Perelman
4 silhouettes.
Circle, square,
triangle, rectangle with curved
ends -- Williams
2 views.
Antilog, Ripley's,
Diagram Group;
3 views.
Madachy, Ranucci,
For
the classic Circle, Square, Triangle, version, the triangle cannot be not
equilateral. Consider a circle,
rectangle, triangle version. If D is
the diameter of the circle and H is the height of the plug, then the rectangle
has dimensions D x H and the triangle has base D
and side S, so S
= Ö(H2 + D2/4). Making the rectangle a square, i.e. H = D,
makes S = DÖ5/2, while making the triangle equilateral,
i.e. S = D, makes H = DÖ3/2.
van Etten. 1624.
Prob.
22 (misnumbered 15 in 1626) (Prob. 20), pp. 19‑20 & figs. opp.
p. 16 (pp. 35‑36): 2 silhouettes --
one circular, the other triangular, rhomboidal or square. (English ed. omits last case.) The 1630 Examen says the author could have
done better and suggests: isosceles
triangle, several scalene triangles, oval or circle, which he says can be done
with an elliptically cut cone and a scalene cone. I am not sure I believe these.
It seems that the authors are allowing the object to fill the hole and
to pass through the hole moving at an angle to the board rather than
perpendicularly as usually understood.
In the English edition the Examination is combined with that of the next
problem.
Prob.
23 (21), pp. 20‑21 & figs. opp. p. 16 (pp. 37‑38): 3
silhouettes -- circle, oval and square or rectangle. The 1630 Examen suggests:
square, circle, several parallelograms and several ellipses, which he
says can be done with an elliptic cylinder of height equal to the major diameter
of the base. The English Examination
says "a solid colume ... cut Ecliptick-wise" -- ??
John II Tradescant
(1608-1662). Musæum Tradescantianum:
Or, A Collection of Rarities Preserved at South-Lambeth neer London By John
Tradescant. Nathaniel Brooke, London,
1656. [Facsimile reprint, omitting the
Garden List, Old Ashmolean Reprints I, edited by R. T. Gunther, on the occasion
of the opening of the Old Ashmolean Museum as what has now become the Museum of
the History of Science, Oxford. OUP,
1925.] John I & II Tradescant were
gardeners to nobility and then royalty and used their connections to request
naval captains to bring back new plants, curiosities and "Any thing that
Is strang". These were accumulated
at his house and garden in south Lambeth, becoming known as Tradescant's Ark,
eventually being acquired by Elias Ashmole and becoming the foundation of the
Ashmolean Museum in Oxford. This
catalogue was prepared by Elias Ashmole and his friend Thomas Wharton, but they
are not named anywhere in the book. It
was the world's first museum catalogue.
P.
37, last entry: "A Hollow cut in wood, that will fit a round, square and
ovall figure."
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. He says square, circle and triangle is in a book in front of him dated
1674. I suspect this must be the 1674
English edition of van Etten, but I don't find the problem in the English
editions I have examined. Perhaps
Dudeney just meant that the idea was given in the 1674 book, though he is
specifically referring to the square, circle, triangle version.
Ozanam. 1725.
Vol. II, prob. 58 & 59, pp. 455‑458 & plate 25* (53 (note
there is a second plate with the same number)). Circle and triangle;
circle and rhombus; circle,
oval, rectangle; circle, oval,
square. Figures are very like van
Etten. See Ozanam-Montucla, 1778.
Ozanam. 1725.
Vol. IV. No text, but shown as
an unnumbered figure on plate 15 (17).
3 silhouettes: circle,
rhombus, rectangle.
Simpson. Algebra.
1745. Section XVIII, prob. XXIX,
pp. 279-281. (1790: prob. XXXVII, pp.
306-307. Computes the volume of an
ungula obtained by cutting a cone with a plane. Cf Riecke, 1867.
Alberti. 1747.
No text, but shown as an unnumbered figure on plate XIIII, opp. p. 218
(112), copied from Ozanam, 1725, vol IV.
3 silhouettes: circle, rhombus,
rectangle.
Ozanam-Montucla. 1778.
Faire passer un même corps par un trou quarré, rond &
elliptique. Prob. 46, 1778: 347-348; 1803: 345-346; 1814: 293. Prob. 45,
1840: 149-150. Circle, ellipse, square.
Catel. Kunst-Cabinet. 1790. Die mathematischen Löcher, p. 16 & fig.
42 on plate II. Circle, square,
triangle.
E. C. Guyot. Nouvelles Récréations Physiques et
Mathématiques. Op. cit. in 6.P.2.
1799. Vol. 2, Quatrième
récréation, p. 45 & figs. 1‑4, plate 7, opp. p. 45. 2 silhouettes: circle & triangle; 3
silhouettes: circle, oval, rectangle.
Bestelmeier.
1801. Item 536: Die 3 mathematischen Löcher. (See also the picture of Item 275, but that
text is for another item.) Square,
triangle and circle.
1807. Item 1126: Tricks includes the square, triangle and circle.
Badcock. Philosophical Recreations, or, Winter
Amusements. [1820]. P. 14, no. 23: How to make a Peg that will
exactly fit three different kinds of Holes.
"Let one of the holes be circular, the other square, and the third
an oval; ...." Solution is a
cylinder whose height equals its diameter.
Jackson. Rational Amusement. 1821.
Geometrical Puzzles.
No.
16, pp. 26 & 86. Circle, square,
triangle, with discussion of the dimensions: "a wedge, except that its base must be
a circle".
No.
29, pp. 30 & 89-90. Circle, oval,
square.
Rational Recreations. 1824.
Feat 19, p. 66. Circle, oval,
square.
Endless Amusement II. 1826?
P. 62: "To make a Peg that
will exactly fit three different kinds of Holes." Circle, oval, square. c= Badcock.
The Boy's Own Book. The triple accommodation. 1828: 419;
1828-2: 424; 1829 (US):
215; 1855: 570; 1868: 677. Circle, square and triangle.
Young Man's Book. 1839.
Pp. 294-295. Circle, oval,
square. Identical to Badcock.
Crambrook. 1843.
P. 5, no. 16: The Mathematical Paradox -- the Circle, Triangle, and
Square. Check??
Family Friend 3 (1850) 60 &
91. Practical puzzle -- No. XII. Circle, square, triangle. This is repeated as Puzzle 16 -- Cylinder
puzzle in (1855) 339 with solution in (1856) 28.
Magician's Own Book. 1857.
Prob. 21: The cylinder puzzle, pp. 273 & 296. Circle, square, triangle. = Book of 500 Puzzles, 1859, prob. 21, pp.
87 & 110. = Boy's Own Conjuring
Book, 1860, prob. 20, pp. 235 & 260.
Illustrated Boy's Own
Treasury. 1860. Practical Puzzles, No. 42, pp. 403 &
442. Identical to Magician's Own Book,
with diagram inverted.
F. J. P. Riecke. Op. cit. in 4.A.1, vol. 1, 1867. Art. 33: Die Ungula, pp. 58‑61. Take a cylinder with equal height and
diameter. A cut from the diameter of
one base which just touches the other base cuts off a piece called an ungula
(Latin for claw). He computes the
volume as 4r3/3. He then makes the symmetric cut to produce
the circle, square, triangle shape, which thus has volume (2π ‑ 8/3) r3. Says he has seen such a shape and a board
with the three holes as a child's toy.
Cf Simpson, 1745.
Magician's Own Book (UK
version). 1871. The round peg in the square hole: To pass a cylinder through three different
holes, yet to fill them entirely, pp. 49-50.
Circle, oval, rectangle; circle
& (isosceles) triangle.
Alfred Elliott. Within‑Doors. A Book of Games and Pastimes for the Drawing
Room. Nelson, 1872. [Toole Stott 251. Toole Stott 1030 is a 1873 ed.]
No. 4: The cylinder puzzle, pp. 27‑28 & 30‑31. Circle, square, triangle.
Mittenzwey. 1880.
Prob. 257, pp. 46 & 97;
1895?: 286, pp. 50 & 99-100;
1917: 286, pp. 45 & 94-95.
Circle, square, triangle.
Will Shortz has a puzzle trade
card with the circle, cross, square problem, c1884.
Tom Tit, vol. 2. 1892.
La cheville universelle, pp. 161-162.
= K, no. 28: The universal plug, pp. 72‑73. = R&A, A versatile peg, p. 106. Circle, square, triangle.
Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. Pp. 238-242:
Captain S's peg puzzle. Circle, square,
triangle.
Hoffmann. 1893.
Chap. X, no. 20: One peg to fit three holes, pp. 344 & 381‑382
= Hoffmann-Hordern, pp. 238-239, with photo. Circle, square, triangle.
Photo on p. 239 shows two examples: one simply a wood board and
pieces; the other labelled The Holes and Peg Puzzle, from Clark's Cabinet of
Puzzles, 1880-1900, but this seems to be just a card box with the holes.
Williams. Home Entertainments. 1914.
The plug puzzle, pp. 103-104.
Circle, square, triangle and rectangle with curved ends. This is the only example of this four-fold
form that I have seen. Nice drawing of
a board with the plug shown in each hole, except the curve on the sloping faces
is not always drawn down to the bottom.
E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in
5.H.1.
The
"cross" plug puzzle, p. 17.
Square, circle and cross.
The
"wedge" plug puzzle, p. 18.
Square, circle and triangle.
Perelman. FMP.
c1935? One plug for three
holes; Further "plug"
puzzles, pp. 339‑340 & 346. 6
simple versions; 3 harder
versions: square, triangle,
circle; circle, square, cross; triangle, square, tee. The three harder versions are also in FFF,
1957: probs. 69-71, pp. 112 & 118-119; 1979: probs. 73‑75, pp. 137 & 144 = MCBF: probs. 73-75, pp. 134-135 &
142-143.
Anonymous [Antilog]. An elevation puzzle. Eureka 19 (Mar 1957) 11 & 19. Front and top views are a square with a
square inside it. What is the side
view? Gives two solutions.
Anonymous. An elevation puzzle. Eureka 21 (Oct 1958) 7 & 29. Front is the lower half of a circle. Plan (= top view) is a circle. What is the side view? Solution is a V shape, but it ought to
be the other way up! Nowadays, one can
buy potato crisps (= potato chips) in this shape.
Joseph S. Madachy. 3‑D in 2‑D. RMM 2 (Apr 1961) 51‑53 &
3 (Jun 1961) 47. Discusses 3
view and 3 silhouette problems.
3
circular silhouettes, but not a sphere.
Square,
circle, triangle.
Ernest R. Ranucci. Non‑unique orthographic
projections. RMM 14 (Jan‑Feb
1964) 50. 3 views such that there
are 10 different objects with these views.
Ripley's Puzzles and Games. 1966.
Pp. 18-19, item 1. Same problem
as Antilog, 1957. Gives one solution.
Cedric A. B. Smith. Simple projections. MG 62 (No. 419) (Mar 1978) 19-25. This is about how different projections
affect one's recognition of what an object is.
He starts with an example with two views and the isometric projection
which is very difficult to interpret.
He gives three other views, each of which is easily interpreted.
The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex,
1984. Problem 114, with Solution at the
back of the book. Front view is a
rectangle with an interior rectangle. Side
view is a rectangle with a rectangular notch on front side. Solution is a short cylinder with a straight
notch in it. This is a fairly classic
problem for engineers but I haven't seen it in print elsewhere.
Marek Penszko. Polish your wits -- 3: Loop the loop. Games 11:2 (Feb/Mar 1987) 28 & 58. Draw lines on a glass cube to produce three
given projections. Problem asks for all
three projections to be the same.
When
assembled, a burr looks like three sticks crossing orthogonally, forming a
'star' with six points at the vertices of an octahedron. Slocum says Wyatt [Puzzles in Wood, 1928,
op. cit. in 5.H.1] is the first to use the word 'burr'. Collins, Book of Puzzles, 1927, p. 135,
calls them "Cluster, Parisian or Gordian Knot Puzzles" and states:
"it is believed that they were first made in Paris, if, indeed, they were
not invented there." Since about
1990, there has been considerable development in new types of burr which use
plates or boards rather than sticks, or whose central volume is subdivided more
(cf in 6.W.1).
See
S&B, pp. 62‑85.
See
also 6.BJ.
Most
of these have three pieces which are rectangular in cross-section (1 x 3 x 5) with
slots of the same size and some of the pieces have notches from the slot to the
outside. When one piece is pushed, it
slides, revealing its notch. When
placed properly, this allows a second piece to slide off and out.
In
the 1990s, a more elaborate type of three piece burr appeared. These have three 3 x 3 x 5
pieces which intersect in a central
3 x 3 x 3 region. Within this region, some of the unit cubes
are not present, which allows sliding of the pieces. Some versions of the puzzle permit twisting of pieces though this
usually requires a bit of rounding of edges and the actual examples tend to
break, so these are not as acceptable.
Crambrook. 1843.
P. 5, no. 4: Puzzling Cross 3 pieces.
This seems likely to be a three piece burr, but perhaps is in 6.W.3 --
?? It is followed by "Maltese Cross 6 pieces".
Edward Hordern's collection has
examples in ivory from 1850-1900.
Hoffmann. 1893.
Chap. III, no. 35: The cross‑keys or three‑piece puzzle, pp.
106 & 139 = Hoffmann-Hordern, pp. 104-105, with photo. One piece has an extra small notch which
does not appear in other versions where the dimensions are better chosen. I have recently acquired an example which appears
identical to the illustrations but does not have the extra notch - this came
from a Jaques puzzle box, c1900, and Dalgety has several examples of such boxes
with the solution, where the puzzle is named The Cross Keys Puzzle (cf
discussion at the beginning of Section 11).
The photo on p. 105 is an assembled version, with verbal instructions,
by Jaques & Son, 1880-1895 (but Jaques was producing them up to at least
c1910). Hordern Collection, p. 67,
shows Le Noeud Mystérieux, 1880‑1905, with a pictorial solution and this
does not have the extra notch.
Benson. 1904.
The cross keys puzzle, pp. 205‑206.
Pearson. 1907.
Part III, no. 56: The cross‑keys, pp. 56 & 127‑128.
Anon. A puzzle in wood. Hobbies
31 (No. 795) (7 Jan 1911) 345. Three
piece burr with small extra notch as in Hoffmann.
Anon. Woodwork Joints. Evans,
London, (1918), 2nd ed., 1919. [I have
also seen a 4th ed., 1925, which is identical to the 2nd ed., except for
advertising pages at the end.]
A mortising puzzle, pp. 197‑199.
Collins. Book of Puzzles. 1927. Pp. 136-137: The
cross‑keys puzzle.
E. M. Wyatt. Three piece cross. Puzzles in Wood, 1928, op. cit. in 5.H.1, pp. 24‑25.
Arthur Mee's Children's
Encyclopedia 'Wonder Box'. The
Children's Encyclopedia appeared in 1908, but versions continued until the
1950s. This looks like 1930s?? 3-Piece Mortise with thin pieces.
A. S. Filipiak. Burr puzzle. Mathematical Puzzles, 1942, op. cit. in 5.H.1, p. 101.
Dic Sonneveld seems to be the
first to begin designing three piece burrs of the more elaborate style, perhaps
about 1985. Trevor Wood has made
several examples for sale.
Bill Cutler. Email announcement to NOBNET on 27 Jan
1999. He has begun analysing the newer
style of three piece burr, excluding twist moves. His first stage has examined cases where the centre cube of the
central region is occupied and the piece this central cube belongs to has no
symmetry. He finds 202 x 109 assemblies (I'm not sure if this is an exact
figure) and there are 33 level-8 examples (i.e. where it takes 8
moves to remove the first piece);
6674 level-7 examples; 73362 level-6 examples. He
thinks this is about 70% of the total and it is already about six
times the number of cases considered for the six piece burr (see 6.W.2).
Bill Cutler. Christmas letter of 4 Dec 1999. Says he has completed the above analysis and
found 25 x 1010 possibilities, which took 225 days on a
workstation. The most elaborate
examples require 8 moves to get a piece out and there are 80 of these. He used one for his IPP19 puzzle. He has a website with many of his results on
burrs, etc.: www.billcutlerpuzzles.com
.
6.W.2. SIX PIECE BURR = CHINESE CROSS
The
usual form of these has six sticks, 2 x 2 x 6 (or 8), which have various
notches in them. In the 1990s, new
forms were introduced, using plates or boards.
One version makes an open frame shape, something like a 3 x 3 x 3
chessboard. In the other, 1 x 4
x 6 boards are paired side by side and the result looks like a classic
six-piece burr with the end rectangle divided lengthwise rather than
crosswise. See also 6.W.7.
Jurgis Baltrušaitis. Anamorphoses ou magie artificielle des effets merveilleux. Olivier Perrin Éditeur, Paris, 1969. On pp. 110-116 & 184 is a discussion of
a 1698 engraving "L'Académie des Sciences et des Beaux Arts" by
Sébastien Leclerc (or Le Clerc). In the
right foreground is an object looking like a six piece burr. James Dalgety discusses this in his Latest
news on oldest puzzles; Lecture to Second Meeting on the History of
Recreational Mathematics, 1 Jun 1996.
This image also exists in a large painted version (950 x 480 mm) which
is more precise and more legible in many details, so it is supposed that the
engraving was done in conjunction with the painting. Though it was normal for a notable painting to be turned into an
engraving, the opposite sometimes happened and Leclerc was a famous
engraver. The painter is unknown. The divisions between the pairs of pieces of
the 'burr' are pretty clear in the engraving, but two of them are not visible
in the painting. The 'burr' is also not
quite correctly drawn, but all in all, it seems pretty convincing. James Dalgety was the first to discover this
picture and he has a copy of the engraving, but has not been able to locate the
painting, though it was in the Bernard Monnier Collection exhibited at the
Musée des Arts Decoratifs in Paris in 1975/76.
Camille Frémontier. Sébastien Leclerc and the British
Encyclopeaedists. Sphæra [Newsletter of
the Museum of the History of Science, Oxford] 6 (Aut 1997) 6-7. Discusses the Leclerc engraving which was
used as the frontispiece to several encyclopedias, the earliest being Chambers
Cyclopaedia of 1728.
Minguet. 1733.
Pp. 103-105 (1755: 51-52; 1822: 122-124; 1864: 103-104). Pieces diagrammed. One plain key piece.
Catel. Kunst-Cabinet. 1790. Die kleine Teufelsklaue, p. 10 & fig. 16
on plate I. Figure shows it assembled
and fails to draw one of the divisions between pieces. Description says it is 6 pieces, 2 inches
long, from plum wood and costs 3 groschen (worth about an English penny of the
time). (See also pp. 9-10, fig. 20 on
plate I for Die grosse Teufelsklaue -- the 'squirrelcage'.)
Bestelmeier. 1801.
Item 147: Die kleine Teufelsklaue.
(Note -- there is another item 147 on the next plate.) Only shows it assembled. Brief text may be copying part of
Catel. See also the picture for item
1099 which looks like a six‑piece burr included in a set of puzzles. (See also Item 142: Die grosse
Teufelsklaue.)
Edward Hordern's collection has
examples, called The Oak of Old England, from c1840.
Crambrook. 1843.
P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these might be here or in 6.W.4
or 6.W.5 -- ??
Magician's Own Book. 1857.
Prob. 1: The Chinese cross, pp. 266-267 & 291. One plain key piece. Not the same as in Minguét.
Landells. Boy's Own Toy-Maker. 1858.
Pp. 137-139. Identical to
Magician's Own Book.
Book of 500 Puzzles. 1859.
1: The Chinese cross, pp. 80-81 & 105. Identical to Magician's Own Book.
A. F. Bogesen (1792‑1876). In the Danish Technical Museum, Helsingør (=
Elsinore) are a number of wooden puzzles made by him, including a 6 piece burr,
a 12 piece burr, an Imperial Scale? and a complex (trick??) joint.
Illustrated Boy's Own
Treasury. 1860. Practical Puzzles, No. 23: The Chinese
Cross, pp. 399 & 439. Identical to
Magician's Own Book, except one diagram in the solution omits two labels.
Boy's Own Conjuring Book. 1860.
Prob. 1: The Chinese cross, pp. 228 & 254. Identical to Magician's Own Book.
Hoffmann. 1893.
Chap. III, no. 36: The nut (or six‑piece) puzzle, pp. 106 &
139‑140 = Hoffmann-Hordern, pp. 104-106. Different pieces than in Minguét and Magician's Own Book.
Dudeney. Prob. 473 -- Chinese cross. Weekly Dispatch (23 Nov &
7 Dec 1902), both p. 13.
"There is considerable variety in the manner of cutting out the
pieces, and though the puzzle has been given in some of the old books, I have
purposely presented it in a form that has not, I believe, been published."
Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the
"Chinese Cross," a puzzle of undoubted Oriental origin that was
formerly brought from China by travellers as a curiosity, but for a long time
has had a steady sale in this country."
Wehman. New Book of 200 Puzzles. 1908.
The Chinese cross, pp. 40-41. =
Magician's Own Book.
Dudeney. The world's best puzzles. 1908.
Op. cit. in 2. P. 779 shows a
'"Chinese Cross" which ... is of great antiquity.'
Oscar W. Brown. US Patent 1,225,760 -- Puzzle. Applied: 27 Jun 1916; patented: 15 May 1917. 3pp + 1p diagrams. Coffin says this is the earliest US patent, with several others
following soon after.
Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. Eastern joint puzzle, pp. 196‑197:
Two versions using different pieces.
Six‑piece joint puzzle, pp. 199‑200. Another version.
Western Puzzle Works, 1926
Catalogue. No. 86: 6 piece Wood
Block. Several other possible versions
-- see 6.W.7.
E. M. Wyatt. Six‑piece burr. Puzzles in Wood, 1928, op. cit. in 5.H.1,
pp. 27‑28. Describes 17 versions
from 13 types of piece.
A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in
5.H.1, pp. 79‑87. 73 versions
from 38 types of piece.
William H. [Bill] Cutler. The six‑piece burr. JRM 10 (1977‑78) 241‑250. Complete, computer assisted, analysis, with
help from T. H. O'Beirne and A. C. Cross.
Pieces are considered as 'notchable' if they can be made by a sequence
of notches, which are produced by two saw cuts and then chiselling out the
space between them. Otherwise viewed,
notches are what could be produced by a wide cutter or router. There are
25 of these which can occur in
solutions. (In 1994, he states that
there are a total of 59 notchable pieces and diagrams all of
them.) One can also have more general
pieces with 'right-angle notches' which would require four chisel cuts -- e.g.
to cut a single 1 x 1 x
1 piece out of a 2 x 2 x 8
rod. Alternatively, one can glue
cubes into notches. There are 369
which can occur in solutions.
(In 1994, he states that there are
837 pieces which produce 2225
different oriented pieces, and he lists them all.) He only considers solid solutions -- i.e.
ones where there are no internal holes.
He finds and lists the 314 'notchable' solutions. There are
119,979 general solutions.
C. Arthur Cross. The Chinese Cross. Pentangle, Over Wallop, Hants., UK, 1979. Brief description of the solutions in the
general case, as found by Cutler and Cross.
S&B, p. 83, describes holey
burrs.
W. H. [Bill] Cutler. Christmas letter, 1987. Sketches results of his (and other's) search
for holey burrs with notchable pieces.
Bill Cutler. Holey 6‑Piece Burr! Published by the author, Palatine,
Illinois. (1986); with addendum, 1988, 48pp. He is now permitting internal holes. Describes holey burrs with notchable pieces,
particularly those with multiple moves to release the first piece.
Bill Cutler. A Computer Analysis of All 6-Piece
Burrs. Published by the author, ibid.,
1994. 86pp. Sketches complete history of the project. (I have included a few details in the
description of his 1977/78 article, above.)
In 1987, he computed all the notchable holey solutions, using about 2
months of PC AT time, finding
13,354,991 assemblies giving 7.4 million solutions. Two of these were level 10 -- i.e. they
require 10 moves to remove the first piece (or pieces), but the highest level
occurring for a unique solution was 5.
After that he started on the general holey burrs and estimated it would
take 400 years of PC AT time -- running at 8 MHz. After some development, the actual time used was about 62.5 PC AT
years, but a lot of this was done on by Harry L. Nelson during idle time on the
Crays at Lawrence Livermore Laboratories, and faster PCs became available, so
the whole project only took about 2½ years, being completed in Aug 1990 and
finding 35,657,131,235 assemblies.
He hasn't checked if all assemblies come apart fully, but he estimates
there are 5.75 billion solutions. He
estimates the project used 45 times the computing power used in the proof of
the Four Color Theorem and that the project would only take two weeks on the
eight RS6000 workstations he now supervises.
Some 70,000 high-level solutions were specifically saved and can be
obtained on disc from him. The highest
level found was 12 and the highest level for a unique solution was 10. See 6.W.1 for a continuation of this
work. He has a website with many of his
results on burrs, etc.:
www.billcutlerpuzzles.com .
Bill Cutler & Frans de
Vreugd. Information leaflet
accompanying their separate IPP22 puzzles, 2002. In 2001, they did an analysis of six-board burrs, of the type
where the boards are paired side by side.
There are 4096 possible such boards, but only 219 usable boards
occur. They looked at all combinations
of six of these and found 14,563,061,989 assemblies. Of these, the highest level found was 13.
6.W.3. THREE PIECE BURR WITH IDENTICAL PIECES
See
S&B, p. 66.
Crambrook. 1843.
P. 5, no. 4: Puzzling Cross 3 pieces.
This seems likely to be a three piece burr, but perhaps is in 6.W.1 --
?? It is followed by "Maltese Cross 6 pieces".
Wilhelm Segerblom. Trick wood joining. SA (1 Apr 1899) 196.
6.W.4. DIAGONAL SIX PIECE BURR = TRICK STAR
This
version often looks like a stellated rhombic dodecahedron. It has two basic forms, one with a key
piece; the other with all pieces identical,
which assembles as two groups of three.
See
S&B, p. 78.
Crambrook. 1843.
P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these belong here or in 6.W.2
or 6.W.5 -- ??
The Youth's Companion. 1875.
[Mail order catalogue.] Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of
Toys, Games & Dolls 1860··1930; DBI
Books, Northfield, Illinois, 1977?, p. 19.
Star Puzzle. The picture does
not show which form it is. Slocum's
Compendium also shows this.
Samuel P. Chandler. US Patent 393,816 -- Puzzle. Applied: 9 Mar 1888; patented: 23 Apr 1888. 1p + 1p diagrams. Coffin says this is the earliest version, but it is more complex
than usual, with 12 pieces, and has a key piece.
John S. Pinnell. US Patent 774,197 -- Puzzle. Applied: 9 Oct 1902; patented: 8 Nov 1904. 2pp + 2pp diagrams. Coffin notes that this extends the idea
to 102
pieces!
William E. Hoy. US Patent 766,444 -- Puzzle‑Ball. Applied: 16 Oct 1902; patented: 2 Aug 1904. 2pp + 2pp diagrams. Spherical version with a key piece.
George R. Ford. US Patent 779,121 -- Puzzle. Applied: 16 May 1904; patented: 3 Jan 1905. 1p + 1p diagrams. With square rods, all identical.
He shows assembly by inserting a last piece rather than joining two
groups of three.
Anon. Simple wood puzzle.
Hobbies 31 (No. 786) (5 Nov 1910) 127.
With key piece.
E. M. Wyatt. Woodwork puzzles. Industrial Arts Magazine 12 (1923) 326‑327. Version with a key piece and square rods.
Collins. Book of Puzzles. 1927. The bonbon or nut
puzzle, pp. 137-139.
Iffland Frères (Lausanne). Swiss Patent 245,402 --
Zusammensetzspiel. Received:
19 Nov 1945; granted: 15 Nov
1946; published: 1 Jul 1947. 2pp + 1p diagrams. Stellated rhombic dodecahedral version with a key piece. (Coffin says this is the first to use this
shape, although Slocum has a version c1875.)
6.W.5. SIX PIECE BURR WITH IDENTICAL PIECES
One form has six identical pieces and
all move outward or inward together.
Another form with flat notched pieces has one piece with an extra notch
or an extended notch which allows it to fit in last, either by sliding or
twisting, but this is not initially obvious.
This form is sometimes made with equal pieces so that it can only be assembled
by force, perhaps after steaming, and it then makes an unopenable money
box. This might be considered under
11.M.
Edward Hordern's collection has
a version with one piece a little smaller than the rest from c1800.
Crambrook. 1843.
P. 5, no. 5: Maltese Cross 6 [pieces], three sorts. Not clear if these belong here or in 6.W.2
or 6.W.4 -- ??
C. Baudenbecher catalogue,
c1850s. Op. cit. in 6.W.7. This has an example of the six equal flat
pieces making an unopenable(?) money box.
F. Chasemore. Some mechanical puzzles. In:
Hutchison; op. cit. in 5.A; 1891, chap. 70, part 1, pp. 571‑572. Item 5: The puzzle box, p. 572. Six U pieces make a uniformly expanding
cubical box.
Hoffmann. 1893.
Chap. III, no.33: The bonbon nut puzzle, pp. 104 & 138
= Hoffmann‑Hordern, pp. 102-103, with photo. One piece has an extra notch to simplify the
assembly. Photo on p. 103 shows an
example, almost certainly by Jaques & Son, 1860-1895.
Burnett Fallow. How to make a puzzle money-box. The Boy's Own Paper 15 (No. 755)
(1 Jul 1893) 638. Equal flat
notched pieces forced together to make an unopenable box.
Burnett Fallow. How to make a puzzle picture-frame. The Boy's Own Paper 16 (No. 815)
(25 Aug 1894) 749. Each corner has
the same basic forced construction as used in the puzzle money-box.
Benson. 1904.
The bonbon nut puzzle, p. 204.
Bartl. c1920. Several versions
on p. 306.
Western Puzzle Works, 1926
Catalogue. Last page shows 20 Chinese
Wood Block Puzzles, High Grade. Some of
these are of the present type.
Collins. Book of Puzzles. 1927. The bonbon or nut
puzzle, pp. 137-139. As in Hoffmann.
Iona & Robert Opie and Brian
Alderson. Treasures of Childhood. Pavilion (Michael Joseph), London,
1989. P. 158 shows a "cluster
puzzle which Professor Hoffman [sic] names the 'Nut (or Six‑piece) Puzzle',
but which is usually called 'The Maltese Puzzle'."
William Altekruse. US Patent 430,502 -- Block-Puzzle. Applied: 3 Apr 1890; patented: 17 Jun 1890. 1p + 1p diagrams. Described in S&B, p. 72.
The standard version has 12 pieces, but variations discovered by Coffin
have 14, 36 & 38 pieces.
Western Puzzle Works, 1926
Catalogue. No. 112: 12 piece Wood
Block. Possibly Altekruse.
See
also 6.BJ for other 3D dissections. I
have avoided repeating items, so 6.BJ should also be consulted if you are
reading this section.
Catel. Kunst-Cabinet. 1790. Die grosse Teufelsklaue, pp. 9-10 & fig.
20 on plate I. 24 piece 'squirrel
cage'. Cost 16 groschen.
Bestelmeier. 1801.
Item 142: Die grosse Teufelsklaue.
The 'squirrelcage', identical to Catel, with same drawing, but
reversed. Text may be copying some of
Catel.
C. Baudenbecher, toy
manufacturer in Nuremberg. Sample book
or catalogue from c1850s. Baudenbecher
was taken over by J. W. Spear & Sons in 1919 and the catalogue is now in
the Spear's Game Archive, Ware, Hertfordshire.
It comprises folio and double folio sheets with finely painted
illustrations of the firm's products.
One whole folio page shows about 20 types of wooden interlocking puzzles,
including most of the types mentioned elsewhere in this section and in 6.W.5
and 6.BJ. Until I get a picture, I
can't be more specific.
The Youth's Companion. 1875.
[Mail order catalogue.]
Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys,
Games & Dolls 1860··1930; DBI
Books, Northfield, Illinois, 1977?, p. 19.
Shows a 'woodchuck' type puzzle, called White Wood Block Puzzle, from
The Youth's Companion, 1875. I can't
see how many pieces it has: 12 or
18?? Slocum's Compendium also shows
this.
Slocum. Compendium.
Shows: "Mystery", Magic "Champion Puzzle" and
"Puzzle of Puzzles" from Bland's Catalogue, c1890.
The
first looks like a 6 piece burr with circular segments added to make it look
like a ball. So it may be a 6 piece
burr in disguise. See also Hoffmann,
Chap. III, no. 38, pp. 107‑108 & 141‑142
= Hoffmann-Hordern, pp. 106-108 =
Benson, p. 205.
The
second is a six piece puzzle, but the pieces are flattish and it may be of the
type described in 6.W.5.
The
third is complex, with perhaps 18 pieces.
Bartl. c1920. Several versions
on pp. 306-307, including some that are in 6.W.5 and some 'Chinese block
puzzles'.
Western Puzzle Works, 1926
Catalogue. Shows a number of burrs and
similar puzzles.
No.
86: 6 piece Wood Block.
No.
112: 12 piece Wood Block. Possibly
Altekruse.
No.
212: 11 piece Wood Block
The
last page shows 20 Chinese Wood Block Puzzles, High Grade. Some of these are burrs.
Collins. Book of Puzzles. 1927. Other cluster
puzzles, pp. 139-142. Describes and
illustrates: The cluster; The cluster of clusters; The gun cluster; The point cluster; The
flat cluster; The cluster (or secret)
table; The barrel; The Ball;
The football. All of these have
a key piece.
Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escher-puzzle; Pythagoras (Amsterdam)
(1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -- ??)
(1992) 12-13. Two Penrose tribars made
into an impossible 5-piece burr.
6.X. ROTATING RINGS OF POLYHEDRA
Generally,
these have edge to edge joints.
'Jacob's ladder' joints are used by Engel -- see 11.L for other forms of
this joint.
I am told these may appear in
Fedorov (??NYS).
Max Brückner. Vielecke und Vielfläche. Teubner, Leipzig, 1900. Section 162, pp. 215‑216 and Tafel
VIII, fig. 4. Describes rings of 2n
tetrahedra joined edge to edge, called stephanoids of the second order. The figure shows the case n = 5.
Paul Schatz. UK Patent 406,680 -- Improvements in or
relating to Boxes or Containers.
Convention date (Germany): 10 Dec 1931;
application date (in UK): 19 Jul 1932;
accepted: 19 Feb 1934. 6pp + 6pp
diagrams. Six and four piece rings of
prisms which fold into a box.
Paul Schatz. UK Patent 411,125 -- Improvements in
Linkwork comprising Jointed Rods or the like.
Convention Date (Germany): 31 Aug 1931;
application Date (in UK): 31 Aug 1932; accepted: 31 May 1934. 3p + 6pp diagrams. Rotating rings of six tetrahedra and linkwork versions of the
same idea, similar to Flowerday's Hexyflex.
Ralph M. Stalker. US Patent 1,997,022 -- Advertising Medium or
Toy. Applied: 27 Apr 1933; patented: 9 Apr 1935. 3pp + 2pp diagrams. "... a plurality of tetrahedron members
or bodies flexibly connected together."
Shows six tetrahedra in a ring and an unfolded pattern for such
objects. Shows a linear form with 14
tetrahedra of decreasing sizes.
Sidney Melmore. A single‑sided doubly collapsible
tessellation. MG 31 (No. 294) (1947)
106. Forms a Möbius strip of three
triangles and three rhombi, which is basically a flexagon (cf 6.D). He sees it has two distinct forms, but
doesn't see the flexing property!! He
describes how to extend these hexagons into a tessellation which has some
resemblance to other items in this section.
Alexander M. Shemet. US Patent 2,688,820 -- Changeable Display
Amusement Device. Applied: 25 Jul
1950; patented: 14 Sep 1954. 2pp + 2pp diagrams. Basically a rotating ring of six tetrahedra,
but says 'at least six'. Gives an
unfolded version or net for making it and a mechanism for flexing it
continually. Cites Stalker.
Wallace G. Walker invented his
"IsoAxis" ® in 1958 while a student at Cranbrook Academy of Art,
Michigan. This is approximately a ring
of ten tetrahedra. He obtained a US
Patent for it in 1967 -- see below. In
1973(?) he sent an example to Doris Schattschneider who soon realised that the
basic idea was a ring of tetrahedra and that Escher tessellations could be
adapted to it. They developed the idea
into "M. C. Escher Kaleidocycles", published by Ballantine in 1977
and reprinted several times since.
Douglas Engel. Flexahedrons. RMM 11 (Oct 1962) 3‑5.
These have 'Jacob's ladder' hinges, not edge‑to‑edge
hinges. He says he invented these in
Fall, 1961. He formed rings of 4, 6, 7, 8
tetrahedra and used a diagonal joining to make rings of 4 and 6 cubes.
Wallace G. Walker. US Patent 3,302,321 -- Foldable
Structure. Filed: 16 Aug 1963; issued: 7 Feb 1967. 2pp + 6pp diagrams.
Joseph S. Madachy. Mathematics on Vacation. Op. cit. in 5.O, (1966), 1979. Solid Flexagons, pp. 81‑84. Based on Engel, but only gives the ring of 6
tetrahedra.
D. Engel. Flexing rings of regular tetrahedra. Pentagon 26 (Spring 1967) 106‑108. ??NYS -- cited in Schaaf II 89 -- write
Engel.
Paul Bethell. More Mathematical Puzzles. Encyclopædia Britannica International,
London, 1967. The magic ring, pp.
12-13. Gives diagram for a
ten-tetrahedra ring, all tetrahedra being regular.
Jan Slothouber &
William Graatsma. Cubics. Octopus Press, Deventer, Holland, 1970. ??NYS. Presents versions of the flexing cubes and the 'Shinsei
Mystery'. [Jan de Geus has sent a
photocopy of some of this but it does not cover this topic.]
Jan Slothouber. Flexicubes -- reversible cubic shapes. JRM 6 (1973) 39‑46. As above.
Frederick George Flowerday. US Patent 3,916,559 -- Vortex Linkages. Filed: 12 Aug 1974 (23 Aug 1973 in UK); issued: 4 Nov 1975. Abstract + 2pp + 3pp diagrams. Mostly shows his Hexyflex, essentially a six
piece ring of tetrahedra, but with just four edges of each tetrahedron
present. He also shows his Octyflex
which has eight pieces. Text refers to
any even number ³ 6.
Naoki Yoshimoto. Two stars in a cube (= Shinsei
Mystery). Described in Japanese
in: Itsuo Sakane; A Museum of Fun;
Asahi Shimbun, Tokyo, 1977, pp. 208‑210.
Shown and pictured as Exhibit V‑1 with date 1972 in: The Expanding Visual World -- A Museum of
Fun; Exhibition Catalogue, Asahi Shimbun, Tokyo, 1979, pp. 102 & 170‑171. (In Japanese). ??get translated??
Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. Pp. 63-66.
Describes Walkers IsoAxis and rotating rings of six and eight tetrahedra.
The
first few examples illustrate what must be the origin of the idea in more straightforward
situations.
Lucca 1754. c1330.
F. 8r, pp. 31‑32. This
mentions the fact that a circumference increases by 44/7 times the increase
in the radius.
Muscarello. 1478.
Ff.
932-93v, p. 220. A circular garden has
outer circumference 150 and the wall is 3½ thick. What is the inner circumference? Takes
π as 22/7.
F.
95r, p. 222. The internal circumference
of a tower is 20 and its wall is 3 thick. What is the outer circumference? Again takes
π as 22/7.
Pacioli. Summa.
1494. Part II, f. 55r, prob.
33. Florence is 5 miles around the
inside. The wall is 3½
braccia wide and the ditch is
14 braccia wide -- how far is it
around the outside? Several other
similar problems.
William Whiston. Edition of Euclid, 1702. Book 3, Prop. 37, Schol. (3.). ??NYS -- cited by "A Lover" and
Jackson, below.
"A Lover of the
Mathematics." A Mathematical
Miscellany in Four Parts. 2nd ed., S.
Fuller, Dublin, 1735. The First Part
is: An Essay towards the Probable
Solution of the Forty five Surprising PARADOXES, in GORDON's Geography,
so the following must have appeared in Gordon.
Part I, no. 73, p. 56.
"'Tis certainly Matter of Fact, that three certain Travellers went
a Journey, in which, Tho' their Heads travelled full twelve Yards more than
their Feet, yet they all return'd alive, with their Heads on."
Carlile. Collection.
1793. Prob. XXV, p. 17. Two men travel, one upright, the other
standing on his head. Who "sails
farthest"? Basically he compares
the distance travelled by the head and the feet of the first man. He notes that this argument also applies to
a horse working a mill by walking in a circle; the outside of the horse travels
about six times the thickness of the horse further than the inside on each
turn.
Jackson. Rational Amusement. 1821.
Geographical Paradoxes, no. 54, pp. 46 & 115-116. "It is a matter of fact, that three
certain travellers went on a journey, in which their heads travelled full
twelve yards more than their feet; and yet, they all returned alive with their
heads on." Solution says this is
discussed in Whiston's Euclid, Book 3, Prop. 37, Schol. (3.). [This first appeared in 1702.]
K. S. Viwanatha Sastri. Reminiscences of my esteemed tutor. In:
P. K. Srinivasan, ed.; Ramanujan Memorial Volumes: 1: Ramanujan -- Letters and
Reminiscences; 2: Ramanujan -- An
Inspiration; Muthialpet High School,
Number Friends Society, Old Boys' Committee, Madras, 1968. Vol. 1, pp. 89-93. On p. 93, he relates that this was a favourite problem of his
tutor, Srinivasan Ramanujan. Though not
clearly dated, this seems likely to be c1908-1910, but may have been up to
1914. "Suppose we prepare a belt
round the equator of the earth, the belt being
2π feet longer, and if we
put the belt round the earth, how high will it stand? The belt will stand
1 foot high, a substantial
height."
Dudeney. The paradox party. Strand Mag. 38 (No. 228) (Dec 1909) 673‑674 (= AM, p. 139).
Anon. Prob. 58. Hobbies 30 (No. 773) (6 Aug 1910) 405 &
(No. 776) (27 Aug 1910) 448.
Double track circular railway, five miles long. Move all rails outward one foot. How much more material is needed? Solution notes the answer is independent of
the length.
Ludwig Wittgenstein was
fascinated by the problem and used to pose it to students. Most students felt that adding a yard to the
rope would raise it from the earth by a negligible amount -- which it is, in
relation to the size of the earth, but not in relation to the yard. See:
John Lenihan; Science in
Focus; Blackie, 1975, p. 39.
Ernest K. Chapin. Loc. cit. in 5.D.1. 1927.
Prob. 5, p. 87 & Answers p. 7.
A yard is added to a band around the earth. Can you raise it 5 inches?
Answer notes the size of the earth is immaterial.
Collins. Book of Puzzles. 1927. The globetrotter's puzzle,
pp. 68‑69. If you walk around the
equator, how much farther does your head go?
Abraham. 1933.
Prob. 33 -- A ring round the earth, pp. 12 & 24 (9 & 112).
Perelman. FMP.
c1935?? Along the equator, pp.
342 & 349. Same as Collins.
Sullivan. Unusual.
1943.
Prob.
20: A global readjustment. Take a wire
around the earth and insert an extra 40 ft into it -- how high up will it be?
Prob.
23: Getting ahead. If you walk around
the earth, how much further does your head go than your feet?
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. Things are seldom what they seem --
No. 42a, 43, 44, pp. 50-51. 42a
and 43 ask how much the radius increases for a yard gain of circumference. No. 44 asks if we add a yard to a rope
around the earth and then tauten it by pulling outward at one point, how far
will that point be above the earth's surface?
Richard I Hess. Puzzles from Around the World. The author, 1997. (This is a collection of 117 puzzles which he published in
Logigram, the newsletter of Logicon, in 1984-1994, drawn from many
sources. With solutions.) Prob. 28.
Consider a building 125 ft wide and a rubber band stretched around the
earth. If the rubber band has to
stretch an extra 10 cm to fit over the building, how tall is the building? He takes the earth's radius as 20,902,851 ft. He gets three trigonometric equations and uses iteration to
obtain 85.763515... ft.
Erwin Brecher &
Mike Gerrard. Challenging
Science Puzzles. Sterling, 1997. [Reprinted by Goodwill Publishing House, New
Delhi, India, nd [bought in early 2000]].
Pp. 38-39 & 77. The M25 is a
large ring road around London. A man
commutes from the south to the north and finds the distance is the same if goes
by the east or the west, so he normally goes to the east in the morning and to
the west in the evening. Recalling that
the English drive on the left, he realised that his right wheels were on the
outside in both journeys and he worried that they were wear out sooner. So he changed and drove both ways by the
east. But he then worried whether the
wear on the tires was the same since the evening trip was on the outer lanes of
the Motorway.
6.Z. LANGLEY'S ADVENTITIOUS ANGLES
Let
ABC be an isosceles triangle with Ð B
= Ð C = 80o.
Draw BD and
CE, making angles 50o and 60o with the base. Then Ð
CED = 20o.
JRM 15 (1982‑83) 150 cites
Math. Quest. Educ. Times 17 (1910) 75.
??NYS
Peterhouse and Sidney Entrance
Scholarship Examination. Jan 1916. ??NYS.
E. M. Langley. Note 644:
A Problem. MG 11 (No. 160) (Oct
1922) 173.
Thirteen solvers, including
Langley. Solutions to Note 644. MG 11 (No. 164) (May 1923) 321‑323.
Gerrit Bol. Beantwoording van prijsvraag No. 17. Nieuw Archief voor Wiskunde (2) 18 (1936) 14‑66. ??NYS.
Coxeter (CM 3 (1977) 40) and Rigby (below) describe this. The prize question was to completely
determine the concurrent diagonals of regular polygons. The
18‑gon is the key to Langley's problem. However Bol's work was not geometrical.
Birtwistle. Math. Puzzles & Perplexities. 1971.
Find the angle, pp. 86-87. Short
solution using law of sines and other simple trigonometric relations.
Colin Tripp. Adventitious angles. MG 59 (No. 408) (Jun 1975) 98‑106. Studies when Ð CED can be determined and all angles are an
integral number of degrees. Computer
search indicates that there are at most
53 cases.
CM 3 (1977) 12 gives 1939 & 1950 reappearances of the
problem and a 1974 variation.
D. A. Q. [Douglas A.
Quadling]. The adventitious angles
problem: a progress report. MG 61 (No. 415)
(Mar 1977) 55-58. Reports on a number
of contributions resolving the cases which Tripp could not prove. All the work is complicated trigonometry --
no further cases have been demonstrated geometrically.
CM 4 (1978) 52‑53 gives
more references.
D. A. Q. [Douglas A.
Quadling]. Last words on adventitious
angles. MG 62 (No. 421) (Oct 1978)
174-183. Reviews the history, reports
on geometric proofs for all cases and various generalizations.
J[ohn]. F. Rigby. Adventitious quadrangles: a geometrical approach. MG 62 (No. 421) (Oct 1978) 183-191. Gives geometrical proofs for almost all
cases. Cites Bol and a long paper of
his own to appear in Geom. Dedicata (??NYS).
He drops the condition that
ABC be isosceles. His adventitious quadrangles correspond to
Bol's triple intersections of diagonals of a regular n-gon.
MS 27:3 (1994/5) 65 has two straightforward letters on the
problem, which was mentioned in ibid. 27:1 (1994/5) 7. One letter cites 1938 and 1955 appearances. P. 66 gives another solution of the
problem. See next item.
Douglas Quadling. Letter: Langley's adventitious angles. MS 27:3 (1994/5) 65‑66. He was editor of MG when Tripp's article
appeared. He gives some history of the
problem and some life of Langley (d. 1933).
Edward Langley was a teacher at Bedford Modern School and the founding
editor of the MG in 1894-1895. E. T.
Bell was a student of Langley's and contributed an obituary in the MG (Oct
1933) saying that Langley was the finest expositor he ever heard -- ??NYS. Langley also had botanical interests and a
blackberry variety is named for him.
Albrecht Dürer. Underweysung der messung mit dem zirckel
uň [NOTE: ň denotes an
n with an overbar.] richtscheyt,
in Linien ebnen unnd gantzen corporen.
Nürnberg, 1525, revised 1538.
Facsimile of the 1525 edition by Verlag Dr. Alfons Uhl, Nördlingen,
1983. German facsimile with English
translation of the 1525 edition, with notes about the 1538 edition: The
Painter's Manual; trans. by Walter L. Strauss; Abaris Books, NY,
1977. Figures 29‑43 (erroneously
printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v -
N-v-r) show nets and pictures of the regular polyhedra, an approximate sphere
(16 sectors by 8 zones), truncated tetrahedron, truncated cube,
cubo-octahedron, truncated octahedron, rhombi‑cubo-octahedron, snub cube,
great rhombi-cubo-octahedron, truncated cubo‑octahedron (having a pattern
of four triangles replacing each triangle of the cubo‑octahedron -- not
an Archimedean solid) and an elongated hexagonal bipyramid (not even regular
faced). (See 6.AT.3 for more
details.) (Panofsky's biography of
Dürer asserts that Dürer invented the concept of a net -- this is excerpted in
The World of Mathematics I 618‑619.)
In the revised version of 1538, figure 43 is replaced by the
icosi-dodecahedron and great rhombi-cubo-octahedron (figures 43 & 43a,
pp. 414‑419 of The Painter's Manual) to make 9 of
the Archimedean polyhedra.
Albrecht Dürer. Elementorum Geometricorum (?) -- the copy of
this that I saw at the Turner Collection, Keele, has the title page missing,
but Elementorum Geometricorum is the heading of the first text page and appears
to be the book's title. This is a Latin
translation of Unterweysung der Messung ....
Christianus Wechelus, Paris, 1532.
This has the same figures as the 1525 edition, but also has page
numbers. Liber quartus,
fig. 29-43, pp. 145-158 shows the same material as in the 1525
edition.
Cardan. De Rerum Varietate. 1557, ??NYS
= Opera Omnia, vol. III, pp. 246-247.
Liber XIII. Corpora, qua
regularia diei solent, quomodo in plano formentur. Shows nets of the regular solids, except the two halves of the
dodecahedron have been separated to fit into one column of the text.
Barbaro, Daniele. La Practica della Perspectiva. Camillo & Rutilio Borgominieri, Venice,
(1569); facsimile by Arnaldo Forni,
1980, HB. [The facsimile's TP doesn't
have the publication details, but they are given in the colophon. Various catalogues say there are several
versions with dates on the TP and colophon varying independently between 1568
and 1569. A version has both dates
being 1568, so this is presumed to be the first appearance. Another version has an undated title in an
elaborate border and this facsimile must be from that version.] Pp. 45-104 give nets and drawings of the
regular polyhedra and 11 of the 13 Archimedean polyhedra -- he omits the two
snub solids.
E. Welper. Elementa geometrica, in usum geometriae
studiosorum ex variis Authoribus collecta.
J. Reppius, Strassburg, 1620.
??NYS -- cited, with an illustration of the nets of the octahedron,
icosahedron and dodecahedron, in Lange & Springer Katalog 163 -- Mathematik
& Informatik, Oct 1994, item 1350 & illustration on back cover, but the
entry gives Trassburg.
Athanasius Kircher. Ars Magna, Lucis et Umbrae. Rome, 1646.
??NX. Has net of a
rhombi-cuboctahedron.
Pike. Arithmetic. 1788. Pp. 458-459. "As the figures of some of these bodies would give but a
confused idea of them, I have omitted them; but the following figures, cut out
in pasteboard, and the lines cut half through, will fold up into the several
bodies." Gives the regular
polyhedra.
Dudeney. MP.
1926. Prob. 146: The cardboard
box, pp. 58 & 149 (= 536, prob. 316, pp. 109 & 310). All
11 nets of a cube.
Perelman. FMP.
c1935? To develop a cube, pp.
179 & 182‑183. Asserts there
are 10
nets and draws them, but two "can be turned upside down and this
will add two more ...." One shape
is missing. Of the two marked as
reversible, one is symmetric, hence equal to its reverse, but the other isn't.
C. Hope. The nets of the regular star‑faced and
star‑pointed polyhedra. MG 35
(1951) 8‑11. Rather technical.
H. Steinhaus. One Hundred Problems in Elementary
Mathematics. (As: Sto Zadań, PWN -- Polish Scientific
Publishers, Warsaw, 1958.) Pergamon
Press, 1963. With a Foreword by M.
Gardner; Basic Books, NY, 1964. Problem
34: Diagrams of the cube, pp. 20 & 95‑96. (Gives all 11 nets.)
Gardner (pp. 5‑6) refers to Dudeney and suggests the four dimensional
version of the problem should be easy.
M. Gardner. SA (Nov 1966) c= Carnival, pp. 41‑54. Discusses the nets of the cube and the
Answers show all 11 of them.
He asks what shapes these
11 hexominoes will form -- they
cannot form any rectangles. He poses
the four dimensional problem; the
Addendum says he got several answers, no two agreeing.
Charles J. Cooke. Nets of the regular polyhedra. MTg 40 (Aut 1967) 48‑52. Erroneously finds 13 nets of the
octahedron.
Joyce E. Harris. Nets of the regular polyhedra. MTg 41 (Winter 1967) 29. Corrects Cooke's number to 11.
A. Sanders &
D. V. Smith. Nets of the
octahedron and the cube. MTg 42 (Spring 1968) 60‑63. Finds
11 nets for the octahedron and
shows a duality with the cube.
Peter Turney. Unfolding the tesseract. JRM 17 (1984‑85) 1‑16. Finds
261 nets of the 4‑cube. (I don't believe this has ever been confirmed.)
Peter Light &
David Singmaster. The nets of
the regular polyhedra. Presented at New
York Acad. Sci. Graph Theory Day X, 213 Nov 1985. In Notes from New York Graph Theory Day X, 23 Nov 1985;
ed. by J. W. Kennedy & L. V. Quintas; New York Acad. Sci., 1986,
p. 26. Based on Light's BSc
project in 1984-1984 under my supervision.
Shows there are 43,380 nets for the dodecahedron and
icosahedron. I may organize this into a
paper, but several others have since verified the result.
H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938. (= Kalejdoskop Matematyczny. Książnica‑Atlas, Lwów and
Warsaw, 1938, ??NX.) Pp. 74-75
describes the dodecahedron and says to see the model in the pocket at the end,
but makes no special observation of the self-rising property. Described in detail with photographs in OUP,
NY, eds: 1950: pp. 161-164; 1960: pp. 209‑212; 1969 (1983): pp. 196-198.
Donovan A. Johnson. Paper Folding for the Mathematics
Class. NCTM, 1957, p. 29, section 66:
Pop-up dodecahedron.
M. Kac. Hugo Steinhaus -- a reminiscence and a
tribute. AMM 81 (1974) 572‑581. Material is on pp. 580‑581, with
picture on p. 581.
A pop‑up octahedron was
used by Waddington's as an advertising insert in a trade journal at the London
Toy Fair about 1981. Pop-up cubes have
also been used.
There
is now a web page devoted to Life run by Bob Wainwright -- address is:
http://members.aol.com/life1ine/life/lifepage.htm
[sic!].
M. Gardner. Solitaire game of "Life". SA (Oct 1970). On cellular automata, self‑reproduction, the Garden‑of‑Eden
and the game of "Life". SA
(Feb 1971). c= Wheels, chap.
20-22. In the Oct 1970 issue, Conway
offered a $50 prize for a configuration which became infinitely large -- Bill
Gosper found the glider gun a month later.
At G4G2, 1996, Bob Wainwright showed a picture of Gosper's telegram to
Gardner on 4 Nov 1970 giving the coordinates of the glider gun. I wasn't clear if Wainwright has this or
Gardner still has it.
Robert T. Wainwright, ed. (12
Longue Vue Avenue, New Rochelle, NY, 10804, USA). Lifeline (a newsletter on Life), 11 issues, Mar 1971 -- Sep
1973. ??NYR.
John Barry. The game of Life: is it just a game? Sunday Times (London) (13 Jun 1971). ??NYS -- cited by Gardner.
Anon. The game of Life. Time
(21 Jan 1974). ??NYS -- cited by
Gardner.
Carter Bays. The Game of Three‑dimensional
Life. Dept. of Computer Science, Univ.
of South Carolina, Columbia, South Carolina, 29208, USA, 1986. 48pp.
A. K. Dewdney. The game Life acquires some successors in
three dimensions. SA 256:2
(Feb 1987) 8‑13. Describes
Bays' work.
Bays has started a quarterly 3‑D
Life Newsletter, but I have only seen one (or two?) issues. ??get??
Alan Parr. It's Life -- but not as we know it. MiS 21:3 (May 1992) 12-15. Life on a hexagonal lattice.
There is quite a bit of classical
history which I have not yet entered.
Magician's Own Book notes there is a connection between the Dido version
of the problem and Cutting a card so one can pass through it, Section
6.BA. There are several relatively modern
surveys of the subject from a mathematical viewpoint -- I will cite a few of
them.
Virgil. Aeneid.
‑19. Book 1, lines 360‑370. (p. 38 of the Penguin edition, translated by
W. F. Jackson Knight, 1956.) Dido came
to a spot in Tunisia and the local chiefs promised her as much land as she
could enclose in the hide of a bull.
She cut it into a long strip and used it to cut off a peninsula and
founded Carthage. This story was later
adapted to other city foundations. John
Timbs; Curiosities of History; With New Lights; David Bogue, London, 1857,
devotes a section to Artifice of the thong in founding cities, pp. 49-50,
relating that in 1100, Hengist, the first Saxon King of Kent, similarly
purchased a site called Castle of the Thong and gives references to Indian,
Persian and American versions of the story as well as several other English
versions.
Pappus. c290.
Synagoge [Collection]. Book V,
Preface, para. 1‑3, on the sagacity of bees. Greek and English in SIHGM II 588‑593. A different, abridged, English version is in
HGM II 389‑390.
The Friday Night Book (A Jewish Miscellany). Soncino Press, London, 1933. Mathematical Problems in the Talmud:
Arithmetical Problems, no. 2, pp. 135-136.
A Roman Emperor demanded the Jews pay him a tax of as much wheat as
would cover a space 40 x 40 cubits.
Rabbi Huna suggested that they request to pay in two instalments of 20 x 20
and the Emperor granted this.
[The Talmud was compiled in the period -300 to 500. This source says he is one of the few
mathematicians mentioned in the Talmud, but gives no dates and he is not
mentioned in the EB. From the text, the
problem would seem to be sometime in the 1-5 C.]
The 5C Saxon mercenary, Hengist
or Hengest, is said to have requested from Vortigern: "as much land as can
be encircled by a thong". He
"then took the hide of a bull and cut it into a single leather thong. With this thong he marked out a certain
precipitous site, which he had chosen with the greatest possible cunning." This is reported by Geoffrey of Monmouth in
the 12C and this is quoted by the editor in:
The Exeter Book Riddles; 8-10C
(the book was owned by Leofric, first Bishop of Exeter, who mentioned it in his
will of 1072); Translated and edited by
Kevin Crossley-Holland; (As: The Exeter
Riddle Book, Folio Society, 1978, Penguin, 1979); Revised ed., Penguin, 1993; pp. 101-102.
Lucca 1754. c1330.
Ff. 8r‑8v, pp. 31‑33.
Several problems, e.g. a city 1
by 24 has perimeter 50
while a city 8 by 8 has perimeter 32 but is 8/3
as large; stitching two sacks together
gives a sack 4 times as big.
Calandri. Arimethrica. 1491. F. 97v. Joining sacks which hold 9
and 16 yields a sack which holds
49!!
Pacioli. Summa.
1494. Part II, ff. 55r-55v. Several problems, e.g. a cord of length 4
encloses 100 ducats worth, how much does a cord of length 10 enclose? Also stitching bags together.
Buteo. Logistica. 1559. Prob. 86, pp. 298-299. If 9 pieces of wood are bundled up by 5½
feet of cord, how much cord is needed to bundle up 4 pieces? 5 pieces?
Pitiscus. Trigonometria. Revised ed., 1600, p. 223.
??NYS -- described in: Nobuo Miura; The applications of trigonometry in
Pitiscus: a preliminary essay; Historia Scientarum 30 (1986) 63-78. A square of side 4 and triangle of
sides 5, 5, 3 have the same perimeter but different areas. Presumably he was warning people not to be
cheated in this way.
J. Kepler. The Six‑Cornered Snowflake, op. cit.
in 6.AT.3. 1611. Pp. 6‑11 (8‑19). Discusses hexagons and rhombic interfaces,
but only says "the hexagon is the roomiest" (p. 11 (18‑19)).
van Etten. 1624.
Prob. 90 (87). Pp. 136‑138
(214‑218). Compares fields 6 x 6
and 9 x 3. Compares 4 sacks of diameter 1 with 1 sack
of diameter 4. Compares 2 water pipes
of diameter 1 with 1 water pipe of diameter 2.
Ozanam. 1725.
Question
1, 1725: 327. Question 3, 1778:
328; 1803: 325; 1814: 276;
1840: 141. String twice as long
contains four times as much asparagus.
Question
2, 1725: 328. If a cord of length 10
encloses 200, how much does a cord of length 8 enclose?
Question
3, 1725: 328. Sack 5 high by 4 across
versus 4 sacks 5 high by 1 across.
c= Q. 2, 1778: 328;
1803: 324; 1814: 276; 1840: 140-141, which has sack 4 high by 6
around versus two sacks 4 high by 3 around.
Question
4, 1725: 328‑329. How much water
does a pipe of twice the diameter deliver?
Les Amusemens. 1749.
Prob.
211, p. 376. String twice as long
contains four times as much asparagus.
Prob.
212, p. 377. Determine length of string
which contains twice as much asparagus.
Prob.
223-226, pp. 386-389. Various problems
involving changing shape with the same perimeter. Notes the area can be infinitely small.
Ozanam‑Montucla. 1778.
Question
1, 1778: 327; 1803: 323-324; 1814: 275-276; 1840: 140. Square versus
oblong field of the same circumference.
Prob.
35, 1778: 329-333; 1803: 326-330; 1814: 277-280; 1840: 141-143. Les
alvéoles des abeilles (On the form in which bees construct their combs).
Jackson. Rational Amusement. 1821.
Geometrical Puzzles.
No.
30, pp. 30 & 90. Square field
versus oblong (rectangular?) field of the same perimeter.
No.
31, pp. 30 & 90-91. String twice as
long contains four times as much asparagus.
Magician's Own Book (UK
version). 1871. To cut a card for one to jump through, p.
124, says: "The adventurer of old,
who, inducing the aborigines to give him as much land as a bull's hide would
cover, and made it into one strip by which acres were enclosed, had probably
played at this game in his youth."
See 6.BA.
M. Zacharias. Elementargeometrie und elementare
nicht-Euklidische Geometrie in synthetischer Behandlung. Encyklopädie der Mathematischen
Wissenschaften. Band III, Teil 1,
2te Hälfte. Teubner, Leipzig,
1914-1931. Abt. 28: Maxima und
Minima. Die isoperimetrische
Aufgabe. Pp. 1118-1128. General survey, from Zenodorus (-1C) and
Pappus onward.
6.AD.1. LARGEST PARCEL ONE CAN POST
New section. I have just added the problem of packing a fishing rod as the
diagonal of a box. Are there older
examples?
Richard A. Proctor. Greatest content with parcels' post. Knowledge 3 (3 Aug 1883) 76. Height + girth £ 6 ft. States that a
cylinder is well known to be the best solution. Either for a cylinder or a box, the optimum has height = 2,
girth = 4, with optimum
volumes 2 and 8/π = 2.54... ft3.
R. F. Davis. Letter:
Girth and the parcel post.
Knowledge 3 (17 Aug 1883) 109-110, item 897. Independent discussion of the problem, noting that length
£ 3½ ft is specified, though this doesn't affect the
maximum volume problem.
H. F. Letter: Parcel post
problem. Knowledge 3 (24 Aug 1883) 126,
item 905. Suppose 'length' means
"the maximum distance in a straight line between any two points on its
surface". By this he means the diameter
of the solid. Then the optimum shape is
the intersection of a right circular cylinder with a sphere, the axis of the
cylinder passing through the centre of the sphere, and this has the 'length'
being the diameter of the sphere and the maximum volume is then 2⅓ ft3.
Algernon Bray. Letter:
Greatest content of a parcel which can be sent by post. Knowledge 3 (7 Sep 1883) 159, item 923. Says the problem is easily solved without
calculus. However, for the box, he says
"it is plain that the bulk of half the parcel will be greatest when [its]
dimensions are equal".
Pearson. 1907.
Part II, no. 20: Parcel post limitations, pp. 118 & 195. Length
£ 3½ ft; length + girth £ 6 ft. Solution is a
cylinder.
M. Adams. Puzzle Book. 1939. Prob. B.86: Packing
a parcel, pp. 79 & 107. Same as
Pearson, but first asks for the largest box, then the largest parcel.
Philip Kaplan. More Posers. (Harper & Row, 1964);
Macfadden-Bartell Books, 1965.
Prob. 18, pp. 27 & 89.
Ship a rifle about 1½ yards long when the post office does not
permit any dimension to be more than
1 yard.
T. J. Fletcher. Doing without calculus. MG 55 (No. 391) (Feb 1971) 4‑17. Example 5, pp. 8‑9. He says only that length + girth £ 6 ft.
However, the optimal box has length 2, so the maximal length restriction
is not critical.
I have looked at the current
parcel post regulations and they say
length £ 1.5m and
length + girth £ 3m, for which the largest box is
1 x ½ x ½, with volume 1/4 m3. The largest cylinder has length
1 and radius 1/π
with volume 1/π m3.
I have also considered the
simple question of a person posting a fishing rod longer than the maximal
length by putting it diagonally in a box.
The longest rod occurs at a boundary maximum, at 3/2 x 3/4 x 0 or 3/2 x 0 x 3/4, so one can post a rod of length 3Ö5/4 = 1.677...
m, which is about 12% longer than
1.5m. In this problem, the use
of a cylinder actually does worse!
6.AE. 6" HOLE THROUGH SPHERE LEAVES CONSTANT VOLUME
Hamnet Holditch. Geometrical theorem. Quarterly J. of Pure and Applied Math. 2
(1858) ??NYS, described by Broman. If a
chord of a closed curve, of constant length
a+b, be divided into two parts
of lengths a, b respectively, the difference between the
areas of the closed curve, and of the locus of the dividing point as the chord
moves around the curve, will be
πab. When the closed curve
is a circle and a = b, then this is the two dimensional version
given by Jones, below. A letter from
Broman says he has found Holditch's theorem cited in 1888, 1906, 1975 and 1976.
Richard Guy (letter of 27 Feb
1985) recalls this problem from his schooldays, which would be late 1920s-early
1930s, and thought it should occur in calculus texts of that time, but could
not find it in Lamb or Caunt.
Samuel I. Jones. Mathematical Nuts. 1932. P. 86. ??NYS.
Cited by Gardner, (SA, Nov 1957) = 1st Book, chap. 12, prob.
7. Gardner says Jones, p. 93, also
gives the two dimensional version: If
the longest line that can be drawn in an annulus is 6" long, what is the
area of the annulus?
L. Lines. Solid Geometry. Macmillan, London, 1935;
Dover, 1965. P. 101, Example
8W3: "A napkin ring is in the form
of a sphere pierced by a cylindrical hole.
Prove that its volume is the same as that of a sphere with diameter
equal to the length of the hole."
Solution is given, but there is no indication that it is new or recent.
L. A. Graham. Ingenious Mathematical Problems and
Methods. Dover, 1959. Prob. 34: Hole in a sphere, pp. 23 & 145‑147. [The material in this book appeared in
Graham's company magazine from about 1940, but no dates are provided in the
book. (??can date be found out.)]
M. H. Greenblatt. Mathematical Entertainments, op. cit. in
6.U.2, 1965. Volume of a modified
bowling ball, pp. 104‑105.
C. W. Trigg. Op. cit. in 5.Q. 1967. Quickie 217: Hole
in sphere, pp. 59 & 178‑179.
Gives an argument based on surface tension to see that the ring surface
remains spherical as the hole changes radius.
Problem has a 10" hole.
Andrew Jarvis. Note 3235:
A boring problem. MG 53 (No.
385) (Oct 1969) 298‑299. He calls
it "a standard problem" and says it is usually solved with a triple
integral (??!!). He gives the standard
proof using Cavalieri's principle.
Birtwistle. Math. Puzzles & Perplexities. 1971.
Tangential
chord, pp. 71-73. 10" chord in an annulus. What is the area of the annulus? Does traditionally and then by letting inner
radius be zero.
The
hole in the sphere, pp. 87-88 & 177-178.
Bore a hole through a sphere so the remaining piece has half the volume
of the sphere. The radius of the hole
is approx. .61 of the radius of the sphere.
Another
hole, pp. 89, 178 & 192.
6" hole cut out of
sphere. What is the volume of the
remainder? Refers to the tangential
chord problem.
Arne Broman. Holditch's theorem: An introductory
problem. Lecture at ICM, Helsinki,
Aug 1978. Broman then sent out
copies of his lecture notes and a supplementary letter on 30 Aug 1978. He discusses Holditch's proof (see above)
and more careful modern versions of it.
His letter gives some other citations.
6.AF. WHAT COLOUR WAS THE BEAR?
A
hunter goes 100 mi south, 100 mi east and 100 mi north and finds himself where
he started. He then shoots a bear --
what colour was the bear?
Square
versions: Perelman; Klamkin, Breault & Schwarz; Kakinuma, Barwell & Collins; Singmaster.
I
include other polar problems here. See
also 10.K for related geographical problems.
"A Lover of the
Mathematics." A Mathematical
Miscellany in Four Parts. 2nd ed., S.
Fuller, Dublin, 1735. The First Part
is: An Essay towards the Probable
Solution of the Forty five Surprising PARADOXES, in GORDON's Geography,
so the following must have appeared in Gordon.
Part I, no. 10, p. 9.
"There is a particular Place of the Earth where the Winds (tho'
frequently veering round the Compas) do always blow from the North Point."
Philip Breslaw (attrib.). Breslaw's Last Legacy; or the Magical
Companion: containing all that is Curious, Pleasing, Entertaining and Comical;
selected From the most celebrated Masters of Deception: As well with Slight of
Hand, As with Mathematical Inventions.
Wherein is displayed The Mode and Manner of deceiving the Eye; as
practised by those celebrated Masters of Mirthful Deceptions. Including the various Exhibitions of those
wonderful Artists, Breslaw, Sieur, Comus, Jonas, &c. Also the Interpretation of Dreams,
Signification of Moles, Palmestry, &c.
The whole forming A Book of real Knowledge in the Art of Conjuration. (T. Moore, London, 1784, 120pp.) With an accurate Description of the Method
how to make The Air Balloon, and inject the Inflammable Air. (2nd ed., T. Moore, London, 1784,
132pp; 5th ed., W. Lane, London, 1791,
132pp.) A New Edition, with great
Additions and Improvements. (W. Lane,
London, 1795, 144pp.) Facsimile from
the copy in the Byron Walker Collection, with added Introduction, etc., Stevens
Magic Emporium, Wichita, Kansas, 1997.
[This was first published in 1784, after Breslaw's death, so it is
unlikely that he had anything to do with the book. There were versions in
1784, 1791, 1792, 1793, 1794, 1795, 1800, 1806, c1809, c1810, 1811,
1824. Hall, BCB 39-43, 46-51. Toole Stott 120-131, 966‑967. Heyl 35-41.
This book went through many variations of subtitle and contents -- the
above is the largest version.]. I will
cite the date as 1784?.
Geographical
Paradoxes.
Paradox
I, p. 35. Where is it noon every half
hour? Answer: At the North Pole in
Summer, when the sun is due south all day long, so it is noon every moment!
Paradox
II, p. 36. Where can the sun and the
full moon rise at the same time in the same direction? Answer: "Under the North Pole, the sun
and the full moon, both decreasing in south declination, may rise in the equinoxial
points at the same time; and under the North Pole, there is no other point of
compass but south." I think this
means at the North Pole at the equinox.
Carlile. Collection.
1793. Prob. CXVI, p. 69. Where does the wind always blow from the north?
Jackson. Rational Amusement. 1821.
Geographical Paradoxes.
No. 7,
pp. 36 & 103. Where do all winds
blow from the north?
No. 8,
pp. 36 & 110. Two places 100
miles apart, and the travelling directions are to go 50
miles north and 50 miles south.
Mr. X [cf 4.A.1]. His Pages.
The Royal Magazine 10:3 (Jul 1903) 246-247. A safe catch. Airship
starts at the North Pole, goes south for seven days, then west for seven
days. Which way must it go to get back
to its starting point? No solution given.
Pearson. 1907.
Part
II, no. 21: By the compass, pp. 18 & 190.
Start at North Pole and go
20 miles southwest. What direction gets back to the Pole the
quickest? Answer notes that it is hard
to go southwest from the Pole!
Part
II, no. 15: Ask "Where's the north?" -- Pope, pp. 117 & 194. Start
1200 miles from the North Pole
and go 20 mph due north by the compass.
How long will it take to get to the Pole? Answer is that you never get there -- you get to the North Magnetic
Pole.
Ackermann. 1925.
P. 116. Man at North Pole
goes 20 miles south and 30 miles west.
How far, and in what direction, is he from the Pole?
Richard Guy (letter of 27 Feb
1985) recalls this problem (I think he is referring to the 'What colour was the
bear' version) from his schooldays in the 1920s.
H. Phillips. Week‑End. 1932. Prob. 8, pp. 12
& 188. = his Playtime Omnibus,
1933, prob. 10: Popoff, pp. 54 & 237.
House with four sides facing south.
H. Phillips. The Playtime Omnibus. Faber & Faber, London, 1933. Section XVI, prob. 11: Polar conundrum, pp.
51 & 234. Start at the North Pole,
go 40
miles South, then 30 miles West.
How far are you from the Pole.
Answer: "Forty miles. (NOT thirty, as one is tempted to
suggest.)" Thirty appears to be a
slip for fifty??
Perelman. FFF.
1934. 1957: prob. 6, pp. 14-15
& 19-20: A dirigible's flight;
1979: prob. 7, pp. 18-19 & 25-27: A helicopter's flight. MCBF: prob. 7, pp. 18-19 & 25-26: A
helicopter's flight.
Dirigible/helicopter starts at Leningrad and goes 500km
N, 500km E, 500km
S, 500km W.
Where does it land? Cf Klamkin
et seq., below.
Phillips. Brush.
1936. Prob. A.1: A stroll at the
pole, pp. 1 & 73. Eskimo living at
North Pole goes 3 mi south and 4 mi east.
How far is he from home?
Haldeman-Julius. 1937.
No. 51: North Pole problem, pp. 8 & 23. Airplane starts at North Pole, goes 30 miles south, then 40 miles
west. How far is he from the Pole?
J. R. Evans. The Junior Week‑End Book. Gollancz, London, 1939. Prob. 9, pp. 262 & 268. House with four sides facing south.
Leopold. At Ease!
1943. A helluva question!, pp.
10 & 196. Hunter goes 10
mi south, 10 mi west, shoots a bear and drags it 10
mi back to his starting point.
What colour was the bear? Says
the only geographic answer is the North Pole.
E. P. Northrop. Riddles in Mathematics. 1944.
1944: 5-6; 1945: 5-6; 1961: 15‑16. He starts with the house which faces south on all sides. Then he has a hunter that sees a bear 100
yards east. The hunter runs 100 yards
north and shoots south at the bear -- what colour .... He then gives the three‑sided walk
version, but doesn't specify the solution.
E. J. Moulton. A speed test question; a problem in
geography. AMM 51 (1944) 216 &
220. Discusses all solutions of the
three-sided walk problem.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. No. 50: A fine outlook, pp. 54-55. House facing south on all sides used by an
artist painting bears!
Leeming. 1946.
Chap. 3, prob. 32: What color was the bear?, pp. 33 & 160. Man walks
10 miles south, then 10
miles west, where he shoots a bear.
He drags it 10 miles north to his base. What color .... He gives only one solution.
Darwin A. Hindman. Handbook of Indoor Games &
Contests. (Prentice‑Hall,
1955); Nicholas Kaye, London,
1957. Chap. 16, prob. 4: The bear
hunter, pp. 256 & 261. Hunter
surprises bear. Hunter runs 200
yards north, bear runs 200 yards east, hunter fires south at bear. What colour ....
Murray S. Klamkin,
proposer; D. A. Breault & Benjamin
L. Schwarz, solvers. Problem 369. MM 32 (1958/59) 220 &
33 (1959/60) 110 & 226‑228. Explorer goes 100 miles north, then east, then south, then
west, and is back at his starting point.
Breault gives only the obvious solution. Schwartz gives all solutions, but not explicitly. Cf Perelman, 1934.
Benjamin L. Schwartz. What color was the bear?. MM 34 (1960) 1-4. ??NYS -- described by Gardner, SA (May 1966) = Carnival, chap.
17. Considers the problem where the
hunter looks south and sees a bear
100 yards away. The bear goes 100 yards east and the
hunter shoots it by aiming due south.
This gives two extra types of solution.
Ripley's Puzzles and Games. 1966.
Pp. 18, item 5. 50 mi N, 1000 mi W,
10 mi S to return to your
starting point. Answer only gives the
South Pole, ignoring the infinitely many cases near the North Pole. Looking at this made me realise that when
the sideways distance is larger than the circumference of the parallel at that
distance from the pole, then there are other solutions that start near the
pole. Here there are three solutions
where one starts at distances 109.2,
29.6 or 3.05 miles from the South Pole,
circling it 1, 2 or 3 times.
Yasuo Kakinuma, proposer; Brian Barwell and Craig H. Collins,
solvers. Problem 1212 -- Variation of
the polar bear problem. JRM 15:3 (1982‑83)
222 & 16:3 (1983-84) 226‑228.
Square problem going one mile south, east, north, west. Barwell gets the explicit quadratic
equation, but then approximates its solutions.
Collins assumes the earth is flat near the pole.
David Singmaster. Bear hunting problems. Submitted to MM, 1986. Finds explicit solutions for the general
version of Perelman/Klamkin's problem.
[In fact, I was ignorant of (or had long forgotten) the above when I remembered
and solved the problem. My thanks to an
editor (Paul Bateman ??check) for referring me to Klamkin. The Kakinuma et al then turned up also.] Analysis of the solutions leads to some
variations, including the following.
David Singmaster. Home is the hunter. Man heads north, goes ten miles, has lunch,
heads north, goes ten miles and finds himself where he started.
Used
as: Explorer's problem by Keith Devlin
in his Micromaths Column; The Guardian (18 Jun 1987) 16 &
(2 Jul 1987) 16.
Used
by me as one of: Spring term puzzles;
South Bank Polytechnic Computer Services Department Newsletter (Spring 1989)
unpaged [p. 15].
Used
by Will Shortz in his National Public Radio program 6? Jan 1991.
Used
as: A walk on the wild side, Games 15:2
(No. 104) (Aug 1991) 57 & 40.
Used
as: The hunting game, Focus 3 (Feb
1993) 77 & 98.
Used
in my Puzzle Box column, G&P, No. 11 (Feb 1995) 19 &
No. 12 (Mar 1995) 41.
Bob Stanton. The explorers. Games Magazine 17:1 (No. 113) (Feb 1993) 61 & 43. Two explorers set out and go 500
miles in each direction. Madge
goes N, W, S, E, while Ellen goes E, S, W, N.
At the end, they meet at the same point. However, this is not at their starting point. How come?
and how far are they from their starting point, and in what
direction? They are not near either
pole.
Yuri B. Chernyak & Robert S.
Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 11, prob. 9: What color was that bear?
(A lesson in non-Euclidean geometry), pp. 97 & 185-191. Camper walks south 2 km, then west 5 km,
then north 2 km; how far is he from his starting point? Solution analyses this and related problems,
finding that the distance x satisfies
0 £ x £ 7.183, noting that there are many minimal cases
near the south pole and if one is between them, one gets a local maximum, so
one has to determine one's position very carefully.
David Singmaster. Symmetry saves the solution. IN: Alfred S. Posamentier & Wolfgang
Schulz, eds.; The Art of Problem Solving: A Resource for the Mathematics
Teacher; Corwin Press, NY, 1996, pp. 273-286.
Sketches the explicit solution to Klamkin's problem as an example of the
use of symmetric variables to obtain a solution.
Anonymous. Brainteaser B163 -- Shady matters. Quantum 6:3 (Jan/Feb 1996) 15 & 48. Is there anywhere on earth where one's
shadow has the same length all day long?
There are several versions of
this. The simplest is moving a ladder
or board around a corner -- here the problem is two-dimensional and the ladder
is thin enough to be considered as a line.
There are slight variations -- the corner can be at a T or +
junction; the widths of the
corridors may differ; the angle may not
be a right angle; etc. If the object being moved is thicker -- e.g.
a table -- then the problem gets harder.
If one can use the third dimension, it gets even harder.
H. E. Licks. Op. cit. in 5.A, 1917. Art. 110, p. 89. Stick going into a circular shaft in the ceiling. Gets
[h2/3 + d2/3)]3/2 for maximum length, where h is
the height of the room and d is the diameter of the shaft. "A simple way to solve a problem which
has proved a stumbling block to many."
Abraham. 1933.
Prob. 82 -- Another ladder, pp. 37 & 45 (23 & 117). Ladder to go from one street to another, of
different widths.
E. H. Johnson, proposer; W. B. Carver, solver. Problem E436. AMM 47 (1940) 569 & 48 (1941) 271‑273. Table going through a doorway. Obtains
6th order equation.
J. S. Madachy. Turning corners. RMM 5 (Oct 1961) 37, 6
(Dec 1961) 61 & 8 (Apr 1962) 56. In 5, he asks for the greatest length of board which can be moved
around a corner, assuming both corridors have the same width, that the board is
thick and that vertical movement is allowed.
In 6, he gives a numerical answer for his original values and asserts
the maximal length for planar movement, with corridors of width w
and plank of thickness t, is
2 (wÖ2 ‑ t). In vol. 8, he says no two solutions have
been the same.
L. Moser, proposer; M. Goldberg and J. Sebastian, solvers. Problem 66‑11 -- Moving furniture
through a hallway. SIAM Review 8 (1966)
381‑382 & 11 (1969) 75‑78 &
12 (1970) 582‑586.
"What is the largest area region which can be moved through a
"hallway" of width one (see Fig. 1)?" The figure shows that he wants to move around a rectangular
corner joining two hallways of width one.
Sebastian (1970) studies the problem for moving an arc.
J. M. Hammersley. On the enfeeblement of mathematical skills
.... Bull. Inst. Math. Appl. 4 (1968)
66‑85. Appendix IV -- Problems,
pp. 83‑85, prob. 8, p. 84.
Two corridors of width 1 at a corner. Show the largest object one can move around it has area < 2 Ö2 and that there is an object of
area ³ π/2 + 2/π
= 2.2074.
Partial
solution by T. A. Westwell, ibid. 5 (1969) 80, with editorial comment thereon
on pp. 80‑81.
T. J. Fletcher. Easy ways of going round the bend. MG 57 (No. 399) (Feb 1973) 16‑22. Gives five methods for the ladder problem
with corridors of different widths.
Neal R. Wagner. The sofa problem. AMM 83 (1976) 188‑189.
"What is the region of largest area which can be moved around a
right‑angled corner in a corridor of width one?" Survey.
R. K. Guy. Monthly research problems, 1969‑77. AMM 84 (1977) 807‑815. P. 811 reports improvements on the sofa
problem.
J. S. Madachy &
R. R. Rowe. Problem 242 --
Turning table. JRM 9 (1976‑77)
219‑221.
G. P. Henderson, proposer; M. Goldberg, solver; M. S. Klamkin, commentator. Problem 427. CM 5 (1979) 77 & 6 (1979) 31‑32 & 49‑50. Easily finds maximal area of a rectangle
going around a corner.
Research news: Conway's sofa problem. Mathematics Review 1:4 (Mar 1991) 5-8 &
32. Reports on Joseph Gerver's almost
complete resolution of the problem in 1990.
Says Conway asked the problem in the 1960s and that L. Moser is the
first to publish it. Says a group at a
convexity conference in Copenhagen improved Hammersley's results to 2.2164.
Gerver's analysis gives an object made up of 18 segments with area 2.2195.
The analysis depends on some unproven general assumptions which seem
reasonable and is certainly the unique optimum solution given those
assumptions.
A. A. Huntington. More on ladders. M500 145 (Jul 1995) 2-5.
Does usual problem, getting a quartic.
The finds the shortest ladder.
[This turns out to be the same as the longest ladder one can get around
a corner from corridors of widths
w and h, so 6.AG is related to
6.L.]
A goat is grazing in a circular field
and is tethered to a post on the edge.
He can reach half of the field.
How long is the rope? There are
numerous variations obtained by modifying the shape of the field or having
buildings within it. In recent years,
there has been study of the form where the goat is tethered to a point on a
circular silo in a large field -- how much area can he graze?
Upnorensis, proposer; Mr. Heath,
solver. Ladies Diary, 1748-49 = T.
Leybourn, II: 6-7, quest. 302. [I have
a reference to p. 41 of the Ladies' Diary.]
Circular pond enclosed by a circular railing of circumference 160
yards. Horse is tethered to a post of
the railing by a rope 160 yards long.
How much area can he graze?
Dudeney. Problem 67: Two rural puzzles -- No. 67: One
acre and a cow. Tit‑Bits 33 (5
Feb & 5 Mar 1898) 355
& 432. Circular field opening onto a small
rectangular paddock with cow tethered to the gate post so that she can graze
over one acre. By skilful choice of
sizes, he avoids the usual transcendental equation.
Arc. [R. A. Archibald]. Involutes of a circle and a pasturage
problem. AMM 28 (1921) 328‑329. Cites Ladies Diary and it appears that it
deals with a horse outside a circle.
J. Pedoe. Note 1477:
An old problem. MG 24 (No. 261)
(Oct 1940) 286-287. Finds the relevant
area by integrating in polar coordinates centred on the post.
A. J. Booth. Note 1561:
On Note 1477. MG 25 (No. 267)
(Dec 1941) 309‑310. Goat tethered
to a point on the perimeter of a circle which can graze over ½, ⅓, ¼ of the area.
Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968.
No. 8:
"Don't fence me in", pp. 87.
Equilateral triangular field of area 120. Three goats tethered to the corners with ropes of length equal to
the altitude. Consider an area
where n goats graze as contributing
1/n to each goat. What area does each goat graze over?
No.
53: Around the silo, pp. 71 & 112-113.
Goat tethered to the outside of a silo of diameter 20 by a rope of
length 10π, i.e. he can just get to the other side of
the silo. How big an area can he
graze? The curve is a semicircle
together with two involutes of a circle, so the solution uses some calculus.
Marshall Fraser. A tale of two goats. MM 55 (1982) 221‑227. Gives examples back to 1894.
Marshall Fraser. Letter:
More, old goats. MM 56 (1983)
123. Cites Arc[hibald].
Bull, 1998, below, says this
problem has been discussed by the Internet newsgroup sci.math some years
previously.
Michael E. Hoffman. The bull and the silo: An application of
curvature. AMM 105:1 (Jan 1998) ??NYS
-- cited by Bull. Bull is tethered by a
rope of length L to a circular silo of radius R.
If L £
πR, then the grazeable area
is L3/3R + πL2/2. This paper considers the problem for general
shapes.
John Bull. The bull and the silo. M500 163 (Aug 1998) 1-3. Improves Hoffman's solution for the circular
silo by avoiding polar coordinates and using a more appropriate variable,
namely the angle between the taut rope and the axis of symmetry.
Keith Drever. Solution 186.5 -- Horse. M550 188 (Oct 2002) 12. A horse is tethered to a point on the
perimeter of a circular field of radius 1.
He can graze over all but
1/π of the area. How long is the rope? This turns out to make the problem almost
trivial -- the rope is Ö2 long and the angle subtended at the tether
is π/2.
S&B,
pp. 146‑147, show several types.
These
are often made in two contrasting woods and appear to be physically
impossible. They will come apart if one
moves them in the right direction. A
few have extra complications. The
simplest version is a square cylinder with dovetail joints on each face --
called common square version below.
There are also cases where one thinks it should come apart, but the wood
has been bent or forced and no longer comes apart -- see also 6.W.5.
See Bogesen in 6.W.2 for a
possible early example.
Johannes Cornelus Wilhelmus
Pauwels. UK Patent 15,307 -- Improved
Means of Joining or Fastening Pieces of Wood or other Material together,
Applicable also as a Toy. Applied: 9
Nov 1887; complete specification: 9 Aug
1888; accepted: 26 Oct 1888. 2pp + 1p diagrams. It says Pauwels is a civil engineer of The Hague. Common square version.
Tom Tit, vol. 2. 1892.
Assemblage paradoxal, pp. 231-232.
= K, no. 155: The paradoxical coupling, pp. 353‑354. Common square version with instructions for
making it by cutting the corners off a larger square.
Emery Leverett Williams. The double dovetail and blind mortise. SA (25 Apr 1896) 267. The first is a trick T‑joint.
T. Moore. A puzzle joint and how to make it. The Woodworker 1:8 (May 1902) 172. S&B, p. 147, say this is the earliest
reference to the common square version -- but see Pauwels, above. "... the foregoing joint will doubtless
be well-known to our professional readers.
There are probably many amateur woodworkers to whom it will be a
novelty."
Hasluck, Paul N. The Handyman's Book. Cassell, 1903; facsimile by Senate (Tiger Books), Twickenham, London, 1998. Pp. 220‑223 shows various joints. Dovetail halved joint with two bevels, p.
222 & figs. 703-705 of pp. 221-222.
"... of but little practical value, but interesting as a puzzle
joint."
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Shows the common
square version "given to me some ten years ago, but I cannot say who first
invented it." He previously
published it in a newspaper. ??look in
Weekly Dispatch.
Samuel Hicks. Kinks for Handy Men: The dovetail
puzzle. Hobbies 31 (No. 790) (3 Dec
1910) 248-249. Usual square dovetail,
but he suggests to glue it together!
Dudeney. AM. 1917. Prob. 424: The
dovetailed block, pp. 145 & 249.
Shows the common square version -- "... given to me some years ago,
but I cannot say who first invented it."
He previously published it in a newspaper. ??as above
Anon. Woodwork Joints, 1918, op. cit. in 6.W.1. A curious dovetail joint, pp. 193, 195. Common square version. Dovetail puzzle joint, pp. 194‑195. A singly mortised T‑joint, with an
unmortised second piece.
E. M. Wyatt. Woodwork puzzles. Industrial‑Arts Magazine 12 (1923) 326‑327. Doubly dovetailed tongue and mortise T‑joint
called 'The double (?) dovetail'.
Sherman M. Turrill. A double dovetail joint. Industrial‑Arts Magazine 13 (1924) 282‑283. A double dovetail right angle joint, but it
leaves sloping gaps on the inside which are filled with blocks.
Collins. Book of Puzzles. 1927. Pp. 134‑135:
The dovetail puzzle. Common square
version.
E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in 5.H.1.
The
double (?) dovetail, pp. 44‑45.
Doubly dovetailed tongue and mortise T‑joint.
The
"impossible" dovetail joint, p. 46.
Common square version.
Double‑lock
dovetail joint, pp. 47‑49. Less
acceptable tricks for a corner joint.
Two‑way
fanned half‑lap joint, pp. 49‑50.
Corner joint.
A. B. Cutler. Industrial Arts and Vocational Education
(Jan 1930). ??NYS. Wyatt, below, cites this for a triple
dovetail, but I could not not find it in vols. 1‑40.
R. M. Abraham. Prob. 225 -- Dovetail Puzzle. Winter Nights Entertainments. Constable, London, 1932, p. 131. (= Easy‑to‑do Entertainments and
Diversions with coins, cards, string, paper and matches; Dover, 1961, p.
225.) Common square version.
Abraham. 1933.
Prob. 304 -- Hexagon dovetail;
Prob. 306 -- The triangular dovetail, pp. 142‑143 (100 &
102).
Bernard E. Jones, ed. The Practical Woodworker. Waverley Book Co., London, nd [1940s?]. Vol. 1: Lap and secret dovetail joints, pp.
281‑287. This covers various
secret joints -- i.e. ones with concealed laps or dovetails. Pp. 286-287 has a subsection: Puzzle
dovetail joints. Common square version
is shown as fig. 28. A pentagonal
analogue is shown as fig. 29, but it uses splitting and regluing to produce a
result which cannot be taken apart.
E. M. Wyatt. Wonders in Wood. Bruce Publishing Co., Milwaukee, 1946.
Double‑double
dovetail joint, pp. 26‑27.
Requires some bending.
Triple
dovetail puzzle, pp. 28‑29. Uses
curved piece with gravity lock.
S&B, p. 146, reproduces the
above Wyatt and shows a 1948 example.
W. A. Bagley. Puzzle Pie.
Op. cit. in 5.D.5. 1944. Dovetail deceptions, p. 64. Common square version and a tapered T joint.
Allan Boardman. Up and Down Double Dovetail. Shown on p. 147 of S&B. Square version with alternate dovetails in
opposite directions. This is
impossible!
I have a set of examples which
belonged to Tom O'Beirne. There is a
common square version and a similar hexagonal version. There is an equilateral triangle version
which requires a twist. There is a
right triangle version which has to be moved along a space diagonal! [One can adapt the twisting method to n-gons!]
Dick Schnacke (Mountain Craft
Shop, American Ridge Road, Route 1, New Martinsville, West Virginia, 26155,
USA) makes a variant of the common square version which has two dovetails on
each face. I bought one in 1994.
There are a great many illusions. This will only give some general studies and
some specific sources, though the sources of many illusions are unknown.
An exhibition by Al Seckel says
there are impossible geometric patterns in a mosaic floor in the Roman villa at
Fishbourne, c75, but it is not clear if this was intentional.
Anonymous 15C French illustrator
of Giovanni Boccaccio, De Claris Mulieribus, MS Royal 16 Gv in the British
Library. F. 54v: Collecting cocoons and
weaving silk. ??NYS -- reproduced in:
The Medieval Woman An Illuminated Book
of Postcards, HarperCollins, 1991. This
shows a loom(?) frame with uprights at each corner and the crosspieces joining
the tops of the end uprights as though front and rear are reversed compared to
the ground.
Seckel, 2002a, below, p. 25 (=
2002b, p. 175), says Leonardo da Vinci created the first anamorphic picture,
c1500.
Giuseppe Arcimboldo
(1537-1593). One of his paintings shows
a bowl of vegetables, but when turned over, it is a portrait. Seckel, 2000, below, fig. 109, pp. 120 &
122 (= 2002b, fig, 107, pp. 118 & 120), noting that this is the first
known invertible picture, but see next entry.
Topsy turvy coin, mid 16C. Seckel, 2002a, fig. 65, p. 80 (omitted in
2002b), shows an example which shows the Pope, but turns around to show the
Devil. Inscription around edge
reads: CORVI MALUM OVUM MALII.
Robert Smith. A Compleat System of Opticks in Four
Books. Cambridge, 1738. He includes a picture of a distant windmill for
which one cannot tell whether the sails are in front or behind the mill,
apparently the first publication of this visual ambiguity. ??NYS -- cited by: Nicholas J. Wade; Visual
Allusions Pictures of Perception;
Lawrence Erlbaum Associates, Hove, East Sussex, 1990, pp. 17 & 25, with a
similar picture.
L. A. Necker. LXI. Observations on some remarkable optical
phœnomena seen in Switzerland; and on an optical phœnomenon which occurs on
viewing a figure of a crystal or geometrical solid. Phil. Mag. (3) 1:5 (Nov 1832) 329-337. This is a letter from Necker, written on 24 May 1832. On pp. 336-337, Necker describes the visual
reversing figure known as the Necker cube which he discovered in drawing
rhomboid crystals. This is also quoted
in Ernst; The Eye Beguiled, pp. 23-24].
Richard L. Gregory [Mind in Science; Weidenfeld and Nicolson, London,
1981, pp. 385 & 594] and Ernst say that this was the first ambiguous figure
to be described.
See Thompson, 1882, in 6.AJ.2,
for illusions caused by rotations.
F. C. Müller-Lyer. Optische Urtheilstusehungen. Arch. Physiol. Suppl. 2 (1889) 263-270. Cited by Gregory in The Intelligent Eye. Many versions of the illusion. But cf below.
Wehman. New Book of 200 Puzzles. 1908.
The cube puzzle, p. 37. A 'baby
blocks' pattern of cubes, which appears to show six cubes piled in a corner one
way and seven cubes the other way. I
don't recall seeing this kind of puzzle in earlier sources, though this pattern
of rhombuses is common on cathedral floors dating back to the Byzantine era or
earlier.
James Fraser. British Journal of Psychology (Jan
1908). Introduces his 'The Unit of
Direction Illusion' in many forms.
??NYS -- cited in his popular article in Strand Mag., see below. Seckel, 2000, below, has several
versions. On p. 44, note to p. 9 (=
2002b, p. 44, note to p. 9), he says Fraser created a series of these illusions
in 1906.
H. E. Carter. A clever illusion. Curiosities section, Strand Mag. 378 (No. 219) (Mar 1909)
359. An example of Fraser's illusion
with no indication of its source.
James Fraser. A new illusion. What is its scientific explanation? Strand Mag. 38 (No. 224) (Aug 1909) 218-221. Refers to the Mar issue and says he
introduced the illusion in the above article and that the editors have asked
him for a popular article on it. 16
illustrations of various forms of his illusion.
Lietzmann, Walther &
Trier, Viggo. Wo steckt der
Fehler? 3rd ed., Teubner, 1923. [The Vorwort says that Trier was coauthor of
the 1st ed, 1913, and contributed most of the Schülerfehler (students'
mistakes). He died in 1916 and
Lietzmann extended the work in a 2nd ed of 1917 and split it into Trugschlüsse
and this 3rd ed. There was a 4th ed., 1937. See Lietzmann for a later version combining
both parts.] II. Täuschungen der
Anschauung, pp. 7-13.
Lietzmann, Walther. Wo steckt der Fehler? 3rd ed., Teubner, Stuttgart, (1950),
1953. (Strens/Guy has 3rd ed., 1963.) (See: Lietzmann & Trier. There are 2nd ed, 1952??; 5th ed, 1969; 6th ed, 1972. Math.
Gaz. 54 (1970) 182 says the 5th ed appears to be unchanged from the 3rd
ed.) II. Täuschungen der Anschauung,
pp. 15-25. A considerable extension of
the 1923 ed.
Williams. Home Entertainments. 1914.
Colour discs for the gramophone, pp. 207-212. Discusses several effects produced by spirals and eccentric
circles on discs when rotated.
Gerald H. Fisher. The Frameworks for Perceptual Localization.
Report of MOD Research Project70/GEN/9617, Department of Psychology, University
of Newcastle upon Tyne, 1968. Good
collection of examples, with perhaps the best set of impossible figures.
Pp.
42‑47 -- reversible perspectives.
Pp.
56‑65 -- impossible and ambiguous figures.
Appendix
6, p.190 -- 18 reversible figures.
Appendix
7, pp. 191‑192 -- 12 reversible silhouettes.
Appendix
8, p. 193 -- 12 impossible figures.
Appendix
14, pp. 202‑203 -- 72 geometrical illusions.
Harvey Long. "It's All In How You Look At
It". Harvey Long & Associates,
Seattle, 1972. 48pp collection of
examples with a few references.
Bruno Ernst [pseud. of J. A. F.
Rijk]. (Avonturen met Onmogelijke
Figuren; Aramith Uitgevers, Holland, 1985.)
Translated as: Adventures with
Impossible Figures. Tarquin, Norfolk,
1986. Describes tribar and many
variations of it, impossible staircase, two‑pronged trident. Pp. 76‑77 reproduces an Annunciation
of 14C in the Grote Kerk, Breda, with an impossible perspective. P. 78 reproduces Print XIV of Giovanni
Battista Piranesi's "Carceri de Invenzione", 1745, with an impossible
4‑bar.
Diego Uribe. Catalogo de impossibilidades. Cacumen (Madrid) 4 (No. 37) (Feb 1986) 9‑13. Good summary of impossible figures. 15 references to recent work.
Bruno Ernst. Escher's impossible figure prints in a new
context. In: H. S. M. Coxeter, et
al., eds.; M. C. Escher -- Art and Science; North‑Holland (Elsevier),
Amsterdam, 1986, pp. 124‑134.
Pp. 128‑129 discusses the Breda Annunciation, saying it is
15C and quoting a 1912 comment by an art historian on it. There is a colour reproduction on
p. 394. P. 130 shows and discusses
briefly Bruegel's "The Magpie on the Gallows", 1568. Pp. 130‑131 discusses and illustrates
the Piranesi.
Bruno Ernst. (Het Begoochelde Oog, 1986?.) Translated by Karen Williams as: The Eye Beguiled. Benedikt Taschen Verlag, Köln, 1992. Much expanded version of his previous book, with numerous new
pictures and models by new artists in the field. Chapter 6: Origins and history, pp. 68-93, discusses and quotes
almost everything known. P. 68 shows a
miniature of the Madonna and Child from the Pericope of Henry II, compiled by
1025, now in the Bayersche Staatsbibliothek, Munich, which is similar in form
to the Breda Annunciation (stated to be 15C).
(However, Seckel, 1997, below, reproduces it as 2©
and says it is c1250.) P. 69 notes that
Escher invented the impossible cube used in his Belvedere. P. 82 is a colour reproduction of Duchamp's
1916-1917 'Apolinère Enameled' - see 6.AJ.2.
Pp. 83-84 shows and discusses Piranesi.
Pp. 84-85 show and discuss Hogarth's 'False Perspective' of 1754. Reproduction and brief mention of Brueghel
(= Bruegel) on p. 85. Discussion of the
Breda Annunciation on pp. 85-86. Pp.
87-88 show and discuss a 14C Byzantine Annunciation in the National Museum,
Ochrid. Pp. 88-89 show and discuss
Scott Kim's impossible four-dimensional tribar.
J. Richard Block &
Harold E. Yuker. Can You Believe
Your Eyes? Brunner/Mazel, NY, 1992. Excellent survey of the field of illusions,
classified into 17 major types -- e.g. ambiguous figures, unstable figures,
..., two eyes are better than one. They
give as much information as they can about the origins. They give detailed sources for the following
-- originals ??NYS. These are also
available as two decks of playing cards.
W. E. Hill. My wife and my mother-in-law. Puck, (6 Nov 1915) 11. [However, Julian Rothenstein & Mel
Gooding; The Paradox Box; Redstone Press, London, 1993; include a reproduction
of a German visiting card of 1888 with a version of this illusion. The English caption by James Dalgety
is: My Wife and my Mother-in-law. Cf Seckel, 1997, below.] Ernst, just above, cites Hill and says he
was a cartoonist, but gives no source.
Long, above, asserts it was designed by E. G. Boring, an American
psychologist.
G. H. Fisher. Mother, father and daughter. Amer. J. Psychology 81 (1968) 274-277.
G. Kanisza. Subjective contours. SA 234:4 (Apr 1976) 48-52. (Kanisza triangles.)
Al Seckel, 1997. Illusions in Art. Two decks of playing cards in case with notes. Deck 1 -- Classics. Works from Roman times to the middle of the 20th
Century. Deck 2 -- Contemporary. Works from the second half of the 20th
Century. Y&B Associates, Hempstead,
NY, 1997. This gives further details on
some of the classic illusions -- some of this is entered above and in 6.AU and
some is given below.
10¨: Rabbit/Duck. Devised by Joseph (but notes say Robert) Jastrow, c1900. Seckel, 2000, below, p. 159 (= 2002b, p.
156), says Joseph Jastrow, c1900.
10§: My Wife and My Mother-in-Law, anonymous,
1888. However, in an exhibition,
Seckel's text implies the 1888 German card doesn't have a title and the title
first occurs on an 1890 US card.
Seckel, 2000, below, p. 122 (= 2002b, p. 120), says Boring took it from
a popular 19C puzzle trading card.
Al Seckel, 2000. The Art of Optical Illusions. Carlton, 2000. 144 well reproduced illusions with brief notes. All figures except 69-70 are included in
Seckel, 2002b.
J. Richard Block. Seeing Double Over 200 Mind-Bending Illusions.
Routledge, 2002. Update of Block
& Yuker, 1992.
Edgar
Rubin. Rubin's Vase. 1921.
This is the illusion where there appears to be a vase, but the outsides
appear to be two face profiles. [Pp.
8-11.] But Seckel, 2000, above, p. 122
(= 2002b, p. 120), says Rubin's inspiration was a 19C puzzle card.
My
wife and my mother-in-law. P. 17 says
Hill's version may derive from a late 1880s advertising postcard for
Phenyo-Caffein (Worcester, Massachusetts), labelled 'My Girl & Her Mother',
reproduced on p. 17.
P. 18
has G. H. Fisher's 1968 triple image, labelled 'Mother, Father and Daughter-in-Law'.
P. 44
says that Rabbit/Duck was devised by Joseph Jastrow in 1888.
Al Seckel, 2002a. More Optical Illusions. Carlton, 2002. 137 well reproduced illusions with brief notes, different than in
Seckel, 2000, above. All figures except
65-66, 86-87, 95-95, 137 are included in Seckel, 2002b, but with different
figure and page numbers.
Al Seckel, 2002b. The Fantastic World of Optical
Illusions. Carlton, 2002. This is essentially a combination of Seckel,
2000, and Seckel, 2002a, both listed above.
The Introduction is revised.
Figures 69-70 of the first book and 65-66, 86-87, 94-95, 137 of the
second book are omitted. The remaining
figures are then numbered consecutively.
The page of Further Reading in the first book is put at the end of this
combined book.
Here I make some notes about
origins of other illusions, but I have fewer details on these.
The Müller-Lyer Illusion -- <->
vs >---< was proposed by Zollner in 1859 and described
by Johannes Peter Müller (1801-1858) & Lyer in 1889. This seems to be a confusion, as the 1889
article is by F. C. Müller-Lyer, cf above.
Lietzmann & Trier, p. 7, date it as 1887.
The Bisection Illusion -- with a
vertical segment bisecting a horizontal segment, but above it -- was described
by Albert Oppel (1831-1865) and Wilhelm Wundt (1832-1920) in 1865.
Zollner's Illusion -- parallel
lines crossed by short lines at 45o, alternately in opposite directions -- was
noticed by Johann K. F. Zollner (1834-1882) on a piece of fabric with a similar
design.
Hering's Illusion -- with
parallel lines crossed by numerous lines through a point between the lines --
was invented by Ewals Hering (1834-1918) in 1860.
I have invented this name as it is
more descriptive than any I have seen.
The object or a version of it is variously called: Devil's Fork; Three Stick Clevis;
Widgit; Blivit; Impossible Columnade;
Trichometric Indicator Support;
Triple Encabulator for Tuned Manifold;
Hole Location Gage; Poiyut; Triple-Pronged Fork with only Two
Branches; Old Roman Pitchfork.
Oscar Reutersvård. Letters quoted in Ernst, 1992, pp. 69-70,
says he developed an equivalent type of object, which he calls impossible
meanders, in the 1930s.
R. L. Gregory says this is due to
a MIT draftsman (= draughtsman) about 1950??
California Technical
Industries. Advertisement. Aviation Week and Space Technology 80:12 (23
Mar 1964) 5. Standard form. (I wrote them but my letter was returned
'insufficient address'.)
Hole location gage. Analog Science Fact • Science Fiction 73:4
(Jun 1964) 27. Classic Two pronged
trident, with some measurements given.
Editorial note says the item was 'sent anonymously for some reason' and
offers the contributor $10 or a two year subscription if he identifies
himself. (Thanks to Peter McMullen for
the Analog items, but he doesn't recall the contributor ever being named.)
Edward G. Robles, Jr. Letter (Brass Tacks column). Analog Science Fact • Science Fiction 74:4
(Dec 1964) 4. Says the Jun 1964 object
is a "three-hole two slot BLIVIT" and was developed at JPL (Jet
Propulsion Laboratory, Pasadena) and published in their Goddard News. He provides a six-hole five-slot BLIVIT, but
as the Editor comments, it 'lacks the classic simple elegance of the Original.' However, a letter of inquiry to JPL resulted
in an email revealing that Goddard News is not their publication, but comes
from the Goddard Space Flight Center. I
have had a response from Goddard, ??NYR.
D. H. Schuster. A new ambiguous figure: a three‑stick
clevis. Amer. J. Psychol. 77 (1964)
673. Cites Calif. Tech. Ind. ad. [Ernst, 1992, pp. 80-81 reproduces this
article.]
Mad Magazine. No. 93 (Mar 1965). (I don't have a copy of this -- has anyone got one for
sale?) Cover by Norman Poiyut (?) shows
the figure and it is called a poiyut.
Miniature reproduction in: Maria
Reidelbach; Completely Mad -- A History of the Comic Book and Magazine; Little,
Brown & Co., Boston, 1991, p. 82.
Shows a standard version. Al
Seckel says they thought it was an original idea and they apologised in the
next issue -- to all of the following!
I now have the relevant issue, No. 95 (Jun 1965) and p. 2 has 15 letters
citing earlier appearances in Engineering Digest, The Airman (official journal
of the U.S. Airforce), Analog, Astounding Science Fact -- Science Fiction (Jun
1964, see above), The Red Rag (engineering journal at the University of British
Columbia), Society of Automotive Engineers Journal (designed by by Gregory
Flynn Jr. of General Motors as Triple Encabulator Tuned Manifold), Popular Mechanics, Popular Science (Jul
1964), Road & Track (Jun 1964).
Other letters say it was circulating at: the Engineering Graphics Lab of the University of Minnesota at
Duluth; the Nevada Test Site; Eastman Kodak (used to check
resolution); Industrial Camera Co. of
Oakland California (on their letterhead).
Two letters give an impossible crate and an impossible rectangular frame
(sort of a Penrose rectangle).
Sergio Aragones. A Mad look at winter sports. Mad Magazine (?? 1965); reprinted in: Mad Power; Signet, NY,
1970, pp. 120‑129. P. 124 shows a
standard version.
Bob Clark, illustrator. A Mad look at signs of the times. Loc. cit. under Aragones, pp. 167‑188. P. 186 shows standard version.
Reveille (a UK weekly magazine)
(10 Jun 1965). ??NYS -- cited by
Briggs, below -- standard version.
Don Mackey. Optical illusion. Skywriter (magazine of North American Aviation) (18 Feb
1966). ??NYS -- cited by Conrad G.
Mueller et al.; Light and Vision; Time-Life Books Pocket Edition, Time-Life
International, Netherlands, 1969, pp. 171 & 190. Standard version with nuts on the ends.
Heinz Von Foerster. From stimulus to symbol: The economy of
biological computation. IN: Sign Image Symbol; ed. Gyorgy Kepes; Studio
Vista, London, 1966, pp. 42-60. On
p. 55, he shows the "Triple-pronged fork with only two branches"
and on p. 54, he notes that although each portion is correct, it is impossible
overall, but he gives no indication of its history or that it is at all new.
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Pp. 17-18 shows the unnamed trident in a
version from Adcock & Shipley (Sales) Ltd., machine tool makers in
Leicester. Cites Reveille, above. Standard versions.
Harold Baldwin. Building better blivets. The Worm Runner's Digest 9:2 (1967) 104‑106. Discusses relation between numbers of slots
and of prongs. Draws a three slot
version and 2 and 4 way versions.
Charlie Rice. Challenge!
Op. cit. in 5.C. 1968. P. 10 shows a six prong, four slot version,
called the "Old Roman Pitchfork".
Roger Hayward. Blivets; research and development. The Worm Runner's Digest 10 (Dec 1968) 89‑92. Several fine developments, including two
interlaced frames and his monumental version.
Cites Baldwin.
M. Gardner. SA (May 1970) = Circus, pp. 3‑15. Says this became known in 1964 and cites Mad
& Hayward, but not Schuster.
D. Uribe, op. cit. above, gives
several variations.
6.AJ.2. TRIBAR AND IMPOSSIBLE STAIRCASE
Silvanus P. Thompson. Optical illusions of motion. Brain 3 (1882) 289-298. Hexagon of non‑overlapping
circles.
Thomas Foster. Illusions of motion and strobic
circles. Knowledge 1 (17 Mar 1882)
421-423. Says Thompson exhibited these
illusions at the British Association meeting in 1877.
Pearson. 1907.
Part II, no. 3: Whirling wheels, p. 3.
Gives Thompson's form, but the wheels are overlapping, which makes it
look a bit like an ancestor of the tribar.
Marcel Duchamp (1887-1968). Apolinère Enameled. A 'rectified readymade' of 1916-1917 which
turned a bedframe in an advertisement for Sapolin Enamel into an impossible
figure somewhat like a Penrose Triangle and a square version thereof. A version is in the Philadelphia Museum of
Art and is reproduced and discussed in Ernst; The Eye Beguiled, p. 82. (Duchamp's 'readymades' were frequently
reproduced by himself and others, so there may be other versions of this.)
Oscar Reutersvård. Omöjliga Figure [Impossible Figures -- In
Swedish]. Edited by Paul Gabriel. Doxa, Lund, (1982); 2nd ed., 1984. This seems to be the first publication of his work, but he has
been exhibiting since about 1960 and some of the exhibitions seem to have had
catalogues. P. 9 shows and discusses
his Opus 1 from 1934, which is an impossible tribar made from cubes. (Reproduced in Ernst, 1992, p. 69 as a
drawing signed and dated 1934. Ernst
quotes Reutersvård's correspondence which describes his invention of the form
while doodling in Latin class as a schoolboy.
A school friend who knew of his work showed him the Penroses' article in
1958 -- at that time he had drawn about 100 impossible objects -- by 1986, he
had extended this to some 2500!) He has
numerous variations on the tribar and the two‑pronged trident. An exhibition by Al Seckel says Reutersvård
had produced some impossible staircases, e.g. 'Visualized Impossible Bach
Scale', in 1936-1937, but didn't go far with it until returning to the idea in
1953.
Oscar Reutersvård. Swedish postage stamps for 25, 50, 75
kr. 1982, based on his patterns from
the 1930s. The 25 kr. has the tribar
pattern of cubes which he first drew in 1934.
(Also the 60 kr.??)
L. S. & R. Penrose. Impossible objects: A special type of visual
illusion. British Journal of Psychology
49 (1958) 31‑33. Presents tribar
and staircase. Photo of model
staircase, which Lionel Penrose had made in 1955. [Ernst, 1992, pp. 71-73, quotes conversation with Penrose about
his invention of the Tribar and reproduces this article. Penrose, like the rest of us, only learned
about Reutersvård's work in the 1980s.]
Anon.(?) Don't believe it. Daily Telegraph (24 Mar 1958) ?? (clipping found in an old
book). "Three pages of the latest
issue of the British Journal of Psychology are devoted to "Impossible
Objects."" Shows both the
tribar and the staircase.
M. C. Escher. Lithograph:
Belvedere. 1958.
L. S. & R. Penrose. Christmas Puzzles. New Scientist (25 Dec 1958) 1580‑1581 & 1597. Prob. 2: Staircase for lazy people.
M. C. Escher. Lithograph:
Ascending and Descending. 1960.
M. C. Escher. Lithograph:
Waterfall. 1961.
Oscar Reutersvård, in 1961,
produced a triangular version of the impossible staircase, called 'Triangular
Fortress without Highest Level'.
Joseph Kuykendall. Letter.
Mad Magazine 95 (Jun 1965) 2. An
impossible frame, a kind of Penrose rectangle.
S. W. Draper. The Penrose triangle and a family of related
figures. Perception 7 (1978) 283‑296. ??NYS -- cited and reproduced in Block,
2002, p. 48. A Penrose rectangle.
Uribe, op. cit. above, gives several
variations, including a perspective tribar and Draper's rectangle.
Jan van de Craats. Das unmögliche Escher-puzzle. (Taken from: De onmogelijke Escher-puzzle; Pythagoras (Amsterdam) (1988).) Alpha 6 (or: Mathematik Lehren / Heft 55 -- ??) (1992) 12-13. Two Penrose tribars made into an impossible
5-piece burr.
This is the illusion seen in
alternatingly coloured staggered brickwork where the lines of bricks distinctly
seem tilted. I suspect it must be
apparent in brickwork going back to Roman times.
The illusion is apparent in the
polychrome brick work on the side wall inside Keble College Chapel, Oxford, by
William Butterfield, completed in 1876 [thanks to Deborah Singmaster for
observing this].
Lietzmann & Trier, op. cit.
at 6.AJ, 1923. Pp. 12-13 has a striking
version of this, described as a 'Flechtbogen der Kleinen'. I can't quite translate this -- Flecht is
something interwoven but Bogen could be a ribbon or an arch or a bower,
etc. They say it is reproduced from an
original by Elsner. See Lietzmann,
1953.
Ogden's Optical Illusions. Cigarette card of 1927. No. 5.
Original ??NYS -- reproduced in:
Julian Rothenstein & Mel Gooding; The Paradox Box; Redstone Press,
London, 1993 AND in their: The Playful Eye; Redstone Press, London,
1999, p. 56. Vertical version of
this illusion.
B. K. Gentil. Die optische Täuschung von Fraser. Zeitschr. f. math. u. naturw. Unterr. 66
(1935) 170 ff. ??NYS -- cited by
Lietzmann.
Nelson F. Beeler & Franklyn
M. Branley. Experiments in Optical
Illusion. Ill. by Fred H. Lyon. Crowell, 1951, p. 42, fig. 39, is a good
example of the illusion.
Lietzmann, op. cit. at 6.AJ,
1953. P. 23 is the same as above, but
adds a citation to Gentil, listed above.
Leonard de Vries. The Third Book of Experiments. © 1965, probably for a Dutch edition. Translated by Joost van de Woestijne. John Murray, 1965; Carousel, 1974. Illusion 10, pp. 58-59, has a clear picture
and a brief discussion.
Richard L. Gregory &
Priscilla Heard. Border locking and the
café wall illusion. Perception 8 (1979)
365‑380. ??NYS -- described by
Walker, below. [I have photos of the
actual café wall in Bristol.]
Jearl Walker. The Amateur Scientist: The café‑wall
illusion, in which rows of tiles tilt that should not tilt at all. SA 259:5 (Nov 1988) 100‑103. Good summary and illustrations.
New section, due to reading Glass's
assertion as to the inventor, who is different than other names that I have
seen.
Don Glass, ed. How Can You Tell if a Spider is Dead? and
More Moments of Science. Indiana Univ
Press, Bloomington, Indiana, 1996. Now
you see it, now you don't, pp. 131-132.
Asserts that Christopher Tyler, of the Smith-Kettlewell Eye Research
Institute, San Francisco, is the inventor of stereograms.
This is like a Necker Cube where all
the edges are drawn as wooden slats in an impossible configuration.
Escher. Man with Cuboid, which is essentially a
detail from Belvedere, both 1958, are apparently the first examples of this
impossible object.
Chuck Mathias. Letter
Mad Magazine 95 (Jun 1965) 2.
Gives an impossible crate.
Jerry Andrus developed his
actual model in 1981 and it appeared on the cover of Omni in 1981. But Al Seckel's exhibition says the first
physical example was The Feemish Crate, due to C. F. Cochran.
Seckel, 2002a, figs. 27 A&B,
pp. 36-37 (= 2002b, figs. 169 A&B, pp. 186-187), shows and discusses
Andrus' crate from two viewpoints.
6.AK. POLYGONAL PATH COVERING N x N LATTICE OF POINTS, QUEEN'S TOURS, ETC.
For
magic circuits, see 7.N.4.
3x3 problem: Loyd (1907), Pearson, Anon., Bullivant,
Goldston, Loyd (1914), Blyth,
Abraham, Hedges, Evans,
Doubleday - 1, Piggins &
Eley
4x4 problem: King,
Abraham, M. Adams, Evans,
Depew, Meyer, Ripley's,
Queen's tours: Loyd (1867, 1897, 1914), Loyd Jr.
Bishop's tours: Dudeney (1932), Doubleday - 2, Obermair
Rook's tours: Loyd (1878), Proctor, Loyd
(1897), Bullivant, Loyd (1914), Filipiak, Hartswick, Barwell,
Gardner, Peters, Obermair
Other versions: Prout,
Doubleday - 1
Trick solutions: Fixx,
Adams, Piggins, Piggins & Eley
Thanks
to Heinrich Hemme for pointing out Fixx, which led to adding most of the
material on trick solutions.
Loyd. ??Le Sphinx (Mar 1867 -- but the Supplement to Sam Loyd and His
Chess Problems corrects this to 15 Nov 1866).
= Chess Strategy, Elizabeth, NJ, 1878, no. or p. 336(??). = A. C. White; Sam Loyd and His Chess
Problems; 1913, op. cit. in 1; no. 40, pp. 42‑43. Queen's circuit on 8 x 8 in 14
segments. (I.e. closed circuit,
not leaving board, using queen's moves.)
No. 41 & 42 of White give other solutions. White quotes Loyd from Chess Strategy, which indicates that Loyd
invented this problem. Tit‑Bits
No. 31 & SLAHP: Touring the chessboard, pp. 19 & 89, give No. 41.
Loyd. Chess Strategy, 1878, op. cit. above, no. or p. 337 (??) (= White, 1913, op. cit. above, no. 43,
pp. 42‑43.) Rook's circuit
on 8 x 8 in 16 segments.
(I.e. closed circuit, not leaving board, using rook's moves, and without
crossings.)
Richard A. Proctor. Gossip column. Knowledge 10 (Dec 1886)
43 & (Feb 1887) 92. 6 x 6 array of cells. Prisoner in one corner can exit from the opposite corner if he
passes "once, and once only, through all the 36 cells." "... take the prisoner into either of
the cells adjoining his own, and back into his own, .... This puzzle is rather a sell,
...." Letter and response [in
Gossip column, Knowledge 10 (Mar 1887) 115-116] about the impossibility of any
normal solution.
Loyd. Problem 15: The gaoler's problem. Tit‑Bits 31 (23 Jan
& 13 Feb 1897) 307 &
363. Rook's circuit on 8 x 8
in 16 segments, but beginning and ending on a central square. Cf The postman's puzzle in the Cyclopedia,
1914.
Loyd. Problem 16: The captive maiden.
Tit‑Bits 31 (30 Jan
& 20 Feb 1897) 325 &
381. Rook's tour in minimal
number of moves from a corner to the diagonally opposite corner, entering each
cell once. Because of parity, this is
technically impossible, so the first two moves are into an adjacent cell and
then back to the first cell, so that the first cell has now been entered.
Loyd. Problem 20: Hearts and darts.
Tit‑Bits 31 (20 Feb,
13 & 20 Mar 1897) 381, 437, 455. Queen's tour on 8 x 8, starting in a
corner, permitting crossings, but with no segment going through a square where
the path turns. Solution in 14
segments. This is No. 41 in
White -- see the first Loyd entry above.
Ball. MRE, 4th ed., 1905, p. 197.
At the end of his section on knight's tours, he states that there are
many similar problems for other kinds of pieces.
Loyd. In G. G. Bain, op. cit. in 1, 1907. He gives the 3 x 3 lattice in four lines as the Columbus Egg
Puzzle.
Pearson. 1907.
Part I, no. 36: A charming puzzle, pp. 36 & 152‑153. 3 x 3
lattice in 4 lines.
Loyd. Sam Loyd's Puzzle Magazine (Apr 1908) -- ??NYS, reproduced
in: A. C. White; Sam Loyd and His Chess
Problems; 1913, op. cit. in 1; no. 56, p. 52.
= Problem 26: A brace of puzzles -- No. 26: A study in naval warfare;
Tit‑Bits 31 (27 Mar 1897) 475
& 32 (24 Apr 1897) 59. = Cyclopedia, 1914, Going into action,
pp. 189 & 364. = MPSL1, prob. 46,
pp. 44 & 138. = SLAHP: Bombs to
drop, pp. 86 & 119. Circuit on 8 x 8
in 14 segments, but with two lines of slope 2. In White, p. 43, Loyd
says an ordinary queen's tour can be started "from any of the squares
except the twenty which can be represented by
d1, d3 and d4." This
problem starts at d1. However I think White must have mistakenly
put down twenty for twelve??
Anon. Prob. 67. Hobbies 31 (No. 782) (8 Oct 1910) 39 &
(No. 785) (29 Oct 1910) 94.
3 x 3 lattice in 4
lines "brought under my notice some time back".
C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV.
No. 1:
The travelling draught‑man, pp. 515 & 520. Rook's circuit on 8 x
8 in 16 segments, different than
Loyd's.
No. 3:
Joining the rings. 3 x 3 in 4 segments.
Will Goldston. More Tricks and Puzzles without Mechanical
Apparatus. The Magician Ltd., London,
nd [1910?]. (BMC lists Routledge &
Dutton eds. of 1910.) (There is a 2nd
ed., published by Will Goldston, nd [1919].)
The nine‑dot puzzle, pp. 127‑128 (pp. 90‑91 in
2nd ed.).
Loyd. Cyclopedia, 1914, pp. 301 & 380. = MPSL2, prob. 133 -- Solve Christopher's egg tricks, pp. 93
& 163 (with comment by Gardner). c=
SLAHP: Milkman's route, pp. 34 & 96.
3 x 3 case.
Loyd. Cyclopedia, 1914, pp. 293 & 379. Queen's circuit on 7 x
7 in
12 segments.
Loyd. The postman's puzzle.
Cyclopedia, 1914, pp. 298 & 379.
Rook's circuit on 8 x 8 array of points, with one point a bit out of
line, starting and ending at a central square, in 16 segments. P. 379 also shows another 8 x 8
circuit, but with a slope 2 line.
See also pp. 21 & 341 and SLAHP, pp. 85 & 118, for two more
examples.
Loyd. Switchboard problem.
Cyclopedia, 1914, pp. 255 & 373.
(c= MPSL2, prob. 145, pp. 102 & 167.) Rook's tour with minimum turning.
Blyth. Match-Stick Magic.
1921. Four-way game, pp.
77-78. 3 x 3 in 4 segments.
King. Best 100. 1927. No. 16, pp. 12 & 43. 4 x 4
in 6 segments, not closed, but easily can be closed.
Loyd Jr. SLAHP. 1928. Dropping the mail, pp. 67 & 111. 4 x 4 queen's tour in 6
segments.
Collins. Book of Puzzles. 1927. The star group puzzle,
pp. 95-96. 3 x 3 in
4 segments.
Dudeney. PCP.
1932. Prob. 264: The fly's tour,
pp. 82 & 169. = 536, prob. 422, pp.
159 & 368. Bishop's path, with
repeated cells, going from corner to corner in
17 segments.
Abraham. 1933.
Probs. 101, 102, 103, pp. 49 & 66 (30 & 118). 3 x 3,
4 x 4 and 6 x 6
cases.
The Bile Beans Puzzle Book. 1933.
No. 4: The puzzled milkman. 3 x
3 array in four lines.
Sid G. Hedges. More Indoor and Community Games. Methuen, London, 1937. Nine spot, p. 110. 3 x 3.
"Of course it can be done, but it is not easy." No solution given.
M. Adams. Puzzle Book. 1939. Prob. C.64: Six
strokes, pp. 140 & 178. 4 x 4 array in
6 segments which form a closed
path, though the closure was not asked for.
J. R. Evans. The Junior Week‑End Book. Op. cit. in 6.AF. 1939. Probs. 30 & 31,
pp. 264 & 270. 3 x 3 &
4 x 4 cases in 4 & 6
segments, neither closed nor staying within the array.
Depew. Cokesbury Game Book.
1939. Drawing, p. 220. 4 x 4
in 6 segments, not closed, not staying within the array.
Meyer. Big Fun Book. 1940. Right on the dot, pp. 99 & 732. 4 x 4
in 6 segments.
A. S. Filipiak. Mathematical Puzzles, 1942, op. cit. in
5.H.1, pp. 50‑51. Same as
Bullivant, but opens the circuit to make a 15 segment path.
M. S. Klamkin, proposer and
solver; John L. Selfridge, further
solver. Problem E1123 -- Polygonal path
covering a square lattice. AMM 61
(1954) 423 & 62 (1955) 124 & 443. Shows
N x N can be done in 2N‑2
segments. Selfridge shows this is
minimal.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Joining the stars, pp. 41 & 129. 5 x 5
array of points. Using a line of
four segments, pass through 17 points.
This is a bit like the 3 x
3 problem in that one must go outside
the array.
R. E. Miller & J. L.
Selfridge. Maximal paths on rectangular
boards. IBM J. Research and Development
4:5 (Nov 1960) 479-486. They study
rook's paths where a cell is deemed visited if the rook changes direction
there. They find maximal such paths in
all cases.
Ripley's Puzzles and Games. 1966.
Pp. 72-73, item 2. 4 x 4 cases with closed solution symmetric both
horizontally and vertically.
F. Gregory Hartswick. In:
H. A. Ripley & F. Gregory Hartswick, Detectograms and Other Puzzles,
Scholastic Book Services, NY, 1969.
Prob. 4, pp. 42‑43 & 82.
Asks for 8 x 8 rook's circuit with minimal turning and
having a turn at a central cell.
Solution gives two such with
16 segments and asserts there
are no others.
Doubleday - 1. 1969. Prob. 60: Test case, pp. 75 & 167. = Doubleday - 4, pp. 83-84.
Two 3 x 3 arrays joined at a corner, looking like the
Fore and Aft board (cf 5.R.3), to be covered in a minimum number of
segments. He does it in seven segments
by joining two 3 x 3 solutions.
Brian R. Barwell. Arrows and circuits. JRM 2 (1969) 196‑204. Introduces idea of maximal length rook's
tours. Shows the maximal length on
a 4 x 4 board is 38 and finds there are 3
solutions. Considers also
the 1 x n board.
Solomon W. Golomb & John L.
Selfridge. Unicursal polygonal paths
and other graphs on point lattices. Pi
Mu Epsilon J. 5 (1970) 107‑117.
Surveys problem. Generalizes
Selfridge's 1955 proof to M x N for which
MIN(2M, M+N‑2) segments
occur in a minimal circuit.
Doubleday - 2. 1971.
Path finder, pp. 95-96. Bishop's
corner to corner path, same as Dudeney, 1932.
James F. Fixx. More Games for the Superintelligent. (Doubleday, 1972); Muller, (1977), 1981. 6.
Variation on a variation, pp. 31 & 87.
Trick solution in three lines, assuming points of finite size.
M. Gardner. SA (May 1973) c= Knotted, chap. 6. Prob. 1: Find rook's tours of maximum length
on the 4 x 4 board. Cites
Barwell. Knotted also cites Peters,
below.
Edward N. Peters. Rooks roaming round regular rectangles. JRM 6 (1973) 169‑173. Finds maximum length on 1 x N
board is N2/2 for
N even; (N‑1)2/2 + N‑1 for
N odd, and believes he has
counted such tours. He finds tours on
the N x N board whose length is a formula that reduces to 4 BC(N+1, 3) ‑ 2[(N‑1)/2]. I am a bit unsure if he has shown that this
is maximal.
James L. Adams. Conceptual Blockbusting. Freeman, 1974, pp. 16-22. 3rd ed., (A-W, 1986), Penguin, 1987, pp.
24-33. Trick solution of 3 x 3
case in three lines, assuming points of finite size, which he says was
submitted anonymously when he and Bob McKim used the puzzle on an ad for a talk
on problem-solving at Stanford. Also
describes a version using paperfolding to get all nine points into a line. The material is considerably expanded in the
3rd ed. and adds several new versions.
From the references in Piggins and Eley, it seems that these all
appeared in the 2nd ed of 1979 -- ??NYS.
Cut
out the 3 x 1 parts and tape them into a straight line.
Take
the paper and roll it to a cylinder and then draw a slanting line on the
cylinder which goes through all nine, largish, points.
Cut
out bits with each point on and skewer the lot with a pencil.
Place
the paper on the earth and draw a line around the earth to go through all nine
points. One has to assume the points
have some size.
Wodge
the paper, with large dots, into a ball and stick a pencil through it. Open up to see if you have won -- if not,
try again!
Use
a very fat line, i.e. as thick as the spacing between the edges of the array.
David J. Piggins. Pathological solutions to a popular
puzzle. JRM 8:2 (1975-76) 128-129. Gives two trick solutions.
Three
parallel lines, since they meet at infinity.
Put
the figure on the earth and use a slanting line around the earth. This works in the limit, but otherwise
requires points of finite size, a detail that he doesn't mention.
No
references for these versions.
David J. Piggins & Arthur D.
Eley. Minimal path length for covering
polygonal lattices: A review. JRM 14:4 (1981‑82) 279‑283. Mostly devoted to various trick solutions of
the 3 x 3 case. They cite Piggins'
solution with three parallel lines.
They say that Gardner sent them the trick solution in 1973 and then cite
Adams, 1979. They give solutions using
points of different sizes, getting both three and two segment solutions and
mention a two segment version that depends on the direction of view. They then give the solution on a sphere,
citing Adams, 1979, and Piggins. They
give several further versions using paper folding, including putting the
surface onto a twisted triangular prism joined at the ends to make the surfaces
into a Möbius strip -- Zeeman calls this a umbilical bracelet or a Möbius bar.
Obermair. Op. cit. in 5.Z.1. 1984.
Prob.
19, pp. 23 & 50. Bishop's path
on 8 x 8 in 17 segments, as in Dudeney, PCP, 1932.
Prob.
41, p. 72. Rook's path with maximal
number of segments, which is 57. [For the
2 x 2, 3 x 3, 4 x 4
boards, I get the maximum numbers are
3, 6, 13.]
Nob Yoshigahara. Puzzlart.
Tokyo, 1992. Section: The wisdom
of Solomon, pp. 40-47, abridged from an article by Solomon W. Golomb in Johns
Hopkins Magazine (Oct 1984).
Classic 3 x 3 problem.
For the 4 x 4 case: 1) find four closed paths; 2), says there are about 30 solutions and
gives 19 beyond the previous 4. Find
the unique 5‑segment closed path on the
3 x 4. Gives 3 solutions on 5 x 5.
10-segment solution on 6 x
6 which stays on the board. Loyd's 1867? Queen's circuit. Queen's circuit on 7 x 7, attributed to
Dudeney, though my earliest entry is Loyd, 1914 -- ??CHECK.
This
has such an extensive history that I will give only a few items.
C. L. Lehmus first posed the problem
to Jacob Steiner in 1840.
Rougevin published the first
proof in 1842. ??NYS.
Jacob Steiner. Elementare Lösung einer Aufgabe über das
ebene und sphärische Dreieck.
J. reine angew. Math. 28 (1844) 375‑379 & Tafel III. Says Lehmus sent it to him in 1840 asking
for a purely geometric proof. Here he
gives proofs for the plane and the sphere and also considers external
bisectors.
Theodor Lange. Nachtrag zu dem Aufsatze in Thl. XIII, Nr.
XXXIII. Archiv der Math. und Physik 15
(1850) 221‑226. Discusses the
problem and gives a solution by Steiner and two by C. L. Lehmus. Steiner also considers the external
bisectors.
N. J. Chignell. Note 1031: A difficult converse. MG 16 (No. 219) (Jul 1932) 200-202. [The author's name is omitted in the article
but appears on the cover.] 'Three
fairly simple proofs', due to: M. J.
Newell; J. Travers, improving J. H.
Doughty, based on material in Lady's and Gentleman's Diary (1859) 87-88 &
(1860) 84-86; Wm. Mason, found by
Doughty, in Lady's and Gentleman's Diary (1860) 86.
H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Section 1.5, ex. 4, p. 16.
An easy proof is posed as a problem with adequate hints in four lines.
M. Gardner. SA (Apr 1961) = New MD, chap. 17. Review of Coxeter's book, saying his brief
proof came as a pleasant shock.
G. Gilbert & D.
MacDonnell. The Steiner‑Lehmus
theorem. AMM 70 (1963) 79‑80. This is the best of the proofs sent to
Gardner in response to his review of Coxeter.
A later source says this turned out to be identical to Lehmus' original
proof!
Léo Sauvé. The Steiner‑Lehmus theorem. CM 2:2 (Feb 1976) 19‑24. Discusses history and gives 22 references,
some of which refer to 60 proofs.
Charles W. Trigg. A bibliography of the Steiner‑Lehmus
theorem. CM 2:9 (Nov 1976) 191‑193. 36 references beyond Sauvé's.
David C. Kay. Nearly the last comment on the Steiner‑Lehmus
theorem. CM 3:6 (1977) 148‑149. Observes that a version of the proof works
in all three classical geometries at once and gives its history.
This
also has an extensive history and I give only a few items.
T. Delahaye and H. Lez. Problem no. 1655 (Morley's triangle). Mathesis (3) 8 (1908) 138‑139. ??NYS.
E. J. Ebden, proposer; M. Satyanarayana, solver. Problem no. 16381 (Morley's theorem). The Educational Times (NS) 61 (1 Feb 1908)
81 & (1 Jul 1908) 307‑308
= Math. Quest. and Solutions from "The Educational Times" (NS)
15 (1909) 23. Asks for various related
triangles formed using interior and exterior trisectors to be shown
equilateral. Solution is essentially
trigonometric. No mention of Morley.
Frank Morley. On the intersections of the trisectors of
the angles of a triangle. (From a
letter directed to Prof. T. Hayashi.)
J. Math. Assoc. of Japan for Secondary Education 6 (Dec 1924) 260‑262. (= CM 3:10 (Dec 1977) 273‑275.
Frank Morley. Letter to Gino Loria. 22 Aug 1934. Reproduced in: Gino
Loria; Triangles équilatéraux dérivés d'un triangle quelconque. MG 23 (No. 256) (Oct 1939) 364‑372. Morley says he discovered the theorem in
c1904 and cites the letter to Hayashi.
Loria mentions other early work and gives several generalizations.
H. F. Baker. Note 1476:
A theorem due to Professor F. Morley.
MG 24 (No. 261) (Oct 1940) 284‑286. Easy proof and reference to other proofs. He cites a related result of Steiner.
Anonymous [R. P.] Morley's trisector theorem. Eureka 16 (Oct 1953) 6-7. Short proof, working backward from the
equilateral triangle.
Dan Pedoe. Notes on Morley's proof of his theorem on
angle trisectors. CM 3:10 (Dec 1977)
276‑279. "... very tentative
... first steps towards the elucidation of his work."
C. O. Oakley & Charles W.
Trigg. A list of references to the
Morley theorem. CM 3:10 (Dec 1977)
281‑290 & 4 (1978) 132. 169 items.
André Viricel (with Jacques
Bouteloup). Le Théorème de Morley. L'Association pour le Développement de la
Culture Scientifique, Amiens, 1993.
[This publisher or this book was apparently taken over by Blanchard as
Blanchard was selling copies with his label pasted over the previous
publisher's name in Dec 1994.] A
substantial book (180pp) on all aspects of the theorem. The bibliography is extremely cryptic, but
says it is abridged from Mathesis (1949) 175
??NYS. The most recent item
cited is 1970.
6.AN. VOLUME OF THE INTERSECTION OF TWO CYLINDERS
Archimedes. The Method:
Preface, 2. In: T. L. Heath; The Works of Archimedes, with a
supplement "The Method of Archimedes"; (originally two works, CUP,
1897 & 1912) = Dover, 1953. Supplement, p. 12, states the result. The proof is lost, but pp. 48‑51
gives a reconstruction of the proof by Zeuthen.
Liu Hui. Jiu Zhang Suan Chu Zhu (Commentary on the
Nine Chapters of the Mathematical Art).
263. ??NYS -- described in Li
& Du, pp. 73‑74 & 85. He
shows that the ratio of the volume of the sphere to the volume of Archimedes'
solid, called mou he fang gai (two square umbrellas), is π/4,
but he cannot determine either volume.
Zu Geng. c500.
Lost, but described in: Li
Chunfeng; annotation to Jiu Zhang (= Chiu Chang Suan Ching) made c656. ??NYS.
Described on pp. 86‑87
of: Wu Wenchun; The out‑in
complementary principle; IN: Ancient China's Technology and Science;
compiled by the Institute of the History of Natural Sciences, Chinese Academy
of Sciences; Foreign Languages Press, Beijing, 1983, pp. 66‑89. [This is a revision and translation of parts
of: Achievements in Science and
Technology in Ancient China [in Chinese]; China Youth Publishing House,
Beijing(?), 1978.]
He
considers the shape, called fanggai, within the natural circumscribed cube and
shows that, in each octant, the part of the cube outside the fanggai has cross
section of area h2 at distance
h from the centre. This is equivalent to a tetrahedron, whose
volume had been determined by Liu, so the excluded volume is ⅓
of the cube.
Li
& Du, pp. 85‑87, and say the result may have been found c480 by Zu
Geng's father, Zu Chongzhi.
Lam Lay-Yong & Shen
Kangsheng. The Chinese concept of
Cavalieri's Principle and its applications.
HM 12 (1985) 219-228. Discusses
the work of Liu and Zu.
Shiraishi Chōchū. Shamei Sampu. 1826. ??NYS -- described
in Smith & Mikami, pp. 233-236.
"Find the volume cut from a cylinder by another cylinder that
intersects is orthogonally and touches a point on the surface". I'm not quite sure what the last phrase
indicates. The book gives a number of
similar problems of finding volumes of intersections.
P. R. Rider, proposer; N. B. Moore, solver. Problem 3587. AMM 40 (1933) 52 (??NX) &
612. Gives the standard proof by
cross sections, then considers the case of unequal cylinders where the solution
involves complete elliptic integrals of the first and second kinds. References to solution and similar problem
in textbooks.
Leo Moser, solver; J. M. Butchart, extender. MM 25 (May 1952) 290 &
26 (Sep 1952) 54. ??NX. Reproduced in Trigg, op. cit. in 5.Q: Quickie
15, pp. 6 & 82‑83. Moser
gives the classic proof that
V = 16r3/3.
Butchart points out that this also shows that the shape has surface
area 16r2.
NOTATION: (a, b, c) denotes the
configuration of a points in
b rows of c
each. The index below covers
articles other than the surveys of Burr et al. and Gardner.
( 5, 2, 3): Sylvester
( 6, 3, 3): Mittenzwey
( 7, 6, 3): Criton
( 9, 8, 3): Sylvester; Carroll;
Criton
( 9, 9, 3): Carroll; Bridges; Criton
( 9, 10, 3): Jackson; Family Friend; Parlour Pastime;
Magician's Own Book;
The Sociable;
Book of 500 Puzzles;
Charades etc.;
Boy's Own Conjuring Book;
Hanky Panky; Carroll; Crompton;
Berkeley & Rowland;
Hoffmann;
Dudeney (1908);
Wehman; Williams; Loyd Jr;
Blyth; Rudin; Young World; Brooke; Putnam; Criton
(10, 5, 4): The Sociable; Book of 500 Puzzles; Carroll;
Hoffmann;
Dudeney (1908);
Wehman; Williams; Dudeney (1917); Blyth; King; Rudin;
Young World; Hutchings
& Blake; Putnam
(10, 10, 3): Sylvester
(11, 11, 3): The Sociable; Book of 500 Puzzles; Wehman
(11, 12, 3): Hoffmann; Williams;
Young World
(11, 13, 3): Prout
(11, 16, 3): Wilkinson -- in Dudeney (1908
& 1917); Macmillan
(12, 4, 5) -- Trick version of a hollow 3 x 3 square with doubled
corners, as in 7.Q: Family Friend (1858); Secret Out;
Illustrated Boy's Own Treasury;
(12, 6, 4): Endless
Amusement II; The Sociable; Book of 500 Puzzles; Boy's Own Book; Cassell's;
Hoffmann; Wehman; Rudin;
Criton
(12, 7, 4) -- Trick version of a
3 x 3 square with doubled
diagonal: Secret Out; Hoffmann (1876); Mittenzwey;
Hoffmann (1893), no. 8
(12, 7, 4): Dudeney
(1917); Putnam
(12, 19, 3): Macmillan
(13, 9, 4): Criton
(13, 12, 3): Criton
(13, 18, 3): Sylvester
(13, 22, 3): Criton
(15, 15, 3): Jackson
(15, 16, 3): The Sociable; Book of 500 Puzzles; H. D. Northrop; Wehman
(15, 23, 3): Jackson
(15, 26, 3): Woolhouse
(16, 10, 4): The Sociable; Book of 500 Puzzles; Hoffmann;
Wehman
(16, 12, 4): Criton
(16, 15, 4): Dudeney (1899, 1902, 1908); Brooke;
Putnam; Criton
(17, 24, 3): Jackson
(17, 28, 3): Endless Amusement II; Pearson
(17, 32, 3): Sylvester
(17, 7, 5): Ripley's
(18, 18, 4): Macmillan
(19, 19, 4): Criton
(19, 9, 5): Endless
Amusement II; The Sociable; Book of 500 Puzzles; Proctor;
Hoffmann; Clark; Wehman;
Ripley; Rudin; Putnam;
Criton
(19, 10, 5): Proctor
(20, 12, 5): trick method: Doubleday - 3
(20, 18, 4): Loyd Jr
(20, 21, 4): Criton
(21, 9, 5): Magician's
Own Book; Book of 500 Puzzles;
Boy's Own Conjuring Book; Blyth; Depew
(21, 10, 5): Mittenzwey
(21, 11, 5): Putnam
(21, 12, 5): Dudeney (1917); Criton
(21, 30, 3): Secret Out; Hoffmann
(21, 50, 3): Sylvester
(22, 15, 5): Macmillan
(22, 20, 4): Dudeney (1899)
(22, 21, 4): Dudeney (1917); Putnam
(24, 28, 3): Jackson; Parlour Pastime
(24, 28, 4): Jackson; Héraud; Benson; Macmillan
(24, 28, 5): Jackson
(25, 12, 5): Endless Amusement II; Young Man's Book; Proctor; Criton
(25, 18, 5): Bridges
(25, 30, 4): Macmillan
(25, 72, 3): Sylvester
(26, 21, 5): Macmillan
(27, 9, 6): The
Sociable; Book of 500 Puzzles; Hoffmann;
Wehman
(27, 10, 6): The Sociable; Book of 500 Puzzles; Wehman
(27, 15, 5): Jackson
(29, 98, 3): Sylvester
(30, 12, 7): Criton
(30, 22, 5): Criton
(30, 26, 5): Macmillan
(31, 6, 6) -- with 7 circles of 6:
The Sociable; Book of 500
Puzzles;
Magician's Own Book (UK version); Wehman
(31, 15, 5): Proctor
(36, 55, 4): Macmillan
(37, 18, 5): Proctor
(37, 20, 5): The Sociable; Book of 500 Puzzles;
Illustrated Boy's Own Treasury; Hanky Panky; Wehman
(49, 16, 7): Criton
Trick versions -- with doubled
counters: Family Friend (1858); Secret Out;
Illustrated Boy's Own Treasury; Hoffmann (1876);
Mittenzwey;
Hoffmann (1893), nos. 8 & 9; Pearson; Home Book ....;
Doubleday - 3. These could also be
considered as in 7.Q.2 or 7.Q.
A different type of
configuration problem is considered by Shepherd, 1947.
Jackson. Rational Amusement. 1821.
Trees Planted in Rows, nos. 1-10, pp. 33-34 & 99-100 and plate IV,
figs. 1-9. [Brooke and others say this
is the earliest statement of such problems.]
1. (9, 10, 3).
Quoted in Burr, below.
"Your aid I want, nine trees to
plant
In rows just half a score;
And let there be in each row three.
Solve this: I ask no
more."
2. (n, n, 3),
He does the case n = 15.
3. (15, 23, 3).
4. (17, 24, 3).
5. (24, 24, 3)
with a pond in the middle.
6. (24, 28, 4).
7. (27, 15, 5)
8. (25, 28, c)
with c = 3, 4, 5.
9. (90, 10, 10) with equal spacing -- decagon with 10 trees on each side.
10.
Leads to drawing square lattice in perspective with two vanishing points, so
the diagonals of the resulting parallelograms are perpendicular.
Endless Amusement II. 1826?
Prob.
13, p. 197. (19, 9, 5). = New Sphinx, c1840, p. 135.
Prob.
14, p. 197. (12, 6, 4). = New Sphinx, c1840, p. 135.
Prob.
26, p. 202. (25, 12, 5). Answer is a
5 x 5 square array.
Ingenious artists, how may I dispose
Of five-and-twenty trees, in just twelve rows;
That every row five lofty trees may grace,
Explain the scheme -- the trees completely
place.
Prob.
35, p. 212. (17, 28, 3). [This is the problem that is replaced in the
1837 ed.]
Young Man's Book. 1839.
P. 239. Identical to Endless
Amusement II.
Crambrook. 1843.
P. 5, no. 15: The Puzzle of the Steward and his Trees. This may be a configuration problem -- ??
Boy's Own Book. 1843 (Paris): 438 & 442, no. 15:
"Is it possible to place twelve pieces of money in six rows, so as to have
four in each row?" I. e. (12, 6, 5).
= Boy's Treasury, 1844, pp. 426 & 429, no. 13. = de Savigny, 1846, pp. 355 & 358, no.
11.
Family Friend 1 (1849) 148 &
177. Family Pastime -- Practical
Puzzles -- 1. The puzzle of the stars.
(9, 10, 3).
Friends
of the Family Friend, pray show
How
you nine stars would so bestow
Ten rows to form -- in each row three
--
Tell
me, ye wits, how this can be?
Robina.
Answer
has
Good-tempered
Friends! here nine stars see:
Ten rows there are, in each row three!
W. S. B. Woolhouse. Problem 39.
The Mathematician 1 (1855) 272.
Solution: ibid. 2 (1856) 278‑280. ??NYS -- cited in Burr, et al., below, who
say he does (15, 26, 3).
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles.
No. 1,
p. 176 (1868: 187). (9, 10, 3).
Ingenious artist pray disclose,
How I nine trees can so dispose,
That these ten rows shall formed be,
And every row consist of three?
No.
12, p. 182 (1868: 192-193). (24, 28,
3), but with a central pond breaking 4
rows of 6 into 8 rows of 3.
Magician's Own Book. 1857.
Prob.
33: The puzzle of the stars, pp. 277 & 300. (9, 10, 3),
Friends one and all, I pray you show
How you nine stars would so bestow,
Ten rows to form -- in each row three --
Tell me, ye wits, how this can be?
Prob.
41: The tree puzzle, pp. 279 & 301.
(21, 9, 5), unequally spaced on
each row. Identical to Book of 500
Puzzles, prob. 41.
The Sociable. 1858.
= Book of 500 Puzzles, 1859, with same problem numbers, but page numbers
decreased by 282.
Prob.
3: The practicable orchard, pp. 286 & 302.
(16, 10, 4).
Prob.
8: The florist's puzzle, pp. 289 & 303-304. (31, 6, 6) with 7 circles
of 6.
Prob.
9: The farmer's puzzle, pp. 289 & 304.
(11, 11, 3).
Prob.
12: The geometrical orchard, p. 291
& 306. (27, 9, 6).
Prob. 17:
The apple-tree puzzle, pp. 292 & 308.
(10, 5, 4).
Prob.
22: The peach orchard puzzle, pp. 294 & 309. (27, 10, 6).
Prob.
26: The gardener's puzzle, pp. 295 & 311.
(12, 6, 4) two ways.
Prob.
27: The circle puzzle, pp. 295 & 311.
(37, 20, 5) equally spaced along
each row.
Prob.
29: The tree puzzle, pp. 296 & 312.
(15, 16, 3) with some bigger
rows. Solution is a 3 x 4
array with three extra trees halfway between the points of the middle
line of four.
Prob.
32: The tulip puzzle, pp. 296 & 314.
(19, 9, 5).
Prob.
36: The plum tree puzzle, pp. 297 & 315.
(9, 10, 3).
Family Friend (Dec 1858)
359. Practical puzzles -- 2. "Make a square with twelve counters,
having five on each side." (12, 4,
5). I haven't got the answer, but
presumably it is the trick version of a hollow square with doubled corners, as
in 7.Q. See Secret Out, 1859 &
Illustrated Boy's Own Treasury, 1860.
Book of 500 Puzzles. 1859.
Prob. 3, 9, 12, 17, 22, 26, 27,
29, 32, 36 are identical to those in
The Sociable, with page numbers decreased by 282.
Prob.
33: The puzzle of the stars, pp. 91 & 114.
(9, 10, 3), identical to
Magician's Own Book, prob. 33.
Prob.
41: The tree puzzle, pp. 93 & 115.
(21, 9, 5), identical to
Magician's Own Book, prob. 41. See
Illustrated Boy's Own Treasury.
The Secret Out. 1859.
To
place twelve Cards in such a manner that you can count Four in every direction,
p. 90. (12, 7, 4) trick of a
3 x 3 array with doubling along
a diagonal. 'Every direction' must
refer to just the rows and columns, but one diagonal also works.
The
magical arrangement, pp. 381-382 = The square of counters, (UK) p. 9. (12, 4, 5) -- trick version. Same as Family Friend & Illustrated
Boy's Own Treasury, prob. 13.
The
Sphynx, pp. 385-386. (21, 30, 3). = Hoffmann, no. 15.
Charades, Enigmas, and
Riddles. 1860: prob. 13, pp. 58 &
61; 1862: prob. 13, pp. 133 &
139; 1865: prob. 557, pp. 105 & 152. (9, 10, 3).
(The 1862 and 1865 have slightly different typography.)
Sir
Isaac Newton's Puzzle (versified).
Ingenious
Artist, pray disclose
How
I, nine Trees may so dispose,
That
just Ten Rows shall planted be,
And
every Row contain just Three.
Boy's Own Conjuring Book. 1860.
Prob.
40: The tree puzzle, pp. 242 & 266.
(21, 9, 5), identical to
Magician's Own Book, prob. 41.
Prob.
42: The puzzle of the stars, pp. 243 & 267. (9, 10, 3), identical to
Magician's Own Book, prob. 33, with commas omitted.
Illustrated Boy's Own
Treasury. 1860.
Prob.
2, pp. 395 & 436. (37, 20, 5), equally spaced on each row, identical to The
Sociable, prob. 27.
Prob.
13, pp. 397 & 438. "Make a
square with twelve counters, having five on each side." (12, 4, 5).
Trick version of a hollow square with doubled corners. Presumably identical to Family Friend,
1858. Same as Secret Out.
J. J. Sylvester. Problem 2473. Math. Quest. from the Educ. Times 8 (1867) 106‑107. ??NYS -- Burr, et al. say he gives (10, 10, 3), (81, 800, 3) and (a, (a‑1)2/8, 3).
Magician's Own Book (UK
version). 1871. The solution to The florist's puzzle (The
Sociable, prob. 8) is given at the bottom of p. 284, apparently to fill
out the page as there is no relevant text anywhere.
Hanky Panky. 1872.
To
place nine cards in ten rows of three each, p. 291. I.e. (9, 10, 3).
Diagram
with no text, p. 128. (37, 20, 5), equally spaced on each line as in The
Sociable, prob. 27.
Hoffmann. Modern Magic. (George Routledge, London, 1876); reprinted by Dover, 1978.
To place twelve cards in rows, in such a manner that they will count
four in every direction, p. 58. Trick
version of a 3 x 3 square with extras on a diagonal, giving a
form of (12, 7, 4). Same as Secret Out.
Lewis Carroll. MS of 1876.
??NYS -- described in: David
Shulman; The Lewis Carroll problem; SM 6 (1939) 238-240.
Given two rows of five dots,
move four to make 5 rows of 4. Shulman
describes this case, following Dudeney, AM, 1917, then observes that since
Dudeney is using coins, there are further solutions by putting a coin on top of
another. He refers to Hoffmann and
Loyd. The same problem is in
Carroll-Wakeling, prob. 1: Cakes in a row, pp. 1-2 & 63, but undated and
the answer mentions the possibility of stacking the counters.
(9, 10, 3). Shulman quotes from Robert T. Philip; Family
Pastime; London, 1852, p. 30, ??NYS, but this must refer to the item in Family
Friend, which was edited by Robert Kemp Philp.
BMC indicates Family Pastime which may be another periodical. Shulman then cites Jackson and Dudeney. Carroll-Wakeling, prob. 2: More cakes in a
row, pp. 3 & 63, gives the problems
(9, 8, 3), (9, 9, 3), (9, 10, 3), undated.
Mittenzwey. 1880.
Prob.
151, pp. 31 & 83; 1895?: 174, pp.
36 & 85; 1917: 174, pp. 33 &
82. (6, 3, 3) in three ways.
Prob.
152, pp. 31 & 83; 1895?: 175, pp.
36 & 85; 1917: 175, pp. 33 &
82. Arrange 16 pennies as a 3 x 3
square so each row and column has four in it. Solution shows a 3 x 3 square with extras on the diagonal -- but
this only uses 12 pennies! So this the
trick version of (12, 7, 4) as in Secret Out & Hoffmann (1876).
Prob.
153, pp. 31 & 83; 1895?: 176, pp.
36 & 85; 1917: 176, pp. 33 &
82. (21, 10, 5).
Cassell's. 1881.
P. 92: The six rows puzzle. =
Manson, 1911, p. 146.
J. J. Sylvester. Problem 2572. Math. Quest. from the Educ. Times 45 (1886) 127‑128. ??NYS -- cited in Burr, below. Obtains good examples of (a, b,
3) for each a. In most cases, this is
still the best known.
Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306
& Three puzzles; Knowledge 9 (Sep 1886) 336-337. (19, 9, 5).
Generalises to (6n+1, 3n, 5).
Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886)
9 & (Dec 1886) 39-40. Gives
several solutions of (19, 9, 5) and asks for (19, 10, 5). Gossip
column, (Feb 1887) 92, gives another solution
William Crompton. The odd half-hour. The Boy's Own Paper 13 (No. 657) (15 Aug 1891) 731-732. Sir Isaac Newton's puzzle (versified). (9, 10, 3).
Ingenious
artist pray disclose
How
I nine trees may so dispose
That
just ten rows shall planted be
And
every row contain just three.
Berkeley & Rowland. Card Tricks and Puzzles. 1892.
Card Puzzles No. IV, p. 3. (9,
10, 3).
Hoffmann. 1893.
Chap. VI, pp. 265‑268 & 275‑281 = Hoffmann-Hordern,
pp. 174-182, with photo.
No. 1:
(11, 12, 3).
No. 2:
(9, 10, 3).
No. 3:
(27, 9, 6).
No. 4:
(10, 5, 4).
No. 5:
(12, 6, 4). Photo on p. 177 shows
L'Embarras du Brigadier, by Mauclair-Dacier, 1891‑1900, which has a board
with a 7 x 6 array of holes and 12 pegs.
The horizontal spacing seems closer than the vertical spacing.
No. 6:
(19, 9, 5).
No. 7:
(16, 10, 4).
No. 8:
(12, 7, 4) -- Trick version of a 3 x
3 square with extras on a diagonal as
in Secret Out, Hoffmann (1876) & Mittenzwey.
No. 9:
9 red + 9 white, form 10 + 8
lines of 3 each. Puts a red and
a white point at the same place, so this is a trick version.
No.
11: (10, 8, 4) -- counts in 8 'directions', so he counts each
line twice!
No.
12: (13, 12, 5) -- with double counting as in no. 11.
No.
15: (21, 30, 3) -- but points must lie on a given figure, which
is the same as in The Secret Out.
Clark. Mental Nuts. 1897, no.
19: The apple orchard; 1904, no. 91:
The lovers' grove. (19, 9,
5). 1897 just has "Place an
orchard of nineteen trees so as to have nine rows of five trees
each." 1904 gives a poem.
I
am required to plant a grove
To
please the lady whom I love.
This
simple grove to be composed
Of
nineteen trees in nine straight rows;
Five
trees in each row I must place,
Or
I shall never see her face.
Cf
Ripley, below.
Dudeney. A batch of puzzles. Royal Magazine 1:3 (Jan 1899) &
1:4 (Feb 1899) 368-372. (22, 20,
4) with trees at lattice points of
a 7 x 10 lattice. Compare with AM,
prob. 212.
Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314-320 &
2:6 (Mar 1900) 598-599
& 3:1 (Apr 1900) 89. (16, 15, 4). Cf 1902.
Dudeney. "The Captain" puzzle corner. The Captain 3:2 (May 1900) 179. This gives a solution of a problem called
Joubert's guns, but I haven't seen the proposal. (10, 5, 4) but wants the
maximum number of castles to be inside the walls joining the castles. Manages to get two inside. = Dudeney; The puzzle realm; Cassell's
Magazine ?? (May 1908) 713-716; no. 6: The king and the castles. = AM, 1917, prob. 206: The king and the
castles, pp. 56 & 189.
H. D. Northrop. Popular Pastimes. 1901. No. 11: The tree
puzzle, pp. 68 & 73. = The
Sociable, no. 29.
Dudeney. The ploughman's puzzle. In:
The Canterbury Puzzles, London Magazine 9 (No. 49) (Aug 1902) 88‑92 &
(No. 50) (Sep 1902) 219.
= CP; 1907; no. 21, pp. 43‑44 & 175‑176. (16, 15, 4). Cf 1899.
A. Héraud. Jeux et Récréations Scientifiques -- Chimie,
Histoire Naturelle, Mathématiques.
Baillière et Fils, Paris, 1903.
P. 307: Un paradoxe mathématique.
(24, 28, 4). I haven't checked
for this problem in the 1884 ed.
Pearson. 1907.
Part
I, no. 77: Lines on an old sampler, pp. 77 & 167. (17, 28, 3).
Part
II, no. 83: For the children, pp. 83 & 177. Trick version of (12, 4,
5), as in Family Friend (1858).
Dudeney. The world's best puzzles. Op. cit. in 2. 1908. He says (9, 10, 3)
"is attributed to Sir Isaac Newton, but the earliest collection of
such puzzles is, I believe, in a rare little book that I possess -- published
in 1821." [This must refer to
Jackson.] Says Rev. Mr. Wilkinson gave (11, 16, 3)
"some quarter of a century ago" and that he, Dudeney,
published (16, 15, 4) in 1897 (cf under 1902 above). He leaves these as problems but doesn't give
their solutions in the next issue.
Wehman. New Book of 200 Puzzles. 1908.
P. 4:
The practicable orchard. (16, 10,
4). = The Sociable, prob. 3.
P. 7:
The puzzle of the stars. (9, 10,
3). = Magician's Own Book, prob. 33.
P. 8:
The apple-tree puzzle. (10, 5, 4). = The Sociable, prob. 17.
P. 8:
The peach orchard puzzle. (27, 10, 6). = The Sociable, prob. 22.
P. 8:
The plum tree puzzle. (9, 10, 3). = The Sociable, prob. 36.
P. 12:
The farmer's puzzle. (11, 11, 3). = The Sociable, prob. 9.
P. 19:
The gardener's puzzle. (12, 6, 4) two ways.
= The Sociable, prob. 26.
P. 26:
The circle puzzle. (37, 20, 5) equally spaced along each row. = The Sociable, prob. 27.
P. 30:
The tree puzzle. (15, 16, 3) with some bigger rows. = The Sociable, prob. 29.
P. 31:
The geometrical orchard. (27, 9,
6). = The Sociable, prob. 12.
P. 31:
The tulip puzzle. (19, 9, 5). = The Sociable, prob. 32.
P. 41:
The florist's puzzle. (31, 6, 6) with seven circles of six. = The Sociable, prob. 8.
J. K. Benson, ed. The Pearson Puzzle Book. C. Arthur Pearson, London, nd [c1910, not in
BMC or NUC]. [This is almost identical
with the puzzle section of Benson, but has 13 pages of different
material.] A symmetrical plantation, p.
99. (24, 28, 4).
Williams. Home Entertainments. 1914.
Competitions with counters, p. 115.
(11, 12, 3); (9, 10,
3); (10, 5, 4).
Dudeney. AM.
1917. Points and lines problems,
pp. 56-58 & 189-193.
Prob.
206: The king and the castles. See The
Captain, 1900.
Prob.
207: Cherries and plums. Two (10, 5, 4)
patterns among 55 of the points of an
8 x 8 array.
Prob.
208: A plantation puzzle. (10, 5,
4) among 45 of the points of a 7 x 7
array.
Prob.
209: The twenty-one trees. (21, 12, 5).
Prob.
210: The ten coins. Two rows of
five. Move four to make (10, 5, 4).
Cf Carroll, 1876. Shows there
are 2400 ways to do this. He shows that
there are six basic solutions of the
(10, 5, 4) which he calls: star, dart, compasses, funnel, scissors,
nail and he describes the smallest arrays on which they can fit.
Prob.
211: The twelve mince-pies. 12 points
at the vertices and intersections of a Star of David. Move four to make (12, 7,
4).
Prob.
212: The Burmese plantation. (22, x,
4) among the points of a 7 x 7
array. Finds x = 21.
Cf 1899.
Prob.
213: Turks and Russians, pp. 58 & 191‑193. Complicated problem leading to
(11, 16, 3) -- cites his Afridi
problem in Tit-Bits and attributes the pattern to Wilkinson 'some twenty years
ago', cf 1908.
Blyth. Match-Stick Magic. 1921.
Four
in line, p. 48. (10, 5, 4).
Three
in line, p. 77. (9, 10, 3).
Five-line
game, pp. 78-79. (21, 9, 5).
King. Best 100. 1927. No. 62, pp. 26 & 54. = Foulsham's no. 21, pp. 9 & 13. (10, 5, 4).
Loyd Jr. SLAHP.
1928. Points and lines puzzle,
pp. 20 & 90. Says Newton proposed (9, 10, 3). Asks for
(20, 18, 4) on a 7 x 7
array.
R. Ripley. Believe It or Not! Book 2. Op. cit. in 5.E,
1931. The planter's puzzle, p. 197,
asks for (19, 9, 5) but no solution is given. See Clark, above, for a better version of
the verse.
"I
am constrained to plant a grove
For
a lady that I love.
This
ample grove is too composed;
Nineteen trees in nine straight
rows.
Five trees in each row I must place,
Or
I shall never see her face."
Rudin. 1936. Nos. 105-108, pp.
39 & 99-100.
No.
105: (9, 10, 3).
No.
106: (10, 5, 4) -- two solutions.
No.
107: (12, 6, 4) -- two solutions.
No.
108: (19, 9, 5).
Depew. Cokesbury Game Book.
1939. The orange grower, p.
221. (21, 9, 5).
The Home Book of Quizzes, Games
and Jokes. Op. cit. in 4.B.1,
1941. P. 147, prob. 1 & 2. Place six coins in an L or
a cross and make two rows of four, i.e.
(6, 2, 4), which is
done by the simple trick of putting a coin on the intersection.
R. H. Macmillan. Letter:
An old problem. MG 30 (No. 289)
(May 1946) 109. Says he believes Newton
and Sylvester studied this. Says he has
examples of (11, 16, 3), (12, 19,
3), (18, 18, 4), (24, 28, 4), (25, 30, 4), (36, 55,
4), (22, 15, 5), (26, 21, 5), (30, 26, 5).
G. C. Shephard. A problem in orchards. Eureka 9 (Apr 1947) 11-14. Given
k points in n‑dimensions, the general problem is
to draw N(k, n) hyperplanes to produce k
regions, each containing one point.
The most common example is k =
7, n = 2, N = 3. [See Section 5.Q
for determining k as a function of n and N.]
The author investigates the question of determining the possible
locations of the seventh point given six points. He gives a construction of a set
T such that being in T is
necessary and sufficient for three such lines to exist.
J. Bridges. Potter's orchard. Eureka 11 (Jan 1949)
30 & 12 (Oct 1949) 17. Start
with an orchard (9, 9, 3). Add 16 trees to make (25, 18, 5). The nine trees are three points in a triangle, with the three
midpoints of the sides and the three points halfway between these. Six of the new trees are one third of the
way along the sides of the original triangle; another six are one third of the
way along the lines joining the midpoints of the original triangle; one point
is the centre of the original triangle and the last three are easily seen.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Thirteen rows of three, pp. 45 &
132. (11, 13, 3).
Young World. c1960.
Pp. 10-11.
Three
coin lines. (9, 10, 3).
Five
coin lines. (10, 5, 4).
Eleven
coin trick. (11, 12, 3).
Maxey Brooke. Dots and lines. RMM 6 (Dec 1961) 51‑55.
Cites Jackson and Dudeney. Says
Sylvester showed that n points can be arranged in at least (n‑1)(n‑2)/6 rows of three. Shows (9, 10, 3) and
(16, 15, 4).
R. L. Hutchings &
J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35. Prob. F.
(10, 5, 4) with points in the
centres of cells of a chess board.
Actually only needs a 7 x 7 board.
Ripley's Puzzles and Games. 1966.
Pp. 18-19, item 4. (17, 7, 5).
Doubleday - 3. 1972.
Count down, pp. 125-126. Start
with a 4 x 4 array of coins. Add four
coins so that each row, column and diagonal has the same number. Solution doubles the coins in the 1, 3, 4, 2
positions in the rows.
S. A. Burr, B. Grünbaum & N.
J. A. Sloane. The orchard problem. Geometria Dedicata 2 (1974) 397‑424. Establishes good examples of (a, b, 3)
slightly improving on Sylvester, and establishes some special better
examples. Gives upper bounds for b in
(a, b, 3). Sketches history and tabulates best values and upper bounds for b in
(a, b, 3), for a = 1 (1) 32.
The
following have the maximal possible value of
b for given a
and c.
(3,
1, 3); (4, 1, 3); (5, 2, 3);
(6, 4, 3); (7, 6, 3); (8, 7, 3);
(9, 10, 3); (10, 12, 3); (11, 16, 3); (12, 19, 3); (16, 37, 3).
The
following have the largest known value of
b for the given a
and c.
(13,
22, 3); (14, 26, 3); (15, 31, 3); (17, 40, 3); (18, 46, 3); (19, 52, 3); (20, 57, 3); (21, 64,
3); (22, 70, 3); (23, 77, 3); (24, 85, 3); (25, 92,
3); (26, 100, 3); (27, 109, 3); (28, 117, 3); (29, 126,
3); (30, 136, 3); (31, 145, 3); (32, 155, 3).
M. Gardner. SA (Aug 1976). Surveys these problems, based on Burr, Grünbaum &
Sloane. He gives results for c = 4.
The
following have the maximal possible value of
b for the given a
and c.
(4,
1, 4); (5, 1, 4); (6, 1, 4);
(7, 2, 4); (8, 2, 4); (9, 3, 4);
(10, 5, 4); (11, 6, 4); (12, 7, 4).
The
following have the largest known value of
b for the given a
and c.
(13,
9, 4); (14, 10, 4); (15, 12, 4); (16, 15, 4); (17, 15,
4); (18, 18, 4); (19, 19, 4); (20, 20, 4).
Putnam. Puzzle Fun.
1978. Nos. 17-23: Bingo
arrangements, pp. 6 & 29-30. (21,
11, 5), (16, 15, 4), (19, 9, 5),
(9, 10, 3), (12, 7, 4), (22, 21, 4), (10, 5, 4).
S. A. Burr. Planting trees. In: The Mathematical
Gardner; ed. by David Klarner; Prindle, Weber & Schmidt/Wadsworth, 1981. Pp. 90‑99. Pleasant survey of the 1974 paper by Burr, et al.
Michel Criton. Des points et des Lignes. Jouer Jeux Mathématiques 3 (Jul/Sep 1991)
6-9. Survey, with a graph showing c
at (a, b). Observes that some solutions have points
which are not at intersections of lines and proposes a more restrictive kind of
arrangement of b lines whose intersections give a
points with c points on each
line. He denotes these with square
brackets which I write as [a, b,
c]. Pictures of (7, 6, 3),
[9, 8, 3], (9, 9, 3), (12, 6, 4),
[13, 9, 4], (13, 12, 3), (13, 22, 3), (16, 12, 4), (19, 19, 4), (19, 19, 5), (20, 21, 4), [21, 12,
5], (25, 12, 5), (30, 12, 7), (30, 22, 5), (49, 16,
7) and mentions of (9, 10, 3),
(16, 15, 4),
6.AO.1. PLACE FOUR POINTS EQUIDISTANTLY = MAKE FOUR TRIANGLES WITH SIX MATCHSTICKS
I am adding the problem of making
three squares with nine matchsticks here a it uses the same thought process --
see Mittenzwey and see the extended discussion at Anon., 1910.
Pacioli. De Viribus.
c1500. Ff. 191r - 192r. LXXX. Do(cumento). commo non e possibile piu
ch' tre ponti o ver tondi spere tocarse in un piano tutti (how it is not
possible for more than three points or discs or spheres to all touch in a
plane). = Peirani 252-253. Says you can only get three discs touching
in the plane, but you can get a fourth so they are all touching by making a
pyramid.
Endless Amusement II. 1826?
Prob. 21, p. 200. "To place
4 poles in the ground, precisely at an equal distance from each other." Uses a pyramidal mound of earth.
Young Man's Book. 1839.
P. 235. Identical to Endless
Amusement II.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 6, p. 178 (1868:
189). Plant four trees at equal
distances from each other.
Frank Bellew. The Art of Amusing. 1866.
Op. cit. in 5.E. 1866: pp. 97-98
& 105-106; 1870: pp. 93‑94
& 101‑102.
Mittenzwey. 1880.
Prob.
161, pp. 32 & 84; 1895?: 184, pp.
37 & 86; 1917: 184, pp. 34 &
83. Use six sticks to make four
congruent triangles. Solution is a
rectangle (should be a square) with its diagonals, but then two of the sticks
have to be longer than the others.
Prob.
163, pp. 32 & 84; 1895?: 186 &
194, pp. 37 & 86-87; 1917: 186
& 194, pp. 34 & 83-84. Use six
equally long sticks to make four congruent triangles -- solution is a
tetrahedron. The two problems in the
1895? are differently phrased, but identical in content, while the first
solution is a picture and the second is a description.
Prob.
171, pp. 33 & 85; 1895?: 195, pp.
38 & 87; 1917: 195, pp. 34 &
84. Use nine equal sticks to make three
squares. Solution is three faces of a
cube.
F. Chasemore. Loc. cit. in 6.W.5. 1891.
Item 3: The triangle puzzle, p. 572.
Hoffmann. 1893.
Chap.
VII, no. 15, pp. 290 & 298 = Hoffmann-Hordern, pp. 195. Four matches.
Chap.
X, no. 19: The four wine glasses, pp. 344 & 381 = Hoffmann-Hordern,
pp. 238‑239, with photo on p. 239 of a version by Jaques & Son,
1870-1900. I usually solve the second
version by setting one glass on top of the other three, but here he wants the
centre of the feet of the glasses to be equally spaced and he turns one glass
over and places it in the centre of the other three, appropriately spaced.
Loyd. Problem 34: War‑ships at anchor. Tit‑Bits 32 (22 May
& 12 Jun 1897) 135 &
193. Place four warships
equidistantly so that if one is attacked, the others can come to assist
it. Solution is a tetrahedron of points
on the earth's oceans.
Parlour Games for
Everybody. John Leng, Dundee &
London, nd [1903 -- BLC], p. 30.
"With 6 matches form 4 triangles of equal size."
Pearson. 1907.
Part III, no. 77: Three squares, p. 77.
Make three squares with nine matches.
Solution is a triangular prism!
Anon. Prob. 66. Hobbies 31 (No. 781) (1 Oct 1910) 2 &
(No. 784) (22 Oct 1910) 68. Use
nine matches to make three squares.
"... the only possible solution" is to make two adjacent
squares with seven matches, then bisect each square to produce a third square
which overlaps the other two.
I
re-invented this problem in Apr 1999 and posted it on NOBNET on 19 Apr
1999. Solution (1) is the idea I had
when I made up the puzzle, but various friends gave more examples and then I
found solution (3).
(1). Arrange the nine matches to form the
following.
_
|_| |_|
|
|
Then 4 is
a square, 9 is a square and 49 is a square.
(2). Use the matches to form a triangular
prism. One may object that this also
makes two triangles.
(3). Make three squares forming three faces of a
cube, all meeting at one corner. Cf
Mittenzwey 171.
(4). Make two adjacent squares with seven of the
matches. Now bisect each of the squares
with a match parallel to the common edge of the squares. This produces a row of four adjacent
half-squares as below. The middle two
form a new square. Here one may object
that the squares are overlapping.
───
───
│
│ │ │ │
───
───
(5). Use the matches to make the figures 0,
1 and 4.
One
can use the matches to make squares whose edge is half the match length, but
one only needs eight matches to make three squares.
There
are other solutions which use the fact that matches have squared off ends and
have square cross-section, but these properties do not hold for paper matches
torn from a matchbook or for other equivalent objects like toothpicks and hence
I don't consider them quite reasonable.
Anon. Prob. 76. Hobbies 31
(No. 791) (10 Dec 1910) 256
& (No. 794) (31 Dec 1910)
318. Make as many triangles as possible
with six matches. From the solution, it
seems that the tetrahedron was expected with four triangles, but many submitted
the figure of a triangle with its altitudes drawn, but only one solver noted
that this figure contains 16 triangles!
However, if the altitudes are displaced to give an interior triangle, I
find 17 triangles!!
Williams. Home
Entertainments. 1914. Tricks with matches: To form four triangles
with six matches, p. 106.
Blyth. Match-Stick
Magic. 1921. Four triangle puzzle, p. 23.
Make four triangles with six matchsticks.
King. Best 100. 1927.
No. 59, pp. 24 & 53. =
Foulsham's no. 20, pp. 8 & 12. Use
six matches to make four triangles.
6.AO.2. PLACE AN EVEN NUMBER ON EACH LINE
See also
section 6.T.
Sometimes the diagonals are considered, but it is not always clear what is intended.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob.
564-31, pp. 254 & 396. From a 6 x 6
array, remove 6 to leave an even number in each row. (The German 'Reihe' can be interpreted as
row or column or both.) If we consider
this in the first quadrant with coordinates going from 1 to 6, the removed
points are: (1,2), (1,3), (2,1), (2,2),
(6,1), (6,3). The use of the sixth
column is peculiar and has the effect of making both diagonals odd, while the
more usual use of the third column would make both diagonals even.
Prob.
583-5, pp. 285 & 403: Von folgenden 36 Punkten sechs zu streichen. As above, but each file ('Zeile') in 'all
four directions' has four or six points.
Deletes: (1,1), (1,2),
(2,2), (2,3), (6,1), (6,3) which makes
one diagonal even and one odd.
Mittenzwey. 1880.
Prob. 154, pp. 31 & 83;
1895?: 177, pp. 36 & 85;
1917: 177, pp. 33 & 82.
Given a 4 x 4 array, remove 6 to leave an even number in
each row and column. Solution removes
a 2 x 3 rectangle from a corner.
[This fails -- it leaves two rows and a diagonal with an odd
number. One can use the idea mentioned
for Leske 564-31 to get a solution with both diagonals also being even.]
Hoffmann. 1893.
Chap. VI, pp. 271-272 & 285 = Hoffmann-Hordern, pp. 186-187.
No.
22: The thirty‑six puzzle. Place
30 counters on a 6 x 6 board so each horizontal and each vertical
line has an even number. Solution
places the six blanks in a
3 x 3 corner in the
obvious way. This also makes the
diagonals have even numbers.
No.
23: The "Five to Four" puzzle.
Place 20 counters on a
5 x 5 board subject to
the above conditions. Solution puts
blanks on the diagonal. This also makes
the diagonals have even number.
Dudeney. The puzzle realm. Cassell's Magazine ?? (May 1908) 713-716. The crack shots. 10 pieces in a 4 x 4 array making the maximal number of even
lines -- counting diagonals and short diagonals -- with an additional
complication that pieces are hanging on vertical strings. The picture is used in AM, prob. 270.
Loyd. Cyclopedia. 1914. The jolly friar's puzzle, pp. 307 &
380. (= MPSL2, no. 155,
pp. 109 & 172. = SLAHP: A
shifty little problem, pp. 64 & 110.)
10 men on a 4 x 4 board -- make a maximal number of even rows,
including diagonals and short diagonals.
This is a simplification of Dudeney, 1908.
King. Best 100. 1927. No. 72, pp. 29 & 56. As in Hoffmann's No. 22, but specifically
asks for even diagonals as well.
The Bile Beans Puzzle Book. 1933.
No. 19: Thirty-six coins. As in
Hoffmann's No. 22, but specifically asks for even diagonals as well.
Rudin. 1936. No. 151, pp. 53-54
& 111. Place 12 counters on a 6 x 6
board with two in each 'row, column and diagonal'. Reading the positions in each row, the
solution is: 16, 34, 25, 25, 34,
16. Some of the short diagonals and
some of the broken diagonals are empty, so he presumably isn't including these,
or he meant to ask for each of these to have an even number of at most two.
M. Adams. Puzzle Book. 1939. Prob. C.179: Even
stars, pp. 169 & 193. Same as Loyd.
Doubleday - 1. 1969.
Prob. 61: Milky Way, pp. 76 & 167.
= Doubleday - 5, pp. 85-86. 6 x
6 array with two opposite corners
already filled. Add ten more counters
so that no row, column or diagonal has more than two counters in it. Reading the positions in each row, the
solution is: 13, 35, 12, 67, 24,
46. Some short diagonals are empty or
have one counter and some broken diagonals have one or four counters, so he
seems to be ignoring them. Hence this
is the same problem as Rudin, but with a less satisfactory solution.
Obermair. Op. cit. in 5.Z.1. 1984. Prob. 37, pp. 38
& 68. 52 men on an 8 x 8
board with all rows, columns and diagonals (both long and short) having
an even number.
6.AP . DISSECTIONS OF A TETRAHEDRON
Richard A. Proctor. Our puzzles; Knowledge 10 (Feb 1887) 83
& Solutions of puzzles; Knowledge 10 (Mar 1887) 108-109. "Puzzle XIX. Show how to cut a regular tetrahedron (equilateral triangular
pyramid) so that the face cut shall be a square: also show how to plug a square
hole with a tetrahedron." Solution
shows the cut clearly.
Edward T. Johnson. US Patent 2,216,915 -- Puzzle. Applied: 26 Apr 1939; patented: 8 Oct 1940. 2pp + 1p diagrams. Described in S&B, p. 46.
E. M. Wyatt. Wonders in Wood. Op. cit. in 6.AI.
1946. Pp. 9 & 11: the
tetrahedron or triangular pyramid. P. 9
is reproduced in S&B, p. 46.
Donovan A. Johnson. Paper Folding for the Mathematics
Class. NCTM, 1957, p. 26, section 62:
Pyramid puzzle. Gives instructions for
making the pieces from paper.
Claude Birtwistle. Editor's footnote. MTg 21 (Winter 1962) 32.
"The following interesting puzzle was given to us recently."
Birtwistle. Math. Puzzles & Perplexities. 1971.
Bisected tetrahedron, pp. 157-158.
Gives the net so one can make a drawing, cut it out and fold it up to
make one piece.
These
dissections usually also work with a tetrahedron of spheres and hence these are
related to ball pyramid puzzles, 6.AZ.
The
first version I had in mind dissects each of the two pieces of 6.AP.1 giving
four congruent rhombic pyramids.
Alternatively, imagine a tetrahedron bisected by two of its midplanes,
where a midplane goes halfway between a pair of opposite edges. This puzzle has been available in various versions
since at least the 1970s, including one from Stokes Publishing Co., 1292
Reamwood Avenue, Sunnyvale, California, 94089, USA., but I have no idea of the
original source. The same pieces are
part of a more complex dissection of a cube, PolyPackPuzzle, which was produced
by Stokes in 1996. (I bought mine from
Key Curriculum Press.)
In
1997, Bill Ritchie, of Binary Arts, sent a quadrisection of the tetrahedron
that they are producing. Each piece is
a hexahedron. The easiest way to
describe it is to consider the tetrahedron as a pile of spheres with four on an
edge and hence 20 altogether.
Consider a planar triangle of six of these spheres with three on an edge
and remove one vertex sphere to produce a trapezium (or trapezoid) shape. Four of these assemble to make the
tetrahedron. Writing this has made me
realise that Ray Bathke has made and sold these 5-sphere pieces as
Pyramid 4 for a few years. However, the
solid pieces used by Binary Arts are distinctly more deceptive.
Len Gordon
produced another quadrisection of the
20 sphere tetrahedron 0 0
using the planar shape at the right. This was c1980?? 0 0
0
David Singmaster. Sums of squares and pyramidal numbers. MG 66 (No. 436) (Jun 1982) 100-104. Consider a tetrahedron of spheres with 2n
on an edge. The quadrisection
described above gives four pyramids whose layers are the squares 1, 4, ..., n2. Hence
four times the sum of the first
n squares is the tetrahedral
number for 2n, i.e. 4 [1 + 4 + ...
+ n2] = BC(n+2, 3).
6.AQ. DISSECTIONS OF A CROSS, T OR H
The
usual dissection of a cross has two diagonal cuts at 45o to the
sides and passing through two of the reflex corners of the cross and yielding
five pieces. The central piece is six-sided,
looking like a rectangle with its ends pushed in and being symmetric. Depending on the relative lengths of the
arms, head and upright of the cross, the other pieces may be isosceles right
triangles or right trapeziums. Removing
the head of the cross gives the usual dissection of the T
into four pieces -- then the central piece is five-sided. Sometimes the central piece is split in
halves. Occasionally the angle of the
cuts is different than 45o. Dissections of an H have the same basic idea
of using cuts at 45o -- the result can be a bit like two Ts
with overlapping stems and the number of pieces depends on the relative
size and positioning of the crossbar of the
H -- see: Rohrbough.
S&B, pp. 20‑21, show
several versions. They say that crosses
date from early 19C. They show a 6‑piece
Druid's Cross, by Edwards & Sons, London, c1855. They show several T‑puzzles
-- they say the first is an 1903 advertisement for White Rose Ceylon Tea, NY --
but see 1898 below. They also show some
H‑puzzles.
Charles Babbage. The Philosophy of Analysis -- unpublished
collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more
details. F. 4r is "Analysis of the
Essay of Games". F. 4v has a cross
cut into 5 pieces in the usual way.
Endless Amusement II. 1826?
Prob. 30, p. 207. Usual five
piece cross. The three small pieces are
equal. = New Sphinx, c1840, pp.
139-140.
Crambrook. 1843.
P. 4.
No.
10: Five pieces to form a Cross.
No.
11: The new dissected Cross.
Without pictures, I cannot tell what
dissections are used??
Boy's Own Book. 1843 (Paris): 435 & 440, no. 2. Usual five piece cross, very similar to
Endless Amusement. One has to make
three pieces of fig. 2. = Boy's
Treasury, 1844, pp. 424 & 428. = de
Savigny, 1846, pp. 353 & 357, no. 1.
Family Friend 2 (1850) 58 &
89. Practical Puzzle -- No. II. = Illustrated Boy's Own Treasury, 1860, No.
32, pp. 401 & 440. Usual five piece
cross to "form that which, viewed mentally, comforts the
afflicted." Three pieces of fig.
1.
Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 7, p. 178-179 (1868:
189). Five piece dissection of a cross,
but the statement of the problem doesn't say which piece to make multiple
copies of.
Magician's Own Book. 1857.
Prob. 17: The cross puzzle, pp. 272 & 295. Usual 5 piece cross, essentially identical to Family Friend,
except this says to "form a cross."
= Book of 500 Puzzles, 1859, prob. 17, pp. 86 & 109. = Boy's Own Conjuring Book, 1860, prob. 16,
pp. 234 & 258.
Charades, Enigmas, and
Riddles. 1860: prob. 33, pp. 60 &
66; 1862: prob. 33, pp. 136 &
143; 1865: prob. 577, pp. 108 & 156. Usual five piece cross, showing all five
pieces.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 584-12, pp.
288 & 406: Ein Kreuz. Begins as the
usual five piece cross, but the central piece is then bisected into two mitres
and the base has two bits cut off to give an eight piece puzzle.
Frank Bellew. The Art of Amusing. 1866.
Op. cit. in 5.E. 1866: pp.
239-240; 1870: pp. 236‑238. Usual five piece cross.
Elliott. Within‑Doors. Op. cit. in 6.V. 1872. Chap. 1, no. 1: The
cross puzzle, pp. 27 & 30. Usual
five piece cross, but instructions say to cut three copies of the wrong piece.
Mittenzwey. 1880.
Prob. 188, pp. 35 & 88;
1895?: 213, pp. 40 & 91;
1917: 213, pp. 37 & 87. This
is supposed to be a 10 piece dissection of a cross obtained by further
dissecting the usual five pieces.
However, pieces 3 & 4 are drawn as trapezoids in the problem and
triangles in the solution and piece 2 in the solution is half the size given in
the problem. Further, pieces 1 & 2
appear equilateral in the problem, but are isosceles right triangles in the
solution. One could modify this to get
a 9 piece version where eight of the pieces are right trapezoids -- four having
edges 1, 1, 2, Ö2 and four having edges Ö2, Ö2, 2Ö2, 2, but the arms would be twice as long as they
are wide. Or one can make the second
four pieces be Ö2, Ö2,
2 isosceles right triangles. In either case, the ninth piece would be a
rectangle.
Lemon. 1890. A card board
puzzle, no. 33, pp. 8 & 98. Usual
five piece cross.
Hoffmann. 1893.
Chap. III, no. 12: The Latin cross puzzle, pp. 93 & 126
= Hoffmann‑Hordern, pp. 82-83, with photo. As in Indoor & Outdoor.
Photo on p. 83 shows two versions: one in metal by Jaques & Sons,
1870-1895; the other in ivory, 1850-1900.
Hordern Collection, p. 59, shows a Druid's Cross Puzzle.
Lash, Inc. -- Clifton, N.J. --
Chicago, Ill. -- Anaheim, Calif. T Puzzle.
Copyright Sept. 1898. 4‑piece T
puzzle to be cut out from a paper card, but the angle of the cuts is
about 35o instead of
45o which makes it
less symmetric and less confusing than the more common version. The resulting T is somewhat wider than
usual, being about 16% wider than it is tall. It advertises: Lash's Bitters The
Original Tonic Laxative. Photocopy sent
by Slocum.
Benson. 1904.
The cross puzzle, pp. 191‑192.
Usual 5 piece version.
Wehman. New Book of 200 Puzzles. 1908.
The cross puzzle, p. 17. Usual 5
piece version.
A. Neely Hall. Op. cit. in 6.F.5. 1918. The T‑puzzle,
pp. 19‑20. "A famous old
puzzle ...." Usual 4‑piece
version, but with long arms.
Western Puzzle Works, 1926
Catalogue. No. 1394: Four pieces to
form Letter T. The notched piece is less symmetric than
usual.
Collins. Book of Puzzles. 1927. The crusader's
cross puzzle, pp. 1-2. The three small
pieces are equal.
Arthur Mee's Children's
Encyclopedia 'Wonder Box'. The
Children's Encyclopedia appeared in 1908, but versions continued until the
1950s. This looks like 1930s?? Usual 5 piece cross.
A. F. Starkey. The
T puzzle. Industrial Arts and Vocational Education 37
(1938) 442. "An interesting
novelty ...."
Rohrbough. Puzzle Craft. 1932. The "H"
Puzzle, p. 23. Very square H --
consider a 3 x 3 board with the top and bottom middle cells
removed. Make a cut along the main
diagonal and two shorter cuts parallel to this to produce four congruent
isosceles right triangles and two odd pentagons.
See Rohrbough in 6.AS.1 for a
very different T puzzle.
6.AR. QUADRISECTED SQUARE PUZZLE
This
is usually done by two perpendicular cuts through the centre. A dissection proof of the Theorem of
Pythagoras described by Henry Perigal (Messenger of Mathematics 2 (1873) 104)
uses the same shapes -- cf 6.AS.2.
The
pieces make a number of other different shapes.
Crambrook. 1843.
P. 4, no. 17: Four pieces to form a Square. This might be the dissection being considered here??
A. Héraud. Jeux et Récréations Scientifiques -- Chimie,
Histoire Naturelle, Mathématiques.
(1884); Baillière, Paris, 1903. Pp. 303‑304: Casse‑tête. Uses two cuts which are perpendicular but
are not through the centre. He claims
there are 120 ways to try to assemble it, but his mathematics is shaky -- he
adds the numbers of ways at each stage rather than multiplying! Also, as Strens notes in the margin of his
copy (now at Calgary), if the crossing is off-centre, then many of the edges have
different lengths and the number of ways to try is really only one. Actually, I'm not at all sure what the
number of ways to try is -- Héraud seems to assume one tries each orientation
of each piece, but some intelligence sees that a piece can only fit one way
beside another.
Handy Book for Boys and
Girls. Op. cit. in 6.F.3. 1892.
P. 14: The divided square puzzle.
Crossing is off-centre.
Tom Tit, vol 3. 1893.
Carré casse-tête, pp. 179-180. =
K, no. 26: Puzzle squares, pp. 68‑69.
= R&A, Puzzling squares, p. 99.
Not illustrated, but described:
cut a square into four parts by two perpendicular cuts, not necessarily
through the centre.
A. B. Nordmann. One Hundred More Parlour Tricks and
Problems. Wells, Gardner, Darton &
Co., London, nd [1927 -- BMC]. No. 77:
Pattern making, pp. 69-70 & 109.
Make five other shapes.
M. Adams. Puzzle Book. 1939. Prob. C.12: The
broken square, pp. 125 & 173. As
above, but notes that the pieces also make a square with a square hole.
6.AS. DISSECTION OF SQUARES INTO A SQUARE
Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. P. 102 asserts that dissections of squares
to various hexagons and heptagons were known c1800 while square to rectangle
dissections were known to Montucla -- though she illustrates the latter with
examples like 6.Y, she must mean 6.AS.5.
6.AS.1. TWENTY 1, 2, Ö5 TRIANGLES MAKE A SQUARE OR FIVE EQUAL SQUARES TO A SQUARE
The
basic puzzle has been varied in many ways by joining up the 20 triangles into various
shapes, but I haven't attempted to consider all the modern variants. A common form is a square with a skew # in
it, with each line joining a corner to the midpoint of an opposite side, giving
the 9 piece version. This has four of
the squares having a triangle cut off.
For symmetry, it is common to cut off a triangle from the fifth square,
giving 10 pieces, though the assembly into one square doesn't need this. See Les Amusemens for details.
Cf
Mason in 6.S.2 for a similar puzzle with twenty pieces.
If
the dividing lines are moved a bit toward the middle and the central square is
bisected, we get a 10 piece puzzle, having two groups of four equal pieces and
a group of two equal pieces, called the Japan square puzzle. I have recently noted the connection of this
puzzle with this section, so there may be other examples which I have not
previously paid attention to -- see:
Magician's Own Book, Book of 500
Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Landells,
Hanky Panky, Wehman.
Les Amusemens. 1749.
P. xxxii. Consider five 2 x 2
squares. Make a cut from a
corner to the midpoint of an opposite side on each square. This yields five 1, 2, Ö5 triangles and five pieces comprising three such triangles. The problem says to make a square from five
equal squares. So this is the 10 piece
version.
Minguet. 1755.
Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Not in 1733 ed. 10 piece version. Also a
15 piece version where triangles
are cut off diagonally opposite corners of each small square leaving
parallelogram pieces as in Guyot.
Vyse. Tutor's Guide. 1771? Prob. 6, 1793: p. 304, 1799: p. 317 &
Key p. 357. 2 x 10 board to be cut into five pieces to make
into a square. Cut into a 2 x 2
square and four 2, 4, 2Ö5 triangles.
Ozanam‑Montucla. 1778.
Avec cinq quarrés égaux, en former un seul. Prob. 18 & fig. 123, plate 15, 1778: 297; 1803: 292-293; 1814: 249-250; 1840:
127. 9 piece version. Remarks that any number of squares can be
made into a square -- see 6.AS.5.
Catel. Kunst-Cabinet. 1790.
Das
mathematische Viereck, pp. 10-11 & fig. 15 on plate I. 10 piece version with solution shown. Notes these make five squares.
Das
grosse mathematische Viereck, p. 11 & fig. 14 on plate I. Cut the larger pieces to give five more 1, 2, Ö5 triangles and five Ö5, Ö5, 2 triangles.
Again notes these make five squares.
Guyot. Op. cit. in 6.P.2.
1799. Vol. 2: première
récréation: Cinq quarrés éqaux étant sonnés, en former un seul quarré, pp. 40‑41
& plate 6, opp. p. 37. 10 piece
version. Suggests cutting another
triangle off each square to give 10 triangles and 5 parallelograms.
Bestelmeier. 1801.
Item 629: Die 5 geometrisch zerschnittenen Quadrate, um aus 5 ein
einziges Quadrat zu machen. As in Les
Amusemens. S&B say this is the
first appearance of the puzzle. Only
shown in a box with one small square visible.
Jackson. Rational Amusement. 1821.
Geometrical Puzzles.
No. 8,
pp. 25 & 84 & plate I, fig. 5, no. 1.
= Vyse.
No.
10, pp. 25 & 84-85 & plate I, fig. 7, no. 1. Five squares to one. Nine
piece version.
Rational Recreations. 1824.
Feat 35, pp. 164-166. Usual 20
piece form.
Manuel des Sorciers. 1825.
Pp. 201-202, art. 18. ??NX Five squares to one -- usual 10 piece form and
15 piece form as in Guyot.
Endless Amusement II. 1826?
[1837
only] Prob. 35, p. 212. 20 triangles to form a square. = New Sphinx, c1840, p. 141, with
problem title: Dissected square.
Prob.
37, p. 215. 10 piece version. = New Sphinx, c1840, p. 141.
Boy's Own Book. The square of triangles. 1828: 426;
1828-2: 430; 1829 (US):
222; 1855: 576; 1868: 676.
Uses 20 triangles cut from a square of wood. Cf 1843 (Paris) edition, below.
c= de Savigny, 1846, p. 272: Division d'un carré en vingt triangles.
Nuts to Crack IV (1835), no.
195. 20 triangles -- part of a long
section: Tricks upon Travellers. The
problem is used as a wager and the smart-alec gets it wrong.
The Riddler. 1835.
The square of triangles, p. 8.
Identical to Boy's Own Book, but without illustration, some consequent
changing of the text, and omitting the last comment.
Crambrook. 1843.
P. 4.
No. 7:
Egyptian Puzzle. Probably the 10 piece
version as in Les Amusemens. See
S&B below, late 19C. Check??
No.
23: Twenty Triangles to form a Square.
Check??
Boy's Own Book. 1843 (Paris): 436 & 441, no. 5:
"Cut twenty triangles out of ten square pieces of wood; mix them together,
and request a person to make an exact square with them." As stated, this is impossible; it should be
as in Boy's Own Book, 1828 etc., qv. =
Boy's Treasury, 1844, pp. 425 & 429.
= de Savigny, 1846, pp. 353 & 357, no. 4. Also copied, with the error, in:
Magician's Own Book, 1857, prob. 29: The triangle puzzle; Book of 500 Puzzles, 1859, prob. 29: The
triangle puzzle; Boy's Own Conjuring
Book, 1860, prob. 28: The triangle puzzle.
c= Hanky Panky, 1872, p. 122.
Magician's Own Book. 1857.
How to
make five squares into a large one without any waste of stuff, p. 258. 9 piece version.
Prob.
29: The triangle puzzle, pp. 276 & 298.
Identical to Boy's Own Book, 1843 (Paris).
Prob.
35: The Japan square puzzle, pp. 277 & 300. Make two parallel cuts and then two perpendicular to the first
two so that a square is formed in the centre.
This gives a 9 piece puzzle, but here the central square is cut by a
vertical through its centre to give a 10 piece puzzle. = Landells, Boy's Own Toy-Maker, 1858, pp.
145-146.
Charles Bailey (manufacturer in
Manchester, Massachusetts). 1858. An Ingenious Puzzle for the Amusement of
Children .... The 10 pieces of Les
Amusemens, with 19 shapes to make, a la tangrams. Sent by Jerry Slocum -- it is not clear if there were actual
pieces with the printed material.
The Sociable. 1858.
Prob.
10: The protean puzzle, pp. 289 & 305-306. Cut a 5 x 1 into 11 pieces to form eight shapes, e.g. a
Greek cross. It is easier to describe the
pieces if we start with a 10 x 2. Then three squares are cut off. One is halved into two 1 x 2
rectangles. Two squares have
two 1, 2, Ö5
triangles cut off leaving triangles of sides 2, Ö5, Ö5.
The remaining double square is almost divided into halves each with
a 1, 2, Ö5 triangle cut off, but these two triangles remain connected along
their sides of size 1, thus giving a 4, Ö5, Ö5
triangle and two trapeziums of sides
2, 2, 1, Ö5. = Book of 500 Puzzles, 1859, prob. 10, pp. 7 & 23-24.
Prob.
42: The mechanic's puzzle, pp. 298 & 317.
Cut a 10 x 2 in five pieces to make a square, as in Vyse. = Book of 500 Puzzles, 1859, prob. 16, pp.
16 & 35.
Book of 500 Puzzles. 1859.
Prob.
10: The protean puzzle, pp. 7 & 23-24.
As in The Sociable.
Prob.
42: The mechanic's puzzle, pp. 16 & 35.
As in The Sociable.
How to
make five squares into a large one without any waste of stuff, p. 72. Identical to Magician's Own Book.
Prob.
29: The triangle puzzle, pp. 90 & 113.
Identical to Boy's Own Book, 1843 (Paris).
Prob.
35: The Japan square puzzle, pp. 91 & 114.
Indoor & Outdoor. c1859.
Part II, prob. 11: The mechanic's puzzle, pp. 130-131. Identical to The Sociable.
Boy's Own Conjuring Book. 1860.
Prob.
28: The triangle puzzle, pp. 238 & 262.
Identical to Boy's Own Book, 1843 (Paris) and Magician's Own Book.
Prob.
34: The Japan square puzzle, pp. 240 & 264. Identical to Magician's Own Book.
Illustrated Boy's Own
Treasury. 1860.
Prob.
9, pp. 396 & 437. [The Japan square
puzzle.] Almost identical to Magician's
Own Book.
Optics:
How to make five squares into a large one without any waste of stuff, p.
445. Identical to Book of 500 Puzzles,
p. 72.
Vinot. 1860. Art. LXXV: Avec
cinq carrés égaux, en faire un seul, p. 90.
Nine piece version.
Leske. Illustriertes Spielbuch für Mädchen. 1864?
Prob.
174, pp. 87-88. Nine piece version.
Prob.
584-6, pp. 287 & 405. Ten piece
version of five squares to one.
Hanky Panky. 1872.
The
puzzle of five pieces, p. 118. 9 piece
version.
Another
[square] of four triangles and a square, p. 120. 10 x 2 into five pieces
to make a square.
[Another
square] of ten pieces, pp. 121-122.
Same as the Japan square puzzles in Magician's Own Book.
[Another
square] of twenty triangles, p. 122.
Similar to Boy's Own Book, 1843 (Paris), but with no diagram and less
text, making it quite cryptic.
Mittenzwey. 1880.
Prob. 175, pp. 33-34 & 85;
1895?: 200, pp. 38 & 87; 1917:
200, pp. 35 & 84. 10 pieces as in
Les Amusemens. See in 6.AS.2 and 6.S.2
for the use of these pieces to make other shapes.
See Mason, 1880, in 6.S.2 for a
similar, but different, 20 piece puzzle.
S&B, pp. 11 & 19, show a
10 piece version called 'Egyptian Puzzle', late 19C?
Lucas. RM2. 1883. Les vingt triangles, pp. 128‑129. Notes that they also make five squares in
the form of a cross.
Tom Tit, vol. 2. 1892.
Diviser un carré en cinq carrés égaux, pp. 147‑148. = K, no. 2: To divide a square into five
equal squares, pp. 12-14. = R&A,
Five easy pieces, p. 105. Uses 9
pieces, but mentions use of 10 pieces.