Sources in recreational mathematics

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Copyright ©2003 Professor David Singmaster

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Last revised on 21 Aug 2000.

Last updated on 4 September 2018.

The version of 4 Aug 1996 is on the following.

Word version made on 21 Mar 2004, but I haven't checked on details, e.g. diacritical marks.

I would be grateful for any information you can provide on the problems sketched below. My assertions should be treated as implied questions   a statement like "The earliest reference is ...." should be considered as "The earliest reference I know is .... Do you know any earlier ones or can you confirm this is the earliest?"

Many of these topics have originated in publications which are difficult to search for   e.g. puzzle columns in newspapers and magazines, medieval and oriental books and manuscripts, unpublished letters, etc. I would be especially grateful for detailed references to such items, and photocopies. In particular, I have found a number of references which are not sufficiently detailed to locate the item of interest. Any help with such items would be appreciated. Any information on translations of sources would be useful.

I would also be grateful for references to biographical and bibliographical articles, especially on particular authors or problems. Interviews, obituaries and photographs are of especial interest.

The sections below cover a number of topics. A very few topics are well enough known that I have no questions (yet). Suggestions for other topics would be useful.

The first version of this letter generated so much response that I had to make this into a computer file. Many thanks to all of you who have helped resolve questions.

This version is ordered in the ad hoc arrangement I have used in my Sources in Recreational Mathematics, presently about 600pp long. The 'Abbreviations and Common References' section of Sources may be attached if the entire document is not.


1. Biographical Items

2. General Puzzle Collections and Surveys

3. General Historical Articles

4. Mathematical Games

5. Combinatorial Recreations

6. Geometric Recreations

7. Arithmetical & Number theoretic Recreations

8. Probability Recreations

9. Logical Recreations

10. Physical Recreations

11. Topological Recreations

I have found a number of topics which have particular geographic connections and I have prepared special letters covering: Oriental queries; Middle Eastern queries (i.e. Egyptian, Babylonian, Indian, Arabic, Persian and Turkish) and Russian queries.

I have some recent books and compilations from the works of Y. I. Perelman. He began writing in 1913 and died in 1943. However, the books are from the late 70s and the 80s. Some of the problems would have been quite new if they appeared about 1920. Hence I am very interested in obtaining information about earlier versions of his works   in any language.

I have a vague reference to an Arabic book: Ketabol Algaz. Can anyone identify this?

I have a recent Spanish edition of a Russian work by Ignatiev, 1908. I'd be interested in seeing an early edition in any language.

I would like to see copies of Our Puzzle Magazine by Sam Loyd.

In the mid-19C, there was a profusion of books, beginning with The Magician's Own Book (1857), many of which are pseudonymous and I have been unable to determine the authors. One version of The Secret Out (1859) says it is based on Le Magicien des Salons (this may be Le Magicien de Société, Delarue, Paris, c1860? - I have seen it advertised in another of their books from 1861), and I have a reference to Le Manuel des Sorciers (various Paris editions from 178?-1825), so perhaps this era was inspired by some earlier French books?? One bibliographer doubts whether Le Magicien des Salons exists. I have recently 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. 3 of 1850 and the volume for Jul Dec 1859 both contain a number of the problems which appear repeatedly and identically in the above cited books. Toole Stott cites 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. Any help in clarifying the bibliographic confusion here would be much appreciated.

I am looking for relevant pictures. I have located pictures of Ahrens, Bachet, Ball, Carroll, Dudeney, Hoffmann, Loyd, Lucas, Phillips, Schubert. Does anyone know of any published picture of Alcuin, Leurechon, Montucla or Ozanam? I have discovered a pencil sketch in a 1696 copy of Ozanam's book which claims to be him.


Y. I. Perelman [Fun with Maths and Physics] describes a Russian MS by the poet V. G. Benediktov (1807 1873), dated c1869, as the first puzzle collection in Russian. Has this ever been published? Does the MS still exist? Where?


Did Part II: Angewandte Mathematik of: Ernst Wölffing; Mathematischer Bücherschatz. Systematisches Verzeichnis der wichtigsten deutschen und ausländischen Lehrbücher und Monographien des 19. Jahrhunderts auf dem Gebiete der mathematischen Wissenschaften; AGM 16, part II, 1903(?) ever appear??


4.A.1. One Pile Game. This is the game where one can take 1 to m at a time from a pile. A version and a general discussion are in De Viribus and in Bachet (Prob. 22), then in Ozanam and Alberti. Loyd's "Blind Luck" may be a version of it?? E. Ducret, Recreations Mathematiques [1900?], calls it 'le jeu du piquet a cheval', and gives some explanation of the name. Les Amusemens (1749) calls it Piquet des Cavaliers and says it was played while riding. When does this name originate?

There are versions using a die (Loyd, 1914) and with a restriction on the number of times you can use a number (Ball, MRE 5th ed., 1911).

There is a 20C(?) version where the winner is the one who takes an odd number of matches in total. I think I recall this in Dudeney??

4.A.1.a. The 31 Game. It is claimed that this was invented by Charles James Fox (1749-1806), but my earliest reference is 1857.

4.A.2. Symmetry Arguments. This refers to the idea that a player can win by playing symmetrically in some way. It appears both in Loyd's Cyclopedia and Dudeney's AM. I am inclined to believe Dudeney may have discovered it. Dudeney gave it as his 500th problem in The Weekly Dispatch (7 Jun 1903) and it occurs in a version of Kayles, ibid., (26 Jan 1902).

Kraitchik (Math. des Jeux & Math. Rec.) gives the problem of the child who agrees to play chess with two masters and bets she will score a point. She plays white against one and black against the other!

4.A.3. Kayles. This occurs in both Dudeney and Loyd. Gardner (MPSL 2) asserts that Dudeney invented it, though it is Loyd's column in Tit Bits.

4.A.4. Nim. Is there any reference prior to Bouton (1901/2)? David Parlett says 16C references to 'Les luettes' seem to suggest a game of the Nim family. Many popular accounts claim it is Chinese but this is unsubstantiated. Ahrens (Naturwissenschaftliche Wochenschrift (1902)) says Bouton admits that Nim is not the same as Fan Tan, as he had claimed in his paper. I have seen descriptions of Fan Tan and it is much different. I have recently seen a letter in MG asserting that Bouton invented the word 'Nim'. Domoryad, Brandreth and Winning Ways assert that Wythoff's Nim is an ancient oriental game.

4.A.5. General Theory. The origin of the result that a game without ties has a strategy for one player or the other seems to be due to Zermelo (1912). Ahrens (A&N, 1918) mentions the idea fairly clearly. Steinhaus attributes the proof using the fact that a first player strategy is the negation of a second player strategy to Mycielski, but without a reference (AMM 72 (1965) 400; see also Kac, AMM 81 (1974) 577). Babbage's MSS of 1820-1860 show some general analysis, including the tree of a game.

4.B.1. Tic Tac Toe = Noughts and Crosses. Are there any ancient references? The earliest mathematical approach, indeed my earliest reference at all, is in Babbage's unpublished MSS The Philosophy of Analysis, c1820, though Wordsworth may refer to it in his The Prelude (1805). The first complete analysis seems to be 1892 in Boys' Own Magazine. A. C. White gives an analysis in British Chess Magazine (Jul 1919).

4.B.1.a. In Higher Dimensions. The first I have is Funkenbusch & Eagle, National Mathematics Mag. (1944) NYR, but Eureka 11 (1949) 5 9 says the game was at Cambridge in 1940 and some people recall playing 4-D versions about that time, while others recall the 3 x 3 x 3 version where you have to get the most rows from the 1930s.

4.B.2. Hex. I have a reference to the Danish newspaper Politiken (26 Dec 1942) - Mogens Larsen has sent me copies, NYR. The Parker Brothers instructions are from about 1952, but I have not yet seen them. Are there any other early Danish references? John Nash also invented the game   did he ever write anything on it? Who originated the strategy stealing argument?

4.B.3. Dots and Boxes. Lucas, RM2, 90 91, calls it Le jeu de l'Ecole Polytechnique and implies it is fairly recent. In his L'Arithmetique Amusante (1895), the editors have included his booklet: La Pipopipette   Nouveau Jeu de Combinaisons (1889). I think he may be asserting that he invented the game ?? Lucas also has an article in La Nature (1889): 'un nouveaux jeu ... dedie aux eleves de l'ecole Polytechnique'. Ahrens, A&N, describes it as recent. Ahrens's version has the boundary lines already in place. This is not clear to me from Lucas' description. Conway told me that it came from the early 19C, but perhaps this is a misremembrance?? Sam Loyd has it as 'The Boxer's Puzzle' in the Cyclopedia and says it is 'from the East'. Has it been played on other lattices?

4.B.4. Sprouts. So far as I know, this appeared first in Gardner's SA column (Jul 1967).

4.B.5. Ovid's Game and Nine Men's Morris. I am gathering information on this type of game.

4.B.6. Phutball. Did Conway describe this anywhere before Winning Ways?

4.B.9. Snakes and Ladders. The 7C Chinese Game of Promotion appears to be a version of this - can anyone provide more details? I am interested in the mathematics - the game can be viewed as a Markov process and the expected length can be worked out. Reference to such work would be appreciated.

4.B.11. Mastermind, etc. Mastermind is said to be a modern development of various older games, but I have no references to such games, except Reddi in JRM (1975).

4.B.13. Mancala Games. I am looking for early material on this - perhaps in anthropological journals??


5.A. The 15 Puzzle. Dudeney ("The Psychology of Puzzle Crazes" (1926)) dates this to 1873. Loyd's Cyclopedia says 'early seventies'. Lucas, Ahrens and Schaaf give 1878. The earliest mathematical articles are in the Amer. J. Math. (1879), though this may not have appeared until early 1880?? The next I have is a New York Times piece on 23 Feb 1880 where it is described as a mental epidemic. Ahrens cites some 1879 items and many 1880 items. I have several popular articles and a solution booklet from 1880. SLAHP says early 80s. The New York Times (22 Mar 1880) says it appeared in Boston a few months ago. Do you know any other contemporary references? Many of the early references in Ahrens are obscure and I would appreciate copies of any that you come across. Edward Hordern and Jerry Slocum now believe that Loyd did not invent the puzzle, but merely the 15 14 problem, but we don't know where or when. The earliest example is one by the Embossing Co. which claims to be patented on 24 Oct 1865, but the patent has not been found. I'd like to get an example or photocopy of it.

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. I call these 'one-piece problems' and I don't know any earlier example.

5.A.2. Three Dimensional Version. P. G. Tait describes this as "conceivable, but scarcely realisable" in PRSE 10 (1880) 664 665. Hordern cites 1889 and 1905 patents for the idea. Rouse Ball, MRE (1st ed., 1892) mentions the idea. Gardner is the first to describe Hein's Bloxbox in SA (Feb 1973). Did Hein write anything? Is there a patent? Are there any earlier references in Europe?

5.A.4. Panex Puzzle. Can anyone provide more details about the origin of this - particularly the date. Also, what is the date of the paper by Manasse, et al. - and was it ever published?

5.B. Crossing Problems. Wolf, goat, cabbages; 3 couples; man, wife and 2 small children are all in Alcuin. Pacioli, De Viribus, says 4 or 5 couples requires a 3 person boat. Tartaglia gives 4 couples, erroneously. Bachet points this out and shows that 4 couples can be done with a 3 person boat. What is the origin of the missionaries and cannibals version? H&S (1927) says it is 'a modern variant'. Jackson (1821) and Mittenzwey (1879?) give equivalent versions with masters and servants. The island in the river version is due to De Fonteney or De Fontenay - can anyone provide the correct spelling of the name?? Pressman and I found more efficient solutions for n couples and an island, but later contemplation reveals that the jealousy conditions are not clear and our solutions are perhaps not acceptable.

5.C. False Coins with a Balance. Schell's Problem E651, AMM 52:1 (Jan 1945) 42 appears to be the source of this problem though it deals with finding at most one light coin among 8 coins in two weighings. There is a letter in MM (1978) stating that Schell said he had invented the problem. However Paul Campbell has corresponded with Schell, who says he did not invent the problem and that he submitted the problem of finding at most one light coin among 26. In this form, it is beginning to look like a development from ternary or weighing with 1, 3, 9, .... If there is known to be one one light coin, it can be located among 3n in n weighings. This version is discussed by Karapetoff in SM 11 (1945) 186 187, where he cites AMM 52, p. 314, but this seems to be an erroneous citation, as I can't find anything in all of vol. 52, except Schell's problem. If there is at most one light coin, it can be located among 3n   1 in n weighings   this is the form submitted by Schell, for n = 3 and simplified to n = 2 by the AMM problem editor (who was he??). Schell says he heard the problem from Walter W. Jacobs, who replied to Campbell that he had heard it by late 1943 and he would try to contact the two people who might have told it to him, but he has not written further. There are rumours that Eilenberg says it dates back to 1939. From the above two forms of the problem, it would be easy to change to one false coin, either heavy or light. Schell solved Problem E712 in AMM 54 (1947) 46 48, which is the 12 coin case with one false coin, but it had appeared elsewhere by then, e.g. in the Graham Dial (Oct 1945), SM (Dec 1945), MG (1946). The general problem is first tackled by R. L. Goodstein in MG (Dec 1945) but required a correction in 1946.

I have been sent a copy of Jack Sieburg; Problem Solving by Computer Logic; Data Processing Magazine – but the date is cut off – can anyone provide details??

5.C.1. Ranking Coins with a Balance. Steinhaus Math. Snapshots, 1950) describes this and cites Schreier (1932, NYS).

5.D.1. Measuring With Jugs. I have reorganised the notation of these problems. Tweedie, MG 23 (1939) 278 282, is the source of the triangular graphical method of solution. Halving 8 using 5 and 3 is in Abbot Albert which seems to be the earliest version. Pacioli, De Viribus, seems to be the first to use any other values, e.g. halving 12 using 7 and 5. Tartaglia seems to be the first to divide in thirds, e.g. divide 24 in thirds using 5, 11, 13. The general problem of what can be obtained from a, b, c with c full to start seems to be first treated by A. Labosne is his 5th ed. of Bachet's Problemes and appears to still be unsolved when c < a+b.

5.D.2. Ruler With Minimal Number of Marks. Dudeney gives this in MP, 1926. Gardner, in his note in 536, says Dudeney invented the problem. I have found Dudeney's Strand articles giving the linear and circular problems in 1920 & 1921 (c= MP, no. 180 & 181). MacMahon has a paper on the infinite case in 1922-23. A. Brauer (1945) seems to be the first paper on the general finite case and is an independent invention based on a resistor problem.

5.D.3. False Coins with a Weighing Scale. I have an AMM problem of 1960 and problems by Fujimura and Hunter in RMM (1961 & 1962).

5.D.4. Timing with Hourglasses. New section   my examples are from 1962 and 1983.

5.D.5. Measure half a barrel. New Section. My earliest example is 1904.

5.E. Euler Circuits and Mazes. My earliest version of the 'five brick puzzle' is 1844. A planar Eulerian graph has a non crossing Euler circuit. When was this discovered? Lewis Carroll used to give such a problem, but I doubt if he had a proof of the general result. There are other examples back to 1826?

5.E.1. Maze Algorithms. BLW discuss this and cite Wiener, Math. Annalen 6 (1873) 29 30, as the first solution. Tremaux gave a simpler solution, but Robin Wilson says they could never find anything by or about Tremaux. I find that he was a telegraph engineer in Paris. Tremaux's algorithm is described in Lucas, Rouse Ball (1st to 11th eds.) and Dudeney, AM.

5.E.2. Memory Wheels = Chain Codes. These were rediscovered by Baudot in 1882, but I have no references before 1956.

5.F.1. Knight's Tours. I have references from R. Wieber, Das Schachspiel in Arabische Literatur..., and van der Linde, Geschichte und Literatur des Schachspiels, back to 1141, but Murray's History of Chess gets back to 9C India. These are very difficult to track down. Any help with these would be appreciated. Kraitchik's Math. des Jeux says the Diderot D'Alembert Encyclopedie asserts that the problem was known very anciently to the Hindus.

5.F.2. Other Hamiltonian Circuits. Was there ever a solid dodecahedral version of Hamilton's Icosian Game? Lucas, RM 2, pp. 208 209, describes a solid wooden version. Ahrens, Mathematische Spiele, 2nd ed., 1910, p. 44, says a Dodekaederspiel is available from Firma Paul Joschkowitz   Magdeburg (.65 mark). This is not in the first ed. and was dropped in later eds. Are there any actual examples anywhere?

5.G.1. Gas, Water and Electricity. Dudeney's AM describes this as being 'as old as the hills', but was the earliest known reference until I found Dudeney's column in Strand Magazine 46 (Jul 1913) 110, but this still says: "It is much older than electric lighting, or even gas ...." In SLAHP (1928), Sam Loyd Jr says he (??) brought out the puzzle about 1900. Dudeney's Strand column says he has recently had four letters from the US about it.

5.H.1. Instant Insanity = The Tantalizer. Moffat's UK Patent (1900) is for 6 cubes considering 4 sides. S&B cites some other early examples, including Schossow (1900) for four cubes and this was applied for about a year before Moffat's application.. Meek's UK Patent (1909) is for 4 cubes. Wyatt's Puzzles in Wood (1928) gives a 6 cube version considering all 6 directions. The graphical method of solution is due to Carteblanche (1947). I am collecting examples of this puzzle and would be grateful for examples or descriptions of any you have.

5.H.2. MacMahon Pieces. Can anyone supply a copy of F. Winter, Das Spiel der 30 Bunte Würfel? (Not in British Library.)

5.I. Latin and Euler Squares. Euler's "Recherches sur une nouvelle espece de Quarres Magiques" (1782) appears to be the modern source. But there are pairs of orthogonal 4 by 4 squares in Ozanam (1725) and Alberti (1747). (The pair in Bachet is due to the 1874 editor.) Ahrens says Latin square amulets go back to medieval Islam (c1200) and a magic square of al Buni, c1200, indicates knowledge of two orthogonal 4 x 4 Latin squares. I have recently found a 7 x 7 Latin square in a Venetian book of 1541 and references to a 4 x 4 Latin square epitaph from Cornwall, 1708, but the references are different - to Meneage parish church, St. Mawgan and to Cunwallow, near Helstone. Can anyone provide clear information on this? Are there any other early western usages of Latin squares??

5.I.1. Eight Queens Problem. Can anyone supply copies of (Berliner) Schachzeitung 3 (Sep 1848) 363 & 4 (Jan 1849). (Not in British Library.)

5.I.2. Colouring Chessboards with no Repeats in a Line. I recall a result that an n x n board can be thus n-coloured if and only if n  1 or 5 (mod 6), but I can't find it.

5.J. Squaring the Square. A. Mąkowski has kindly provided a translation of Z. Moron's 1925 paper. (M. Goldberg has also done a translation.) There is a Russian book by I. M. Yaglom on the subject   is it worth translating? Federico's survey article in Graph Theory and Related Topics, has its earliest reference being Dehn (1903). I find that Dudeney's 'Lady Isabel's Casket' appeared in Strand Mag. 7 (Jan 1902) 584. It later appeared as prob. 40 in his CP (1907). Both Sprague and Tutte cite this problem as a source.

5.J.1. Mrs Perkins's Quilt. This is the problem of cutting a square into squares, possibly equal. Loyd and Dudeney both give it. Gardner thinks Loyd was first.

5.J.2. Cubing the Cube. This was apparently first posed by S. Chowla (1939).

5.K. Derangements. This is derived in de Montmort (1713) who had mentioned the results without proof in the 1708 ed.

5.K.1. Deranged Boxes of A, B and A+B. New section   my example is 1962.

5.K.2. Other Logic Puzzles Based on Derangements. These typically have a butcher, a baker and a brewer whose surnames are Butcher, Baker and Brewer, but no one has the profession of his surname.

5.M. Six People at a Party = Ramsey Theory. The earliest source seems to be the Putnam exam of 1953. Who posed the problem? It is a special case of a result in the Erdös & Szekeres paper of 1935.

5.N. Jeep or Explorer's Problem   Crossing a Desert. Ball (5th ed., 1911) describes two versions, one of which is in Pearson (1907). The other, more common, form appears in Dudeney, MP. The earliest mathematical paper appears to be Helmer's Rand report (Dec 1946). There is a version in Alcuin, but the answer is obscure. Folkerts corrects an error, but it still assumes camels only eat while loaded - or perhaps they are abandoned and not reused. Pacioli gives 4 versions in De Viribus, but Agostini's description does not give the solutions. Both these want to get the most goods across a finite desert, where the transport eats some of the goods, so this is not quite the same as the 20C versions.

Mahavira and Sridhara (c900) give problems where a porter gets part of the load, but they are concerned with interpolating from a given distance and payment to find the payment for a different distance.

5.O. Tait's Counter Puzzle: HHHHTTTT.. to HTHTHTHT.. , by Moving Coins in Pairs. Tait, in his review of Listing's

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