Sources page biographical material

INDEX

( 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.
4.A.1.a. THE 31 GAME
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  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.
4.A.2. SYMMETRY ARGUMENTS
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.

4.A.3. KAYLES
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.
4.A.4. NIM
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.