Sources page biographical material

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.