Sources page biographical material

4.B.1.a IN HIGHER DIMENSIONS
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 p_r(k) of r-sections of a kⁿ board by means of a recurrence p_r(k) = [p_r-1(k+2) - p_r-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.