Gazette 31 Vol 3 - Australian Mathematical Society

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160 Norman Do theoretical draw and “?” indicates that the result is merely conjectured but not actually proven. n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 k = 1 W D D D D D D D D D k = 2 W W D D D D D D D D k = 3 W W W W D? D? D? D D D k = 4 W W W W W? W? W? D? D? D? k = 5 W W W W? W? W? W? W? W? D? k = 6 W W W W? W? W? W? W? W? W? These results appear in Golomb and Hales’ article on hypercube tic-tac-toe [3] as well as the following two conjectures which seem intuitively obvious but have so far resisted proof. (1) If the n k game is a draw, then the n k−1 game is a draw. (2) If the n k game is a draw, then the (n + 1) k game is a draw. Remember that a necessary condition for a pairing strategy to exist is that the number of cells is at least� twice the number of winning paths. As we have shown, this is equivalent to n ≥ . Hales and Jewett conjectured that drawing strategies exist for the second � 2 k√ 2−1 player whenever this inequality holds. Golomb and Hales subsequently noted the interesting 2 fact that there is a fairly accurate linear approximation to the expression k√ . 2−1 where a = ak = k√ 2. 2 k√ = 2 2 − 1 ak − 1 a − 1 = 2(1 + a + a2 + · · · + a k−1 ) ≈ 2 � k 0 = 2k log e 2 , a t dt = 2 ak − 1 log e a From this observation, it is tempting to make the conjecture that � � � � 2 2k k√ = . 2 − 1 loge 2 In fact, a computer program can easily verify the conjecture to be true for thousands, even millions of terms, before reaching the incredibly large value of k = 6, 847, 196, 937, when the conjecture first fails. Furthermore, this is the only failure until we reach the second counterexample at k = 27, 637, 329, 632. In fact, the theory of diophantine approximation can be used to prove that these anomalies can only occur when k is the denominator of a 2 continued fraction convergent for loge 2 . Let us now finish with a tic-tac-toe-inspired number theory problem for which, I believe, the answer is unknown. For how many positive integer values of k does the equation � � � � 2 2k k√ = 2 − 1 loge 2 fail to be true?
References Mathellaneous 161 [1] E.R. Berlekamp, J.H. Conway, and R.K. Guy, Winning Ways for your Mathematical Plays, Volume 3, 2nd Edition (A. K. Peters Ltd Wellesley MA 2003). [2] M. Gardner, Harary’s generalized ticktacktoe, in: The Colossal Book of Mathematics (W.W. Norton and Company New York 2002), 493–503. [3] S. W. Golomb and A. W. Hales, Hypercube tic-tac-toe, in: More Games of No Chance (MSRI Publications 2002), 167–182. [4] R.K. Guy, J.L. Selfridge, T.G.L. Zetters, S10 — 8 (or more) in a row, The American Mathematical Monthly 87 (1980), 575–576. [5] A. W. Hales and R. I. Jewett, Regularity and positional games, Transactions of the American Mathematical Society 106 (1963), 222–229. [6] H. Harborth and M. Seemann, Snaky is a paving winner, Bull. Inst. Combin. Appl. 19 (1997), 71–78. [7] O. Patashnik, Qubic: 4 × 4 × 4 tic-tac-toe, Mathematics Magazine 53 (1980), 202–216. Department of Mathematics and Statistics, The University of Melbourne, VIC 3010 E-mail: N.Do@ms.unimelb.edu.au
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Gazette 31 Vol 3 - Australian Mathematical Society

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