Drawn k-in-a-row games

Abstract

Wu and Huang (2005) [12] and Wu et al. (2006) [13] presented a generalized family of k-in-a-row games, called Connect(m, n, k, p, q). Two players, Black and White, alternately place p stones on an m x n board in each turn. Black plays first, and places q stones initially. The player who first gets k consecutive stones of his/her own horizontally, vertically, or diagonally wins. Both tie the game when the board is filled up with neither player winning. A Connect(m, n, k, p, q) game is drawn if neither has any winning strategy. Given p, this paper derives the value k(draw)(p), such that Connect(m, n, k, p, q) games are drawn for all k >= k(draw)(p), m >= 1, n >= 1, 0 <= q <= p, as follows. (1) k(draw)(p) = 11. (2) For all p >= 3, k(draw)(p) = 3p + 3d - 1, where d is a logarithmic function of p. So, the ratio k(draw)(P)/P is approximately 3 for sufficiently large p. The first result was derived with the help of a program. To our knowledge, our k(draw)(p) values are currently the smallest for all 2 <= p <= 1000. (C) 2011 Elsevier B.V. All rights reserved.