On Drawn K-In-A-Row Games

Abstract

In 2005, Wu and Huang [9] presented a generalized family of k-in-a-row games. The current paper simplifies the family to Connect(k, p). Two players alternately place p stones on empty squares of an infinite board in each turn. The player who first obtains k consecutive stones of his\' own horizontally, vertically, diagonally wins. A Connect(k, p) game is drawn if both have no winning strategy. Given p, this paper derives the value k(draw)(p), such that Connect(k(draw)(p), p) is drawn, as follows. (1) k(draw)(2) = 11. (2) For all p >= 3, k(draw)(p) = 3p+3d+8, where d is a logarithmic function of p. So, the ratio k(draw)(p)/p is approximate to 3 for sufficiently large p. To our knowledge, our k(draw)(p) are currently the smallest for all 2 <= p <= 1000, except for p = 3.