On drawn k-in-a-row games

  • Authors:

  • Sheng-Hao Chiang;I-Chen Wu;Ping-Hung Lin

  • Affiliations:

  • National Experimental High School at Hsinchu Science Park, Hsinchu, Taiwan;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan

  • Venue:
  • ACG'09 Proceedings of the 12th international conference on Advances in Computer Games
  • Year:
  • 2009

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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 kdraw(p), such that Connect(kdraw(p), p) is drawn, as follows. (1) kdraw(2) = 11. (2) For all p ≥ 3, kdraw(p) = 3p+3d+8, where d is a logarithmic function of p. So, the ratio kdraw(p)/p is approximate to 3 for sufficiently large p. To our knowledge, our kdraw(p) are currently the smallest for all 2 ≤ p 1000, except for p = 3.