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:
  • Theoretical Computer Science
  • Year:
  • 2011

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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 mxn 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"d"r"a"w(p), such that Connect(m, n, k, p, q) games are drawn for all k=k"d"r"a"w(p), m=1, n=1, 0@?q@?p, as follows. (1) k"d"r"a"w(p)=11. (2) For all p=3, k"d"r"a"w(p)=3p+3d-1, where d is a logarithmic function of p. So, the ratio k"d"r"a"w(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"d"r"a"w(p) values are currently the smallest for all 2@?p