## 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×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.

Original language | English |
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Pages (from-to) | 4558-4569 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 412 |

Issue number | 35 |

DOIs | |

State | Published - 12 Aug 2011 |

## Keywords

- Connect6
- Hypergraphs
- k-in-a-row games