## Abstract

When solving k-in-a-Row games, the Hales-Jewett pairing

strategy [4] is a well-known strategy to prove that specic positions are (at most) a draw. It requires two empty squares per possible winning line (group) to be marked, i.e., with a coverage ratio of 2.0.

In this paper we present a new strategy, called Set Matching. A matching set consists of a set of nodes (the markers), a set of possible winning lines (the groups), and a coverage set indicating how all groups are covered after every rst initial move. This strategy needs less than two markers per group. As such it is able to prove positions in k-in-a-Row games to be draws, which cannot be proven using the Hales-Jewett pairing strategy.

We show several ecient congurations with their matching sets. These include Cycle Congurations, BiCycle Congurations, and PolyCycle Congurations involving more than two cycles. Depending on conguration, the coverage ratio can be reduced to 1.14.

Many examples in the domain of solving k-in-a-Row games are given, including the direct proof (not based on search) that the empty 4 x 4 board is a draw for 4-in-a-Row.

strategy [4] is a well-known strategy to prove that specic positions are (at most) a draw. It requires two empty squares per possible winning line (group) to be marked, i.e., with a coverage ratio of 2.0.

In this paper we present a new strategy, called Set Matching. A matching set consists of a set of nodes (the markers), a set of possible winning lines (the groups), and a coverage set indicating how all groups are covered after every rst initial move. This strategy needs less than two markers per group. As such it is able to prove positions in k-in-a-Row games to be draws, which cannot be proven using the Hales-Jewett pairing strategy.

We show several ecient congurations with their matching sets. These include Cycle Congurations, BiCycle Congurations, and PolyCycle Congurations involving more than two cycles. Depending on conguration, the coverage ratio can be reduced to 1.14.

Many examples in the domain of solving k-in-a-Row games are given, including the direct proof (not based on search) that the empty 4 x 4 board is a draw for 4-in-a-Row.

Original language | English |
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Title of host publication | Advances in Computer Games. ACG 2017 |

Editors | Mark Winands, H. Jaap van den Herik, Walter Kosters |

Publisher | Springer, Cham |

Pages | 38-50 |

ISBN (Electronic) | 978-3-319-71649-7 |

ISBN (Print) | 978-3-319-71648-0 |

DOIs | |

Publication status | Published - 2017 |

### Publication series

Series | Lecture Notes in Computer Science |
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Volume | 10664 |

ISSN | 0302-9743 |