The pairing strategies of the 9-in-a-row game

Abstract

One of the most useful strategies for proving Breaker’s win in Maker-Breaker Positional Games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the k-in-a-row game, the case k = 9 is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by Hales and Jewett.

In this paper we show that there are other winning pairings for the 9-in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no “irregular”, non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3-dimensional k-in-a-row game.