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 kin 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 kin a row game.

The Structure of Pairing Strategies for k-in-a-row Type Games

“…Figure 3 we get two different good 8-toric pairings. In [6], Corollary 8 states that all 16-toric good pairings of H 9 are just a combination of two (or more) different 8-toric ones.…”

“…In the case of k = 9 -which is the main subject of this paper-there are only highly symmetric (8-or 16-toric) good domino pairings of the board (see Győrffy et al [6]). An 8-toric pairing means that the pairing is an extension of an 8×8 square, while domino pairing means that all pairs consist of only neighboring (vertically, horizontally or diagonally) squares called dominoes.…”

“…An 8-toric pairing means that the pairing is an extension of an 8×8 square, while domino pairing means that all pairs consist of only neighboring (vertically, horizontally or diagonally) squares called dominoes. We know from [6] that all 16-toric good pairings can be deduced from two (or more) different 8-toric pairings.…”

“…However, according to Proposition 1 there might be O(n) anomalies even in a good pairing of H 9 . But this conjecture was shown to be false (see Section 6 of[6]), so all good pairings for H 9 are anomaly-free.Definition 4. A pairing of the infinite board is k-toric if it is a repetition of a k × k square, where k is the smallest possible value.…”