Almost optimal pairing strategy for Tic-Tac-Toe with numerous directions

Details Almost optimal pairing strategy for Tic-Tac-Toe with numerous directions Almost optimal pairing strategy for Tic-Tac-Toe with numerous directions

2010-05-29

arxiv.org/abs/1005.5469v1

Mathematics

Combinatorics

Scientific paper

We show that there is an $m=2n+o(n)$, such that, in the Maker-Breaker game played on $\Z^d$ where Maker needs to put at least $m$ of his marks consecutively in one of $n$ given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg who showed that such a pairing strategy exits if $m\ge 3n$. A simple argument shows that $m$ has to be at least $2n+1$ if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.

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