Infinite Hex is a draw

In this second joint paper with Prof. Joel David Hamkins, we expand the results achieved for infinite Hex in chapter 2 of my MSc dissertation, and present new open questions.

You can read this article at arXiv:2201.06475. (revised and expanded in December 2022)

Abstract

We introduce the game of infinite Hex, extending the familiar finite game to natural play on the infinite hexagonal lattice. Whereas the finite game is a win for the first player, we prove in contrast that infinite Hex is a draw—both players have drawing strategies.

Meanwhile, the transfinite game-value phenomenon, now abundantly exhibited in infinite chess and infinite draughts, regrettably does not arise in infinite Hex; only finite game values occur. Indeed, every game-valued position in infinite Hex is intrinsically local, meaning that winning play depends only on a fixed finite region of the board. This latter fact is proved under very general hypotheses, establishing the conclusion for all simple stone-placing games.