Infinite Hex is a draw
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.
Cite
@article{arxiv.2201.06475,
title = {Infinite Hex is a draw},
author = {Joel David Hamkins and Davide Leonessi},
journal= {arXiv preprint arXiv:2201.06475},
year = {2023}
}
Comments
28 pages, 36 figures. Commentary and inquires can be made at http://jdh.hamkins.org/infinite-hex-is-a-draw