  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.
