The recent question https://mathoverflow.net/questions/27944 of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an infinite board decidable? In other words, given a position on an infinite board (say $\mathbb{Z}\times \mathbb{Z}$, though now pawn promotion is not possible) with finitely many men, say with White to move, is there an algorithm to determine whether White can checkmate Black (or prevent Black from checkmating White) against any Black defense?