Draws, Zugzwangs, and PSPACE-Completeness in the Slither Connection Game

Two features set Slither apart from other connection games. Previously played stones can be relocated and some stone configurations are forbidden. We show that the interplay of these peculiar mechanics with the standard goal of connecting opposite edges of a board results in a game with a few properties unexpected among connection games, for instance, the existence of mutual Zugzwangs. We also establish that, although there are positions where one player has no legal move, there is no position where both players lack a legal move and that the game cannot end in a draw. From the standpoint of computational complexity, we show that the game is pspace-complete, the relocation rule can indeed be tamed so as to simulate a hex game on a Slither board.