PSPACE-Completeness of Bloxorz and of Games with 2-Buttons

Abstract

Bloxorz is an online puzzle game where players move a $1 \times 1 \times 2$ block by tilting it on a subset of the two dimensional grid, that also features switches that open and close trapdoors. The puzzle is to move the block from its initial position to an upright position on the goal square. We show that the problem of deciding whether a given Bloxorz level is solvable is $\mathsf{PSPACE}$-complete and that this remains so even when all trapdoors are initially closed or all trapdoors are initially open. We also answer an open question of Viglietta [6], showing that 2-buttons are sufficient for $\mathsf{PSPACE}$-hardness of general puzzle games. We also examine the hardness of some variants of Bloxorz, including variants where the block is a $1 \times 1 \times 1$ cube, and variants with single-use tiles.