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

TL;DR

This paper proves that solving Bloxorz levels is PSPACE-complete, even with all trapdoors initially open or closed, and demonstrates that 2-buttons suffice for PSPACE-hardness in general puzzle games.

Contribution

It establishes the PSPACE-completeness of Bloxorz and shows 2-buttons are enough for PSPACE-hardness in puzzle games, answering open questions in the field.

Findings

Bloxorz level solvability is PSPACE-complete.

PSPACE-completeness holds even with all trapdoors initially open or closed.

2-buttons suffice for PSPACE-hardness in general puzzle games.

Abstract

Bloxorz is an online puzzle game where players move a 1 by 1 by 2 block by tilting it on a subset of the two dimensional grid. Bloxorz features switches that open and close trapdoors. The puzzle is to move the block from its initial position to an upright position on the destination square. We show that the problem of deciding whether a given Bloxorz level is solvable is 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, showing that 2-buttons are sufficient for PSPACE-hardness of general puzzle games. We also examine the hardness of some variants of Bloxorz, including variants where the block is a 1 by 1 by 1 cube, and variants with single-use tiles.