Cardinal Grid Slime Trail is PSPACE-Complete

2026-08-05

AI Summary

This study resolves an open problem concerning the computational complexity of the grid-based Slime Trail puzzle game. Focusing on the cardinal four-directional variant, the authors establish its PSPACE-completeness by presenting a novel reduction from Quantified Boolean Formula (QBF), which introduces specialized gadgets that satisfy both parity and degree constraints inherent to the integer lattice. The construction leverages the regular structure of the grid and naturally extends—via a 45-degree rotation—to the eight-directional version. Consequently, this work proves that both predominant gameplay variants are PSPACE-complete, thereby filling a longstanding gap in the theoretical understanding of this classic puzzle game’s complexity landscape.

Abstract

Slime Trail is a two-player combinatorial game in which the players alternately move a shared token to an adjacent vertex, permanently removing each vertex the token leaves, while attempting to reach a goal node. Ferland and Burke (2017) proved that Slime Trail is PSPACE-complete on arbitrary planar graphs and asked whether the same holds for the grid version actually used in play. We resolve this open problem by proving that Cardinal Grid Slime Trail, that is, Slime Trail on a square grid with four-directional movement, is PSPACE-complete. We adapt their QBF reduction to the grid setting, designing grid-compatible gadgets that respect the degree-4 bound and the parity constraints of the integer lattice. We further show the construction extends, under a 45-degree rotation, to the eight-directional variant.

Problem

• Slime Trail
• PSPACE-complete
• grid
• combinatorial game
• computational complexity

Innovation

• PSPACE-completeness
• Slime Trail
• grid gadgets
• QBF reduction
• combinatorial game