Complexity of 2D Snake Cube Puzzles

Abstract

Given a chain of $HW$ cubes where each cube is marked "turn $90^{\circ }$" or "go straight", when can it fold into a $1\times H\times W$ rectangular box? We prove several variants of this (still) open problem NP-hard: (1) allowing some cubes to be wildcard (can turn or go straight); (2) allowing a larger box with empty spaces (simplifying a proof from CCCG 2022); (3) growing the box (and the number of cubes) to $2\times H\times W$ (improving a prior 3D result from height $8$ to $2$); (4) with hexagonal prisms rather than cubes, each specified as going straight, turning $60^{\circ }$, or turning $120^{\circ }$; and (5) allowing the cubes to be encoded implicitly to compress exponentially large repetitions.

24 pages, 20 figures. Short version published at 36th Canadian Conference on Computational Geometry (CCCG 2024)