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 NPhard: (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.
 Publication:

arXiv eprints
 Pub Date:
 July 2024
 DOI:
 10.48550/arXiv.2407.10323
 arXiv:
 arXiv:2407.10323
 Bibcode:
 2024arXiv240710323M
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Computational Geometry
 EPrint:
 24 pages, 20 figures. Short version published at 36th Canadian Conference on Computational Geometry (CCCG 2024)