Complexity of 2D Snake Cube Puzzles

TL;DR

This paper investigates the computational complexity of folding 2D Snake Cube puzzles into rectangular boxes, proving NP-hardness for various puzzle variants and extensions, including wildcard cubes, larger boxes, 3D growth, hexagonal prisms, and implicit encoding.

Contribution

It establishes NP-hardness results for multiple variants of Snake Cube puzzles, extending previous work and exploring new dimensions like hexagonal prisms and implicit encoding.

Findings

NP-hardness proven for puzzles with wildcards and larger boxes

Complexity results for 3D growth from height 8 to 2

NP-hardness established for hexagonal prism puzzles and implicit encoding

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.