Complexity of 2D Snake Cube Puzzles
<|reference_start|>Complexity of 2D Snake Cube Puzzles: 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.<|reference_end|>
@article{mit hardness group2024complexity,
    title={Complexity of 2D Snake Cube Puzzles},
    author={MIT Hardness Group, Nithid Anchaleenukoon, Alex Dang, Erik D. Demaine,
  Kaylee Ji, Pitchayut Saengrungkongka},
    journal={arXiv preprint arXiv:2407.10323},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.10323},
    primaryClass={cs.CC cs.CG}
}