An Exact 93-Move Solution to a Nonlocal Crystal-Exchange Puzzle
Abstract
We determine the minimum number of directional commands needed to solve alevel of Order of the Sinking Star, modeled as a crystal-exchangepuzzle on an 11*11 board. A command swapsthe player with the nearest crystal in the selected row or column, regardlessof distance; only an empty ray permits an ordinary unit walk. Two gates admitthe player to the exit exactly while all sixteen crystals occupy their targetcells. We give an explicit 93-command solution and exclude every shortersolution by an exact, cost-bounded breadth-first search. The search representsa position by its seventeen occupied cells and one active cell. Its admissibleheuristic partitions command costs between a compressed core abstraction andresidual motion, strengthened by a necessary activation detour and a cornervisit condition. The completed computation retained 605,349,215 occupied setsand 1,863,925,136 states. We prove the reductions and pruning conditions,describe the exact storage scheme, and provide source code, execution logs,and a reproduction driver. The result is a computer-assisted proof for thespecified instance, with an explicit implementation trust boundary.
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BibTeX
@misc{wang2026exact,
title = {{An Exact 93-Move Solution to a Nonlocal Crystal-Exchange Puzzle}},
author = {Zilai Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29493},
eprint = {2610.29493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29493},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - An Exact 93-Move Solution to a Nonlocal Crystal-Exchange Puzzle
AU - Zilai Wang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29493
UR - https://arxiv.org/abs/2610.29493
AB - We determine the minimum number of directional commands needed to solve alevel of Order of the Sinking Star, modeled as a crystal-exchangepuzzle on an 11*11 board. A command swapsthe player with the nearest crystal in the selected row or column, regardlessof distance; only an empty ray permits an ordinary unit walk. Two gates admitthe player to the exit exactly while all sixteen crystals occupy their targetcells. We give an explicit 93-command solution and exclude every shortersolution by an exact, cost-bounded breadth-first search. The search representsa position by its seventeen occupied cells and one active cell. Its admissibleheuristic partitions command costs between a compressed core abstraction andresidual motion, strengthened by a necessary activation detour and a cornervisit condition. The completed computation retained 605,349,215 occupied setsand 1,863,925,136 states. We prove the reductions and pruning conditions,describe the exact storage scheme, and provide source code, execution logs,and a reproduction driver. The result is a computer-assisted proof for thespecified instance, with an explicit implementation trust boundary.
ER -