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NP-Completeness Proofs of All or Nothing and Water Walk Using the T-Metacell Framework

Published: October 24, 2025 | arXiv ID: 2510.21938v1

By: Pakapim Eua-anant , Papangkorn Apinyanon , Thunyatorn Jirachaisri and more

Potential Business Impact:

Solves tricky puzzles by finding a hidden path.

Business Areas:
Water Natural Resources

All or Nothing and Water Walk are pencil puzzles that involve constructing a continuous loop on a rectangular grid under specific constraints. In this paper, we analyze their computational complexity using the T-metacell framework developed by Tang and MIT Hardness Group. We establish that both puzzles are NP-complete by providing reductions from the problem of finding a Hamiltonian cycle in a maximum-degree-3 spanning subgraph of a rectangular grid graph.

Country of Origin
🇹🇭 Thailand

Page Count
8 pages

Category
Computer Science:
Computational Complexity