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

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.