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

Abstract

All or Nothing, Water Walk, and Remembered Length 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 the puzzles are NP-complete by providing reductions; the first two puzzles, from the problem of finding a Hamiltonian cycle in a maximum-degree-3 spanning subgraph of a rectangular grid graph, and the last, from the problem of finding a Hamiltonian cycle in a required-edge directed rectangular grid graph.