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

Pakapim Eua-anant; Papangkorn Apinyanon; Thunyatorn Jirachaisri; Nantapong Ruangsuksriwong; Suthee Ruangwises

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arXiv · 2510.21938

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

Pakapim Eua-anant·Papangkorn Apinyanon·Thunyatorn Jirachaisri·Nantapong Ruangsuksriwong·Suthee Ruangwises

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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.

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Pakapim Eua-anant, Papangkorn Apinyanon, Thunyatorn Jirachaisri, Nantapong Ruangsuksriwong, Suthee Ruangwises. 2025-10-24. NP-Completeness Proofs of All or Nothing, Water Walk, and Remembered Length Using the T-Metacell Framework. https://arxiv.org/abs/2510.21938

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