NP-Completeness Proofs of All or Nothing, Water Walk, and Remembered Length Using the T-Metacell Framework
Computational Complexity
2026-01-27 v2
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.
Cite
@article{arxiv.2510.21938,
title = {NP-Completeness Proofs of All or Nothing, Water Walk, and Remembered Length Using the T-Metacell Framework},
author = {Pakapim Eua-anant and Papangkorn Apinyanon and Thunyatorn Jirachaisri and Nantapong Ruangsuksriwong and Suthee Ruangwises},
journal= {arXiv preprint arXiv:2510.21938},
year = {2026}
}
Comments
15 pages, 10 figures; Remembered Length proof added