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

TL;DR
This paper proves that the puzzles All or Nothing, Water Walk, and Remembered Length are NP-complete by using the T-metacell framework to establish reductions from Hamiltonian cycle problems in grid graphs.
Contribution
It introduces NP-completeness proofs for three puzzles using the T-metacell framework, expanding computational complexity understanding of these puzzles.
Findings
All or Nothing and Water Walk are NP-complete via Hamiltonian cycle reductions.
Remembered Length is NP-complete through directed grid graph Hamiltonian cycle reduction.
The T-metacell framework effectively proves NP-completeness for these puzzles.
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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