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

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.