Near-rectangular P-positions in three-row Chomp with deficit at most six

We classify the losing positions (n,n,n-g) in three-row Chomp for every integer n>g and 1<=g<=6. There are exactly eight such positions. The proof combines a counting bound for the row at which a value becomes constant with a uniform bound obtained from Zeilberger's published low-row formulas. The resulting argument excludes all n>=35; the remaining cases are supplied by published exact data. The proof uses the diagonal/row-start partition in Erez Sheiner's 2026 preprint, arXiv:2605.23837v2. The eight examples were already known. This note presents the complete classification as a corollary of existing theory and elementary additional arguments, without a claim of priority. Version 1.0. The deposit contains the English manuscript and its self-contained LaTeX source. AI-tool disclosure: OpenAI Codex with language-model assistance was used in research exploration, proof organization, literature comparison, and manuscript preparation. The author supplied the research objective, direction, and supervision. No AI tool is listed as an author.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-08
DOI
https://doi.org/10.5281/zenodo.23241865
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

Near-rectangular P-positions in three-row Chomp with deficit at most six

Gaoqiang Liu
Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

Near-rectangular P-positions in three-row Chomp with deficit at most six

Gaoqiang Liu
preprint en

Abstract

We classify the losing positions (n,n,n-g) in three-row Chomp for every integer n>g and 1<=g<=6. There are exactly eight such positions. The proof combines a counting bound for the row at which a value becomes constant with a uniform bound obtained from Zeilberger's published low-row formulas. The resulting argument excludes all n>=35; the remaining cases are supplied by published exact data. The proof uses the diagonal/row-start partition in Erez Sheiner's 2026 preprint, arXiv:2605.23837v2. The eight examples were already known. This note presents the complete classification as a corollary of existing theory and elementary additional arguments, without a claim of priority. Version 1.0. The deposit contains the English manuscript and its self-contained LaTeX source. AI-tool disclosure: OpenAI Codex with language-model assistance was used in research exploration, proof organization, literature comparison, and manuscript preparation. The author supplied the research objective, direction, and supervision. No AI tool is listed as an author.

Zenodo (CERN European Organization for Nuclear Research)
University of Bayreuth (DE)
Advanced Combinatorial Mathematics
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Near-rectangular P-positions in three-row Chomp with deficit at most six — Gaoqiang Liu · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS