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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.23837 |
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| _version_ | 1866917525304901632 |
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| author | Sheiner, Erez |
| author_facet | Sheiner, Erez |
| contents | Chomp was introduced by Gale in 1974. In the same paper, Gale reported that the 3 x n games had been completely analyzed for n <= 100, with a unique winning first move in every case, and asked whether winning first moves are unique in general. Although the general uniqueness statement is false, we prove that the three-row uniqueness phenomenon suggested by Gale's computations holds for all n: every 3 x n Chomp rectangle has exactly one winning opening move. This settles the three-row case of Gale's 52-year-old first-move uniqueness question.
The proof is carried out in the two-variable recurrence introduced by Brouwer, Horvath, Molnar-Saska, and Szabo for the function f(q,r) whose values encode the P-positions. The main local ingredient is a rightmost-hole principle: if a value p is absent from the set C(q,r) but belongs to all corresponding sets C(t,r) for q < t < p, then all intermediate values q+1,...,p-1 are forced to belong to C(q,r). This separates the diagonal values from the starts of constant rows, and yields a partition of the positive integers into the two possible types of winning opening moves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23837 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unique Winning Opening Move in Three-Row Chomp Sheiner, Erez Combinatorics Primary 91A46 Chomp was introduced by Gale in 1974. In the same paper, Gale reported that the 3 x n games had been completely analyzed for n <= 100, with a unique winning first move in every case, and asked whether winning first moves are unique in general. Although the general uniqueness statement is false, we prove that the three-row uniqueness phenomenon suggested by Gale's computations holds for all n: every 3 x n Chomp rectangle has exactly one winning opening move. This settles the three-row case of Gale's 52-year-old first-move uniqueness question. The proof is carried out in the two-variable recurrence introduced by Brouwer, Horvath, Molnar-Saska, and Szabo for the function f(q,r) whose values encode the P-positions. The main local ingredient is a rightmost-hole principle: if a value p is absent from the set C(q,r) but belongs to all corresponding sets C(t,r) for q < t < p, then all intermediate values q+1,...,p-1 are forced to belong to C(q,r). This separates the diagonal values from the starts of constant rows, and yields a partition of the positive integers into the two possible types of winning opening moves. |
| title | Unique Winning Opening Move in Three-Row Chomp |
| topic | Combinatorics Primary 91A46 |
| url | https://arxiv.org/abs/2605.23837 |