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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.03119 |
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| _version_ | 1866917560998428672 |
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| author | Oh, Seunghwan Schmitt, John R. Wang, Xianzhi |
| author_facet | Oh, Seunghwan Schmitt, John R. Wang, Xianzhi |
| contents | In 1976 Martin Gardner posed the following problem: ``What is the smallest number of [queens] you can put on an [$n \times n$ chessboard] such that no [queen] can be added without creating three in a row, a column, or a diagonal?'' The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least $n$, except in the case when $n$ is congruent to $3$ modulo $4$, in which case one less may suffice. When $n>1$ is odd, Gardner conjectured the lower bound to be $n+1$. We prove this conjecture in the case that $n$ is congruent to 1 modulo 4. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem Oh, Seunghwan Schmitt, John R. Wang, Xianzhi Combinatorics 05D40 In 1976 Martin Gardner posed the following problem: ``What is the smallest number of [queens] you can put on an [$n \times n$ chessboard] such that no [queen] can be added without creating three in a row, a column, or a diagonal?'' The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least $n$, except in the case when $n$ is congruent to $3$ modulo $4$, in which case one less may suffice. When $n>1$ is odd, Gardner conjectured the lower bound to be $n+1$. We prove this conjecture in the case that $n$ is congruent to 1 modulo 4. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica. |
| title | Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem |
| topic | Combinatorics 05D40 |
| url | https://arxiv.org/abs/2401.03119 |