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arXiv · 2401.03119

Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem

Seunghwan Oh·John R. Schmitt·Xianzhi Wang
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Abstract

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.

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Seunghwan Oh, John R. Schmitt, Xianzhi Wang. 2024-01-06. Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem. https://arxiv.org/abs/2401.03119

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