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

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.