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

A 1976 question of Martin Gardner asks for the minimum size of a placement of queens on an $n\times n$ chessboard that is maximal with respect to the property of ‘no-3-in-a-line’. The work of Cooper, Pikhurko, Schmitt and Warrington showed that this number is at least $n$ in the cases that $n⁄\equiv 3(mod4)$, and at least $n-1$ in the case that $n\equiv 3(mod4)$. When $n>1$ is odd, Gardner conjectured the lower bound to be $n+1$. We prove this conjecture in the case that $n\equiv 1(mod4)$. The proof relies heavily on a recent advancement to the Combinatorial Nullstellensatz for zero-sum grids due to Bogdan Nica.