Martin Gardner's minimum no-3-in-a-line problem

TL;DR

This paper investigates the minimum number of queens needed on an n×n chessboard to prevent adding any more without forming three in a line, using algebraic and elementary combinatorial methods.

Contribution

It provides new lower bounds for the minimum number of queens needed, employing the Combinatorial Nullstellensatz and elementary proofs for specific cases.

Findings

Minimum number is at least n for most n

Fewer queens may suffice when n ≡ 3 mod 4

Elementary proof provided for even n cases

Abstract

In Martin Gardner's October, 1976 Mathematical Games column in Scientific American, he posed the following problem: "What is the smallest number of [queens] you can put on a board of side n such that no [queen] can be added without creating three in a row, a column, or a diagonal?" We use the Combinatorial Nullstellensatz to prove 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. A second, more elementary proof is also offered in the case that n is even.