Martin Gardner's Minimum No-3-in-a-Line Problem

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 an [n × n chessboard] such that no [queen] can be added without creating three in a row, a column, or, except in the case when n is congruent to 3 modulo 4, in which case one less may suffice.” We use the Combinatorial Nullstellensatz to prove that this number is at least n. A second, more elementary proof is also offered in the case that n is even.