Martin Gardner's minimum no-3-in-a-line problem
Combinatorics
2014-03-10 v2
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.
Cite
@article{arxiv.1206.5350,
title = {Martin Gardner's minimum no-3-in-a-line problem},
author = {Alec S. Cooper and Oleg Pikhurko and John R. Schmitt and Gregory S. Warrington},
journal= {arXiv preprint arXiv:1206.5350},
year = {2014}
}
Comments
11 pages; lower bound in main theorem corrected to n-1 (from n) in the case of n congruent to 3 mod 4, minor edits, added journal reference