Queens in exile: non-attacking queens on infinite chess boards (Q2309222)

Summary: Number the cells of a (possibly infinite) chessboard in some way with the numbers \(0,1,2, \ldots \). Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of size \(\mathbb{Z} \times \mathbb{Z}\) numbered along a square spiral, and an infinite single-quadrant chessboard (of size \(\mathbb{N} \times \mathbb{N})\) numbered along antidiagonals. We give a fairly complete solution in the first case, based on the Tribonacci word. There are connections with combinatorial games.