A classical problem of Gauss (ca.1850) asks if eight queens can be placed on a chessboard so that no queen can attack any other. Some experimenting shows that the answer is positive. Furthermore, it is known that, for every integer \(n \ge 4\), one can place \(n\) queens on an \(n \times n\) chessboard (with the obvious meaning) so that no queen can attack any other. This can be stated, using the terms “rows” and “columns” as for matrices, as:
Theorem 1: For every \(n \ge 4\) one can place queens on an \(n \times n\) chessboard so that there is a queen in every row and in every column and no queen can attack any other.
This raises the question of what happens if we allow the chessboard to be infinite.
Problem: Can one place infinitely many queens on an infinite chessboard so that there is a queen in every row and in every column and no queen can attack any other? Critically, how would one place the queens?
Submit your solutionSolution will be available on 21st October 2024. Citations will be added with the solution so as to not spoil the problem.
Hint
The solution is not unique, but there is a simple way to place the queens. Think small, maybe start with 5 by 5, and work your way up.
Submitted by Sammy Webb.