Nontrivial solutions to a checkerboard problem

Abstract

The squares of an $m\times n$ checkerboard are alternately colored black and red. It has been shown that for every pair $m,n$ of positive integers, it is possible to place coins on some of the squares of the checkerboard (at most one coin per square) in such a way that for every two squares of the same color the numbers of coins on neighboring squares are of the same parity, while for every two squares of different colors the numbers of coins on neighboring squares are of opposite parity. All solutions to this problem have been what is referred to as trivial solutions, namely, for either black or red, no coins are placed on any square of that color. A nontrivial solution then requires at least one coin to be placed on a square of each color. For some pairs $m,n$ of positive integers, however, nontrivial solutions do not exist. All pairs $m,n$ of positive integers are determined for which there is a nontrivial solution.