Question 170
All the grids of an $m \times n$ chessboard ($m \ge 3, n \ge 3$) are colored either red or blue. Two adjacent grids (with a common side) are called a *good couple* if they are of different colors. Suppose that there are $S$ good couples, explain how to determine whether $S$ is odd or even. Does it depend on certain specific color grids? (Reasoning is required.) (posed by Feng Yuefeng)