CWMO 2004 Problem 2

All the grids of a $m\times n$ chess board ( $m,n\geq 3$ ), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by some specific color grids? Justify your answers.