Let $K_n$ be the complete graph on $n$ vertices, $n>3$, and let $C$ be an edge 2-coloring of $K_n$ with edge colors red and blue. Let a 4-circuit with exactly three red edges or exactly three blue edges be called a mostly monochromatic circuit (mostly red, and mostly blue). Let M be the number of mostly monochromatic 4-circuits in $K_n$. Our problem is the following:
Problem Show that if we switch the color of an edge in the complete graph, the change in M is even.