The problem where one square of each is colour is removed is much more interesting. It's not obvious to me why it should always be possible to solve that proble

The problem where one square of each is colour is removed is much more interesting. It's not obvious to me why it should always be possible to solve that problem, regardless of which squares are removed, but apparently it is. I wonder if it can be proved without resorting to looking at classes of cases (removed squares right next to each other, removed squares with two squares between them, etc)?