The Hamiltonian path starts in black and ends in white

Consider a checkerboard grid with an even number of squares on each side.Every path through it will alternate black and white.These grids will all also have an even total number of squares because an even number times and even number is even.So a Hamiltonian path on an even-sided grid that starts on black will have to end on white. And one that starts on white will have to end on black.