blueplanet200's proof can fit in a sentence or two. So sounds like you came up with an entirely different strategy. I'm curious what it is (doesn't matter if it's not as elegant).

spoiler for how to prove it:

think of colors of chessboard (black/white alternating) and what colors a domino piece will always cover.

Nice.

I was confused by the L-shaped shading in the top-right corner of the diagram, though.