The chessboard without two opposite corner squares can't be covered
with 31 dominoes because these squares are of the same color! When we
remove them, the mutilated board of 62 squares will contain two more
squares of one color than of the other.
Each domino always covers two adjacent squares of the chessboard which are
always of opposite color. So when you cover (in ANY possible way) 60
squares with 30 dominoes, you will have two squares left, but,
unfortunately, of the same color. These two squares are not of opposite
color, so they can't be adjacent in any way. Therefore the last domino
can't cover them.