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.