COVERING THE CHESSBOARD
On the left is a chessboard and a supply of dominoes. Each domino is large enough to cover two adjacent squares of the board exactly, and it is easy to see how you can place 32 dominoes to cover all 64 squares of the board.
On the right is the same chessboard, except that the top-left and bottom-right squares have been removed. Is it still possible to cover the board with dominoes, this time using only 31, so that every square is covered exactly once?