A chessboard is composed of 64 small squares arranged in an 8×8 large square. A domino laid on the board will cover two of those squares. Therefore 32 dominoes can be laid out on the chessboard to cover all 64 squares. Suppose that you cut out two squares from the chessboard at diagonally opposite corners. Now there are 62 squares. Is it possible to arrange 31 dominoes to cover these 62 squares? Discuss and explain your answer. (Hint: Look closely at a real chessboard.)
It cannot be done. Use the colors of the squares to help you see why. Red and black squares alternate on the board in every direction, so a domino always covers one red square and one black square. Thus 31 dominoes can cover 31 red and 31 black squares. But if you remove two squares at opposite corners, they’re the same color. Thus your 31 dominoes must cover 32 red and 30 black squares (or 30 red and 32 black squares), and there’s no way they can do that.