Suppose a standard 8×8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares?

The puzzle is impossible. Any way you would place a domino would cover one white square and one black square. A group of 31 dominoes would cover 31 white and 31 black squares of an unmutilated chessboard, leaving one white and one black square uncovered. The directions had you remove diagonally opposite corner squares, and such squares are always either both black or both white.