The mutilated chessboard

Two diagonally opposite corner squares are cut from an 8 \times 8 chessboard, leaving 62 squares. Can the rest be covered exactly by 31 dominoes, each covering two adjacent squares?

Colour the board as a chessboard. What does a single domino always cover?

Solution. No. Every domino covers one black and one white square, so 31 dominoes cover 31 of each. But opposite corners have the same colour, so the cut board has 32 squares of one colour and 30 of the other.

The colouring is an invariant: something every allowed move preserves. Ralph Gomory showed the converse is also neat. Remove any one black and one white square and a tiling always exists: the board’s squares form a single closed loop, and the two holes cut it into two even-length paths.