Solution
The Mutilated Chessboard
Show the problem again
Two diagonally opposite corner squares are removed from a chessboard, leaving 62 squares. Can the remaining board be tiled exactly by 31 dominoes? Answer yes or no, and be ready to prove it.
Worked solution
No, it is impossible. Color the board in the standard checkerboard pattern. Opposite corners share a color, so the mutilated board has 32 squares of one color and 30 of the other. Every domino covers exactly one square of each color, so 31 dominoes cover 31 of each: a mismatch. No tiling exists.
Source: Posed by Max Black (1946); the colouring argument is commonly credited to Ralph Gomory. Statement written for AxiomIQ.