Puzzle: You’ve got an 8×8 checkerboard and a bunch of dominoes that each fit nicely on two squares of the checkboard. You can easily tile the entire checkerboard with these dominoes. Now say that you remove two squares, one at one corner and the other at the opposite corner. You’re left with 62 squares. Can you tile this with the dominoes? If so, show how. If not, prove why not.

Hint: Use the colors of the squares.

Answer: You can’t.

Solution: Every domino will sit on one white square and one black square. Squares at opposite corners are of the same color, so after removing two opposite corners you’re left with 30 white squares and 32 black ones (or vice versa). At best you can put down 30 dominoes, each of which will use up one white and one black square, but after that you’ll be left with two squares of the same color. These can’t be next to each another, so your last domino won’t go anywhere.