The mutilated chessboard

Question

Take a standard 8 	imes 8 chessboard and remove the two squares at opposite corners. This leaves 62 squares. You have a supply of dominoes, each of which covers exactly two adjacent squares (horizontally or vertically). Can you tile the mutilated board with 31 dominoes, covering every remaining square exactly once? Prove your answer.

Accepted Answer

No, it is impossible. Here is an airtight proof by coloring invariant. Color the board in the usual alternating black-and-white pattern. A standard board has 32 black and 32 white squares. Key fact about the removed squares. Opposite corners of a chessboard are always the same color. (The corners of a chessboard alternate; diagonally opposite corners match.) Say both removed corners are white. Then the mutilated board has 32 	ext{ black} \quad	ext{and}\quad 32 - 2 = 30 	ext{ white squares}. Key fact about dominoes. Every domino covers two adjacent squares, and adjacent squares are always o…

The mutilated chessboard

Your answer

More Brainteasers questions