# Why is the following problem impossible to solve?

Consider a chessboard ( an $$8\times8$$ board with 64 squares) and two opposite corner squares are removed.

Why can't you cover the rest of the board with domino tiles of 2 $$\times$$1?

What's the proof behind it?

I've read the solution on Gomory's theorem when the 2 opposite colors are removed from the chessboard. But I'm still curious behind the reason why this isn't possible.

• Try covering a $4 \times 4$ board to find the reason. Sep 26, 2017 at 16:15
• Every tile covers one white square and one black square. Your board has 32 black squares and 30 white squares. Sep 26, 2017 at 16:15
• But is that the only reason why it's not solvable. Sep 26, 2017 at 16:16
• @AnonymousI I am sure there are other proofs that show it is not solvable ... but this one is super straightforward! Sep 26, 2017 at 16:18
• @AnonymousI that is the proof. It's a proof by contradiction. We could make it a little more formal, but it's still the outline of the proof. Sep 26, 2017 at 16:21