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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.

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    $\begingroup$ Try covering a $4 \times 4$ board to find the reason. $\endgroup$
    – Math Lover
    Sep 26, 2017 at 16:15
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    $\begingroup$ Every tile covers one white square and one black square. Your board has 32 black squares and 30 white squares. $\endgroup$
    – Doug M
    Sep 26, 2017 at 16:15
  • $\begingroup$ But is that the only reason why it's not solvable. $\endgroup$ Sep 26, 2017 at 16:16
  • $\begingroup$ @AnonymousI I am sure there are other proofs that show it is not solvable ... but this one is super straightforward! $\endgroup$
    – Bram28
    Sep 26, 2017 at 16:18
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    $\begingroup$ @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. $\endgroup$
    – Dando18
    Sep 26, 2017 at 16:21

1 Answer 1

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Every time you place a domino, two tiles of different color are covered: one white and one black. If you remove two white squares, then you're left with 30 white squares and 32 black squares, which means one domino would have to cover 2 black squares. We know this is impossible!

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