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.
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!