Covering problem (board and dominoes)
We cut off some squares from the bottom row (rank) of a 2k x 2k board. We should prove that we can cover the board with 2 x 1 dominoes if and only if the number of the cut black and white squares are equal.
As far as I know proving 'only if' part is easy because a dominoe always cover one white square and one black square. There was even (2k*2k) square, so it is obvious than we should have cut off equal black as white squares. (If we had cut of more black squares, there would have been more white squares, so we couldn't cover the board.) So I proved that it is necessity. -Am I right?
But how can we prove the 'if' part? I don't see why it is a possibility?
I would really appreciate any help.