8x8 checkers board

Question

Given a standard 8 x 8 checker board, remove two squares from opposite ends of a diagonal. Suppose
that a rectangular domino exactly covers two squares of the board. You must either find a way to cover
every square that remains on the board with dominoes (you have 32 dominoes) without any overlap or
prove that this is impossible.

Answer 1

The answer is that it's not possible.  But the reason is tricky...

 

Each domino will go on two adjacent squares, and adjacent squares are always different colors.  So no matter how many dominoes you put down, you will always cover and equal number of red and black squares.

 

But opposite corners are the same color.  So when you remove those two, you have an UNEQUAL number of red and black -- either 30 red and 32 black, or 30 black and 32 red.

 

So there's no way to cover all of the remaining squares with dominoes that have to cover one red and one black at a time.