Fill the board with dominoes Imagine a 8x8 cell board that is missing two cells at the opposite corners, a domino takes up exactly two cells. How can you fill the board with dominoes so that none overlap or hang off the edge? !enter image description here

This is a well-known problem - it even has its own wikipedia article: Mutilated chessboard problem.

Here you can find a few similar but more challenging problems for trominos.

Problems like this are usually solved using some type of coloring argument, as explained in Theo's comment and the above links.