The Notched Checkerboard

Strategies and Hints

  1. The squares are either side-by-side or one is on top of the other.

  2. The coordinates of a pair must be either (a, b) and (a + 1, b) if the squares are side by side, or (a, b) and (a, b + 1) if one square is on top of the other.

  3. One square is black and the other is white.

  4. The 31 dominoes must cover 31 black squares. If you can show that the notched checkerboard does not have 31 black squares, you can prove it is impossible to cover the board with the 31 dominoes.

Solution

It is impossible to cover the board with the 31 dominoes. The two squares removed from the corners are the same color. So, the notched checkerboard either has 30 black squares and 32 white squares or it has 32 black and 30 white. In either case, the board cannot be covered by the 31 dominoes because they would cover 31 of each color. The solution can also be expressed in terms of ordered pairs.