Key idea. Count black and white squares, not just the total area.

1. An \(8\times8\) chessboard has 32 black and 32 white squares.

2. Two opposite corners have the same colour. Removing them leaves 30 squares of that colour and 32 of the other.

3. Each domino covers two squares sharing a side. On a chessboard these always have different colours, so every domino covers one black and one white square.

4. Any collection of dominoes therefore covers equal numbers of the two colours. It cannot cover a region with 30 of one and 32 of the other.

Remember. An even area is necessary for a domino tiling, but colour counts can reveal another obstruction.