Dominoes on a mutilated chessboard

Idea. Every domino covers exactly one black and one white square.

Colouring. Colour the board like a chessboard. Two opposite corners, for instance $(1,1)$ and $(8,8)$, have the same colour: $1+1$ and $8+8$ are both even.

Counting. The full board has $32$ black and $32$ white squares. Removing two squares of the same colour leaves $32$ of one colour and $30$ of the other.

Contradiction. Any tiling by $31$ dominoes covers exactly $31$ black and $31$ white squares. Impossible.

Answer. No.

Remark. If the two removed squares have different colours, a tiling always exists (Gomory's theorem).

Lesson. Colouring: choose a colouring in which every tile covers a prescribed number of squares of each colour, then compare with the colour counts of the region.