[This other answer](http://mathoverflow.net/a/9381) shows that an 8x8 board with opposite squares removed cannot be tiled with dominoes, as they are of the same "colour". But what if two squares of *opposite* colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and [this is his proof][1].

![Image of board][2]


(Imagine A and B are the squares removed.) The image is from Honsberger's _Mathematical Gems I_.


  [1]: http://shreevatsa.files.wordpress.com/2010/03/tiling-gomory.png
  [2]: https://i.sstatic.net/Yh9nT.png