Before the two squares are removed from the checkerboard,
the board contains 32 black and 32 white squares--that is,
the number of black squares <em>equals</em> the number of white.
However, after removing the upper left and lower right squares, the number of white squares is 30--two less than the number of black.
Each play of a domino covers one black and one white square.
Thus, each play leaves the board still unbalanced--leaves two more black than white squares to be covered.
When the white squares have all been covered, two black squares still remain uncovered.
In other words, no sequence of plays of dominoes exists that precisely covers the modified checkerboard.
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The key to finding this solution is the word <em>checkerboard.</em>
The notion of a checkerboard suggests, to a person, black and white squares.
If the puzzle had instead been stated as the problem of a real estate developer, the likelihood of finding the black-and-white argument would be small.
If the checkerboard had two squares removed, but squares of opposite color, the black-and-white argument would be of no use.
If ten squares were removed, the black-and-white argument might not help.
The color of the squares on the board has, of course, nothing to do with the fact that the board can be covered or that it cannot.