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Imagine a 8x8 cell board, but missing two cell at the opposite corners, a domino take up exactly two cell, how to fill the board with dominoes so that none overlap or hang off the edge?

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Color it as a chessboard and count the number of white and black squares to see that it is impossible. – t.b. Aug 23 '11 at 7:27
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Before you ask "How can you fill the chessboard?" you should ask "Can you fill the chessboard?" – Aaron Aug 23 '11 at 7:29

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This is a well-known problem - it even has its own wikipedia article: Mutilated chessboard problem.

Here you can find a few similar but more challenging problems for trominos.

Problems like this are usually solved using some type of coloring argument, as explained in Theo's comment and the above links.

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