Consider a chess board with two of the diagonally opposite corners removed. Is it possible to cover the board with
pieces of domino whose size is exactly two board squares?
Solution
No, it's not possible. Two diagonally opposite squares on a chess board are of the same color.
Therefore, when these are removed, the number of squares of one color exceeds by 2 the number of
squares of another color. However, every piece of domino covers exactly two squares and these are
of different colors. Every placement of domino pieces establishes a 1-1correspondence between the set of white squares and the set of black squares. If the two sets have
different number of elements, then, by the Pigeonhole Principle, no 1-1 correspondence between the
two sets is possible.
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