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Knight through the board

Problem:
Starting from the top-left corner of the chessboard, can we move a knight to the bottom-right corner such that every square is visited once?

Solution:
No. Each step, the square the knight is in switches color. There are 63 (i.e. odd) steps to make, hence the last square must be of a different color from that of the starting point. However the top-left corner and the bottom-right corner have the same color. Contradiction.

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