Suppose you have a checkerboard with two opposite corner squares removed, like this:
Can you move a pawn on the board only horizontally and vertically, one square at a time, and have it touch every square exactly once?
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Sign up to join this communitySuppose you have a checkerboard with two opposite corner squares removed, like this:
Can you move a pawn on the board only horizontally and vertically, one square at a time, and have it touch every square exactly once?
Every move the pawn makes it switches from a white to a black square or vice versa. Therefore it must either touch an equal number of white and black squares with an even number of moves, or one more square of either color with an odd number of moves. (A move including initially placing it on the board.)
Because there are two more white squares than black squares (32 and 30) there is no way it can touch each square exactly once.
The same argument applies to show that the Knight's Tour cannot be completed on this board either.