This isn't homework or anything, this is just a question I randomly ran into. I have no idea where to start though. I'm supposed to prove (or disprove) this for 4x4, 5x5 and 6x6 boards.
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Look up the knights tour. It is how the knight can move around the chess board landing in each spot once. Since this is done on a 64 square board and is even numbered I would think it could be done on the 4x4 and 6x6, but not the 5x5. Used the same method of the knights tour to run your experiments.