Given an infinite chessboard, how many moves can a knight make such that it returns to its starting square, but doesn't visit any other square more than once? It seems clear to me that the minimum number of moves is 4, and that there is no maximum. I can also construct sequences of length 6, 8, etc. But are there any sequences of odd length?
I think my question is equivalent to asking whether, on an infinite knight's graph, there exist any cycles of length 2k + 1 for k > 2.
You could think of this problem as one in linear algebra. You have a "basis" of moves you can make. Which linear combinations of them (over $\mathbb{Z}) give you the zero vector? Please note that a lot of the above may need to be taken quite loosely.