We know from a theorem by Schwenk that for any (m x n) chess board with $m \leq n$ it is always possible to create a knight's tour unless one or more of these three conditions are met:
- m and n are both odd
- m = 1,2 or 4
- m = 3 and n = 4, 6 or 8
When I was about 15 years old, I played a game that I had "invented" myself. Years later, I realised this game was actually a generalisation of the Knight's tour problem. The generalisation is that the knight can jump of the edge, and reappear on the opposite side of the board. From there, it can complete its knight move.
Here is an example of what I mean. The red numbers indicate the trajectory of a knight that traverses in the conventional manner. The knight begins at tile 1 and ends in tile 3. The green trajectory indicates the manner in which the knight can travel in the new generalised Knight's Tour problem. Again, it begins at tile 1 and ends at tile 3.
I was wondering whether, with these new abilities, the knight can make a Knight's Tour under the conditions mentioned in Schwenk's Theorem.
Please note that the knight can only reappear on the other side of the board once every move. Furthermore, it can only do it when it is moving in a direction perpendicular to the edge of the board.