"A sheet has a height (h) of at least 2 and a width (w) of at least 2. There are (w)x(h) tiles on the sheet. A valid path is one that moves adjacent to the current tile (forward, backwards, left, right, but not diagonal). Show that if either w or h is an even number, then there exists a path (P) that touches every single tile exactly once and ends up at the same tile it started at. Then prove that if both w or h is odd, then this path (P) can not exist."
I'm stumped. I'm not sure how to prove this. I have drawn it out and can see that it holds, but i'm not sure how to prove it generally for any number of w and h. We are told not to use induction on this question, and that was the only method i was thinking could work. Any help would be greatly appreciated.