This question was posed to me, and I was surprised that I could not find a solution (as I thought that all rook tours [open or closed] were possible). Starting from a8, could a rook visit every square on the board once, ending on f3?
I tried a few times, with a few different strategies, but I always ended up missing one square.
It's really easy to burn pairs of rows or columns, so the problem space could be reduced...
...but at some point (4x4), I was able to convince myself that it is impossible (at least at this size and state):
...but it might be possible that shaving off column or row pairs is also discarding a solution?
The rule is better said that you can visit each square "once and only once." Every time you move to a non-adjacent square, you're crossing a square for a second time and breaking the rule.
If you can only move to black squares from white squares then for a sequence with an even number of terms if you start on a white square you have to end on a black square. Because the problem states you have to start and end on white it's impossible to do without crossing over squares.