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?
I may not be understanding you correctly. Can you elaborate? Feel free to use coordinates.
I am afraid that I didnโt apply any rigor here and thought I solved it. Been playing with it for a bit and see there is likely not a solution. Good fun for me.
Seems this is related to Hamiltonian path problems. The issue is there is always one square that canโt be picked up. Why could this be?
The 2x2 case is impossible, I suspect it is related in some way.