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?
If you take the rooks movement, and reduce each longer move to a series of single steps a1a2a3a4 for example, this becomes a Hamiltonian problem. The corners have 2 nodes, the edges three, and the core has four. Black and white indicates even or uneven distance in nodes.
Ahh yes sorry my bad I mixed up hamiltonian and hilbert in my head