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.

The 2x2 case is impossible, I suspect it is related in some way.

Seems this is related to Hamiltonian path problems. The issue is there is always one square that can’t be picked up. Why could this be?

I am afraid that I didn’t 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.

Rooks don’t move diagonal. I was able to do this exercise in my head on about the third try however. Start with big laps round the outside x2, then column by column to finish

I figure the last 3% of details to have the situation go perfectly are within the scope of luck and preparation. I typical start big and focus down as progress battles the solution.

Very empirical!