Shortest rook tour

A closed tour visits every square once and returns to the start. Tour a small board with single steps along ranks and files, then ask for the fewest moves when a straight run counts as one move.

A closed single-step tour exists only when the number of squares is even, because the path alternates colours and must return to its starting colour. Counting straight runs instead of single steps, a snake (boustrophedon) path sweeps each rank in one run and steps between ranks, giving a short move count. Comparing tours reveals a lower bound you cannot beat.