A rook makes a closed tour on a chessboard of m ranks and n files, visiting each square exactly once and returning to the starting point. The tour consists of horizontal segments and vertical segments. It seems that the total length of the horizontal segments is divisible by 4 if and only if m is not divisible by 4 and n is odd, but I can't prove this yet. Can someone prove this?
(The length of a segment is measured between centers of squares, for example if the rook moves from d8 to f8 that's a horizontal segment of length 2.)