A rook makes a closed tour on a chessboard of m ranks and n files, visiting each square once, and returning to the starting point. The tour consists of horizontal segments and vertical segments. I am trying without success to prove the following result: 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. (The length of a segment is measured between centers of squares, so, for example, if the rook moves from d8 to f8 that's a horizontal segment whose length is 2.)
Any help in solving this problem would be appreciated.

A Wazir is a single-step rook.

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