Closed rook tour on a chessboard

A rook makes a closed tour on a chessboard of 4n rows and 4n columns, passing over each square once, returning in the last move to the initial position. The tour is made of horizontal segments and vertical segments. Prove: The total length of the horizontal segments can't be equal to the total length of the vertical segments. [the length of a segment is measured from square center to square center, so the total length of all the segments is 16n^2 units, where 1 unit = side length of each square; I claim the total length of the horizontal (or vertical) segments cannot be 8n^2 units.]

Originally posted by David113
A rook makes a closed tour on a chessboard of 4n rows and 4n columns, passing over each square once, returning in the last move to the initial position. The tour is made of horizontal segments and vertical segments. Prove: The total length of the horizontal segments can't be equal to the total length of the vertical segments. [the length of a segment is mea ...[text shortened]... quare; I claim the total length of the horizontal (or vertical) segments cannot be 8n^2 units.]

so what, exactly, is the question?
🙂😉😕😉😛

"Prove: The total length of the horizontal segments can't be equal to the total length of the vertical segments."