Problem

Source: Spring 2005 Tournament of Towns Senior A-Level #6

Tags: Chessboard, Chess rook



A lazy rook can only move from a square to a vertical or a horizontal neighbour. It follows a path which visits each square of an $8 \times 8$ chessboard exactly once. Prove that the number of such paths starting at a corner square is greater than the number of such paths starting at a diagonal neighbour of a corner square. (7 points)