A chess king travels over an entire chess board, passing through every square once, and without intersecting his path, gets back to the initial square. Show that the king has made at most 36 diagonal moves.
The king must visit the 28 perimeter squares in order; otherwise he will create a portion of the board which is inaccessible for him. However, he can not travel from one square to a neighboring one using only diagonal moves. Therefore he must make at least 28 horizontal/vertical moves and at most 64 - 28 = 36 diagonal moves.