Professor Erasmus has found a special way of moving a rook across a standard $8\times8$ chessboard that he modestly calls the "Professor-Erasmus-rook-tour". The professor claims that in this tour, the rook makes $64$ moves that visit all the $64$ squares of the chessboard. Every move is from some square to a horizontally or vertically adjacent square, and the very last move brings the rook finally back to its starting square. Exactly $32$ of these moves are in horizontal direction, and $32$ are in vertical direction.
Has the professor once again made one of his mathematical blunders, or do such rook tours indeed exist?
