The Knight’s Grand Circuit

Place a knight on any square of a standard 8×8 chessboard. It then makes ordinary knight moves: two squares in one direction and one square at right angles. The starting square counts as visited. Can the knight travel through the board so that every one of the 64 squares is visited exactly once? It may finish anywhere, and it does not need to return to its starting square. No square may be landed on twice. Decide whether such a route exists. A convincing solution should do more than guess: either explain why the task is impossible or provide a valid method or route showing that it can be done.