KöMaL 2019 A745

We have attached a clock hand to every face of a convex polyhedron. Each hand always points towards a neighboring face (two faces of the polyhedron are neighbors if they share an edge). At the end of every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.