On a regular polyhedron a particle moves from a vertex with equal probability to one of the neighbouring vertices taking a unit time. Let it start at a vertex to make a tour around. How long does it take on an average to visit every vertex at least once? Probabilistic calculations show that the expected time is 5.5 for a tetrahedron, 248/21 for an octahedron and 1996/95 for a cube. The last case needs a complicated transition schema with simultaneous equations. For an icosa‐hedron we only know that the expected time will fall between 37 and 38, which is estimated by 4000 times of simulated trials. A simple device is sufficient for programming the simulation.