A flea sits on one vertex of a regular tetrahedron. He hops continually from one vertex to another, resting for a minute between hops and choosing vertices without bias. Prove that, counting the first hop, we’d expect him to return to his starting point after four hops.
This appeared originally in the Journal of Recreational Mathematics in July 1969, posed by David L. Silverman. Benjamin L. Schwartz of McLean, Va., sent in three different arguments, the simplest of which is purely heuristic. Imagine that the flea is on a dodecahedron (20 vertices), and that he hops around it for a long period of time, say 10,000 hops. He’ll occupy each vertex for approximately the same time, in this case 500 visits. “Then obviously, his average time between calls at that vertex is 20, the number of vertices.”
More formal proofs are possible; see R. Robinson Rowe, “Random Hops on Polyhedral Edges and Other Networks,” Journal of Recreational Mathematics 4:2 [April 1971], 124-130, as well as “Solutions to Problems” in that issue.
