Challenge: Another game was marketed with the board for The Icosian Game as in Figure 12.22 (but laid out over the head of a mushroom-like object; see https://puzzlemuseum.com/month/picm02/200207icosian.htm). It was called The Traveler's Dodecahedron: A Voyage Round the World. In this case, two players start at some peg of the graph and walk to make a path of length four pegs. The goal is to complete a Hamilton circuit starting with this walk. Can it always be done?
