Icosian Game

The Icosian game, also called the Hamiltonian game (Ball and Coxeter 1987, p. 262; Knuth 2025, pp. 1-2), is the problem of finding a Hamiltonian cycle along the edges of an dodecahedron, i.e., a walk through the graph such that every vertex is visited a single time, no edge is visited twice, and the ending point is the same as the starting point (left figure). The puzzle was distributed commercially as a pegboard with holes at the nodes of the dodecahedral graph. The Icosian Game was invented in 1857 by William Rowan Hamilton. Hamilton sold it to a London game dealer in 1859 for 25 pounds, and the game was subsequently marketed in Europe in a number of forms (Gardner 1957). The 30 solutions corresponding to the 30 Hamiltonian cycles of the dodecahedral graph are illustrated above.

A graph having a Hamiltonian cycle, i.e., on which the Icosian game may be played, is said to be a Hamiltonian graph. While the skeletons of all the Platonic solids and Archimedean solids (i.e., the Platonic graphs and Archimedean graphs, respectively) are Hamiltonian, the same is not necessarily true for the skeletons of the Archimedean duals, as shown by Coxeter (1946) and Rosenthal (1946) for the rhombic dodecahedron (Gardner 1984, p. 98).

Wolfram (2022) analyzed the icosian game as a multicomputational process, including through the use of multiway and branchial graphs. In particular, the multiway graph for the icosian game begins as illustrated above.