The Icosian game, also called the Hamiltonian game (Ball and Coxeter 1987, p. 262), 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.
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.