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The Icosian game, also called the Hamiltonian game (Ball and Coxeter 1987, p. 262), is the problem of finding a Hamiltonian circuit along the edges of an dodecahedron, i.e., a path 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, illustrated above (right figure). 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).
A graph having a Hamiltonian circuit, 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).
