Icosian Game
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 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 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).
SEE ALSO: Dodecahedral Graph,
Dodecahedron,
Hamiltonian
Cycle,
Hamiltonian Graph,
Herschel
Graph,
Polyhedral Graph
REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 262-266, 1987.
Coxeter, H. S. M. "Problem E 711." Amer. Math. Monthly 53,
156, 1946.
Dalgety, J. "The Icosian Game." http://puzzlemuseum.com/month/picm02/200207icosian.htm.
Gardner, M. "Mathematical Games: About the Remarkable Similarity between the Icosian Game and the Towers of Hanoi." Sci. Amer. 196, 150-156,
May 1957.
Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University
of Chicago Press, 1984.
Hamilton, W. R. Quart. J. Math., 5, 305, 1862.
Hamilton, W. R. Philos. Mag. 17, 42, 1884.
Harary, F. Graph
Theory. Reading, MA: Addison-Wesley, p. 4, 1994.
Herschel, A. S. "Sir Wm. Hamilton's Icosian Game." Quart.
J. Pure Applied Math. 5, 305, 1862.
Lucas, E. Récréations mathématiques, Vol. 2. Paris: Gauthier-Villars, pp. 201
and 208-255, 1891.
MacTutor Archive. "Mathematical Games and Recreations." http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Mathematical_games.html#49.
Pegg, E. Jr. "The Icosian Game, Revisited." Mathematica
J. 310-314, 11, 2009.
Rosenthal, A. "Solution to Problem E 711: Sir William Hamilton's Icosian Game."
Amer. Math. Monthly 53, 593, 1946.
Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.
Reading, MA: Addison-Wesley, p. 198, 1990.
Tutte, W. T. "On Hamiltonian Circuits." J. London Math. Soc. 21,
98-101, 1946.
Referenced on Wolfram|Alpha:
Icosian Game
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