The game's object is to find a cycle among along the edges of a dodecahedron, returning to its starting vertex after visiting each vertex a single time. In a two-player version of the game, one player starts by choosing five consecutive vertices along the cycle, and the other player must complete the cycle. Cycles that visit every vertex in a graph came to be known as Hamiltonian cycles because of Hamilton's work with this puzzle.^[1]

One version of the game took the form of a flat wooden board inscribed with a planar graph with the same combinatorial structure as the dodecahedron, with holes for numbered pegs to be placed at its vertices. The cycle found by game players was indicated by the consecutive numbering of the pegs.^[2] Another version used a solid dodecahedron.^[3]

The game was too easy to play to achieve much popularity.^[3][4]

At the time of his invention of the icosian game, William Rowan Hamilton was the Andrews Professor of Astronomy at Trinity College Dublin and Royal Astronomer of Ireland, and was already famous for his work on Hamiltonian mechanics and his invention of quaternions.^[5]

The motivation for Hamilton was the problem of understanding the symmetries of the dodecahedron and icosahedron, two dual polyhedra that have the same symmetries as each other. For this purpose he also invented icosian calculus, a system of non-commutative algebra which he used to compute these symmetries.^[6]

The name of the icosian game comes from the fact that the icosahedron has twenty faces, the dodecahedron has twenty vertices, and any cycle through all the vertices of the dodecahedron has twenty edges. Icosa is a Greek root meaning twenty. ^[4][7] On a dodecahedron with labeled vertices, there are 30 different ways that these vertices could be connected to each other to form a Hamiltonian cycle. However, without the labels, the resulting cycles are all symmetric to each other under rotations and reflections of the dodecahedron.^[8]

Both the icosian calculus and the icosian game were outlined by Hamilton in a series of letters to his friend John T. Graves in late 1856.^[6] After exhibiting the game at the 1857 Dublin meeting of the British Association for the Advancement of Science,^[9][10] Hamilton sold its publishing rights to Jaques and Son, a London-based toy and game manufacturing company.^[6][9]

This company marketed Hamilton's game beginning in 1859 under the lengthy title The Travellers Dodecahedron, or a voyage around the world, and the Icosian Game, invented by Sir William Rowan Hamilton, Royal Astronomer of Ireland; forming a new and highly amusing game for the drawing room, particularly interesting to students in mathematics of illustrating the principles of the Icosian Calculus.^[2]

Several other versions of the game were sold in Europe.^[8] However, it was not a commercial success.^[3][4][6] Hamilton received only a £25 licensing fee from Jaques and Son for his invention.^[8]

Although Hamilton invented the icosian game independenly, he was not the first to study Hamiltonian cycles. Knight's tours on chessboards, another puzzle based on Hamiltonian cycles, go back to the 9th century, both in India and in mathematics in the medieval Islamic world.^[11] At about the same time as Hamilton, Thomas Kirkman in England was also studying Hamiltonian cycles on polyhedra.^[12] Hamilton visited Kirkman in 1861, and presented him with a copy of the icosian game.^[6] Despite this related work, some of which was much earlier, Hamiltonian cycles came to be named for Hamilton and for his work on the icosian game.^[1]

Puzzles like Hamilton's icosian game, based on finding Hamiltonian cycles in planar graphs, continue to be sold as smartphone apps.^[13] Additionally, Maker-Breaker games based on Hamiltonian cycles continue to be an object of study in modern combinatorial game theory.^{[14][15][16][17][18][19]}

• Hunt the Wumpus, another game played on the graph of a dodecahedron
• Seven Bridges of Königsberg, a puzzle of finding a cycle through all edges of a graph

• Weisstein, Eric W., "Icosian game", MathWorld
• Puzzle Museum article with pictures