Math history · Puzzle

The Icosian game

By Melon Tools · 24 September 2026

In 1856 the Irish mathematician William Rowan Hamilton invented a puzzle: travel along the edges of a dodecahedron, visit each of its 20 corners exactly once, and arrive back where you started. It was sold as a toy, it flopped, and it gave its name to one of the most important problems in computer science.

Hamilton's rule is the Starwalk mode in Between Stars, a calm puzzle game for iPhone. Free on the App Store.

Play Hamilton's puzzle

This is the dodecahedron flattened onto the page, the same kind of drawing that was printed on Hamilton's wooden game board. Tap a star to start, then tap a connected star to travel there. Visit all 20, then step back onto your first star to close the round trip. The second tab is Hamilton's two-player version: the first five stars are already chosen, and you have to finish the trip.

The rules

A dodecahedron is the solid made of twelve regular pentagons. It has 20 corners and 30 edges, and three edges meet at every corner. The Icosian game asks for a route along the edges that:

1. visits every one of the 20 corners exactly once, and
2. ends on a corner next to the one it started on, so the route closes into a round trip.

Unlike in a one line puzzle, you do not have to use every edge. Only ten of the 30 are left unused by a finished tour. Mathematicians now call such a round trip a Hamiltonian cycle, and a route that visits every point once without returning a Hamiltonian path.

Hamilton, a toy maker and £25

By 1856 Hamilton was already famous. He was Andrews Professor of Astronomy at Trinity College Dublin, Royal Astronomer of Ireland, and the inventor of the quaternions. The game grew out of his work on the symmetries of the dodecahedron, which he described with an algebra he called the icosian calculus. The routes around the dodecahedron were a playful side result.

He sold the idea to Jaques and Son, a London toy and game maker, which marketed it from 1859. There were two versions: a flat wooden board with holes for numbered pegs, and a partly flattened dodecahedron with fixed pegs and a string to mark the route. They were sold under the names The Icosian Game and The Travellers Dodecahedron, or a voyage around the world. Hamilton received a licensing fee of just £25.

It was not a commercial success. The game was simply too easy to become popular, and the numbers below show why.

Why it was too easy

We searched every possible route on the dodecahedron by computer:

• There are exactly 30 different round trips if you ignore where you start and which way you go, the same number Wikipedia gives.
• There are 1,620 routes that visit all 20 corners without the requirement to get back home.
• Hamilton's two-player version, where one player picks the first five corners and the other must finish, is almost impossible to lose. Every opening of five connected corners can be completed to a round trip, in either two or four ways.

A puzzle that the second player can always finish is a nice pastime, but it does not make a best-selling parlour game.

Between Stars takes Hamilton's rule and gives it harder networks. In Starwalk, your aura has to reach every star exactly once. The Star Atlas has a hundred geometric patterns in ten chapters, including a chapter of solids, and one level is a tribute to Hamilton's puzzle. Every level is checked by a solver, so each one has a solution.

See both game modes

Hamilton was not the first

The problem carries Hamilton's name, but the English clergyman and mathematician Thomas Kirkman had studied round trips over the corners of solids a year earlier, and he had even found a polyhedron on which no such trip is possible. Hamilton visited Kirkman in 1861 and gave him a copy of the Icosian game.

From toy to one of the hardest problems

On the dodecahedron, a round trip is easy to find. In general, it is not. For an arbitrary network there is no known quick method to decide whether a Hamiltonian cycle exists. The problem is NP-complete, and in 1972 Richard Karp put it on his list of 21 problems that are, in a precise sense, among the hardest in computer science. Its best-known relative is the travelling salesman problem, which asks for the shortest round trip through a set of cities.

That is the big difference to the older puzzle about using every line once. For that one, Euler found a simple counting rule in 1735, told in our article on the Seven Bridges of Königsberg and put to work in our guide to one line puzzles. For visiting every point once, no such rule exists. The famous exception is the chessboard, where the knight's tour can be found quickly thanks to the board's regular shape.

Tips for Hamilton-style puzzles

1. Protect the lonely points. A point with only two unvisited neighbours left must be entered through one and left through the other. Once that happens, both of those connections are fixed.
2. Don't cut the network in two. If your route splits the unvisited points into two separate groups, you can never reach both. Check before each move.
3. Leave the way home open. For a round trip, at least one neighbour of your starting point must stay unvisited until the very end.
4. Think in faces. On the dodecahedron every finished tour splits the twelve pentagons into two strips of six. Keeping pentagons together in bands helps.

Visit every star, once

Between Stars is a calm puzzle game for iPhone with two modes: Starwalk, where you visit every star once like Hamilton's travellers, and Bridges, where you cross every connection once like Euler's walkers. No ads, no timers, undo as often as you like, and every level is solvable.

About the game

Free to download for iPhone with iOS 16 or later. No ads, no subscription.

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