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# Icosian Game

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Age 11 to 14

Challenge Level

- Problem
- Student Solutions

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only- returning to the vertex you started at.

This is quite difficult to visualise in 3D so this problem involves drawing 2D skeletons of the five Platonic solids (known as Schlegel graphs) and using these to try to answer the challenge we have set you below.

Schlegel graphs are very useful in studying molecular structure. It is as if all the edges were made of elastic and they have been stretched out to make the skeleton flat. Here is a cube and its Schlegel graph:

Can you sketch the skeletons on paper for the tetrahedron, octahedron, dodecahedron and icosahedron? When you think you have a skeleton - you can check your own answer by clicking to reveal the Schlegel graphs.

The problem is in two parts:

· to find the paths along the edges of the solids that enable you to visit every vertex and

· to describe the strategies you have used and what you discovered. For example: is the path unique?

These paths are called Hamiltonian Circuits after the Irish mathematician William Hamilton who also invented this game in 1857.

Can you number the vertices, edges and faces of a tetrahedron so that the number on each edge is the mean of the numbers on the adjacent vertices and the mean of the numbers on the adjacent faces?

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?