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Tetrahedron Faces

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

Face Painting

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.


Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

A Chain of Eight Polyhedra

Age 7 to 11
Challenge Level

A Chain of Eight Polyhedra

These two 3-D shapes, the tetrahedron and the octahedron have the same 2-D shape, an equilateral triangle, as their faces.

Tetrahedron equilateral triangle and octahedron




Can you arrange the shapes below in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it? (The faces do not have to be the same size.)

eight polyhedra: cube, pentagonal based pyramid, pentagonal prism, square based prism, hexagonal prism, cuboid, hexagonal based prism, triangular prism

How many ways can you find to make a loop (a closed chain) using all the shapes so that each one shares a face (or faces) that are the same shape as the one that follows it?

Why do this problem?

This problem is an excellent opportunity to engage children in talking about and describing 3D shapes and relating them to the shapes of their 2D faces.

These cards might be useful for learners to use in pairs or groups.

Key questions

Why not list the shape of the faces of each 3D shape?
Can you begin to link pairs of shapes together?
Can you see another square/hexagon etc?
If you have found a chain, can you find a loop?
If you have found a loop, how many different ones can you find?

Possible extension

Learners could add some more 3D shapes and try to make a longer chain. Perhaps they could draw the shapes as in the problem.

Possible support

Suggest using these cards and listing some of the shapes of their faces. This list of faces that are given in the hints might be helpful.