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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.


Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?


Age 7 to 11 Challenge Level:


Consider a cube and paths along the edges of the cube.

Mark one vertex red.


Colour other vertices red if they can be reached from a red vertex by travelling along an even number of edges of the cube.

Colour vertices blue if they can be reached by travelling along an odd number of edges from a red vertex.

Is it possible to have vertices which are both red and blue at the same time (call these redblue vertices)?

Now do the same for a tetrahedron.

Do the same for other solids, for example the octahedron, dodecahedron and icosahedron, and prisms with different cross sections. Remember the paths must be along the edges of the solids. Decide how to record what you find. What property does the solid need if it is to have redblue vertices?

Why do this problem?

This activity is very good to have pupils explore some special properties of solid shapes. It is particularly good in that the solutions do not appear very obviously.

Possible approach

Introduce the pupils gradually to the recognition of what are vertices and what are edges. Most pupils will then be needing to handle the shapes unless they are gifted in visualisation.

Key questions

Tell me about your shape.
What vertices have you used?

Possible extension

Extensions are included in the last part of the activity.

Possible support

Shapes made from pipe-cleaners and straws can be a useful aid.