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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?


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?


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?

Face Painting

Age 7 to 11
Challenge Level

We received this solution from someone who didn't give their name:

In a tetrahedron any two faces have a common edge so no two faces can be the same colour. A tetrahedron needs 4 colours. If we start by colouring one face, then the 3 faces adjoining it need 3 more colours.

A cube needs at least 3 colours because 3 faces meet at a point. Three colours are sufficient because each pair of opposite faces can be painted in one of the 3 colours.

An octahedron needs 2 colours. At each vertex 4 faces meet and they can be painted in alternate colours.

A dodecahedron needs at least 4 colours because if we start by colouring one face then we have to use 3 more colours to paint the faces around it. The net shows how the dodecahedron can be painted with 3 faces of each colour so 4 colours are sufficient.

dodecahedron net coloured with four colours
An icosahedron needs at least 3 colours because we have to use 3 colours to paint the 5 faces around each vertex. Three colours are sufficient as shown in the net. Net of icosahedron