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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?


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?

Tetrahedron Faces

Age 7 to 11 Challenge Level:

Why do this problem?

This problem offers opportunities for children to visualise a 3D shape. It also encourages conjectures and justifications.

Possible approach

To begin with, encourage children to make conjectures about the number of different tetrahedra, asking pairs of children to come up with justifications for their ideas.

Having polydron available for children to make tetrahedra would be helpful. Alternatively, stickers could be stuck on ready-made tetrahedra. Children might also want to draw nets so isometric dotty paper would be useful.

Key questions

Where could the other colours go?
Can you find another way?
How do you know the tetrahedra are different?
How will you record your tetrahedra?