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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?

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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.

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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?

Cut Nets

Age 7 to 11 Challenge Level:

Cut Nets

The net of a cube has been cut into two. It could be put together in several ways so that it could be folded into a cube.


cube net cut in half

Here are the nets of $9$ solid shapes. Each one of these has been cut into $2$ pieces, like the net of the cube.


Can you see which pieces go together?

Why do this problem?

This problem makes children think hard about the nets of various polyhedra. It is not enough to know roughly what they look like. The number and shape of the faces and the way they are connected are important. It will help to develop children's visualisation skills and increase their familiarity with appropriate vocabulary. It provides an excellent opportunity for describing properties of 3D shapes.

Possible approach

You could start by talking about properties of some solid shapes with the group. One way to do this might be to hold up two shapes and ask them to talk about their similarities and differences. Depending on the children's experience you may need to explain what a net is, but they don't necessarily need to have worked extensively with nets in order to have a go at this problem.

You could introduce the problem on an interactive whiteboard or data projector but it would be helpful for children to have a copy of this sheet in pairs and to work together so that they are able to talk through their ideas with a partner. Scissors and sticky tape would be useful!

Some children will relish the challenge of discovering for themselves which shapes have been used but some may work better knowing which they were to find. The shapes are: a cube, cuboid, tetrahedron, square pyramid, pentagonal pyramid, hexagonal pyramid, triangular prism, pentagonal prism, trapezoid prism.

Key questions

How do you think these two pieces could fit together?
Which piece could go with this one?
What shape are the faces on a ...?

Possible extension

Learners could make some nets of these or other polyhedra and possibly cut them to make puzzles for a friend.

Possible support

You could start some children off with just three shapes using, for example, pieces $B$, $E$, $F$, $K$, $L$ and $M$.