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Here you see the front and back views of a dodecahedron which is a solid made up of pentagonal faces.

Using twenty of the numbers from $1$ to $25$, each vertex has been numbered so that the numbers around each pentagonal face add up to $65$.

The number F is the number of faces of the solid.

Can you find all the missing numbers?

You might like to make a dodecahedron and write the numbers at the vertices.

Why do this problem?

Visualising is a very important mathematical skill. Representing 3d shapes in a 2d form is a sophisticated form of visualising. This problem offers opportunities to visualise, and to use deductive reasoning with small numbers.

Possible approach

If your pupils are used to visualising, you may wish to go straight into the problem and see how far they can get. Display the pictures and ensure that everyone understands the problem, then after a little time draw the pupils together and ask for any useful statements or observations they can make. At this stage you may want to suggest that they could choose to use a 3d model if that would help, but you might also want to challenge them to do it 'in their heads'.

If your pupils are not used to visualising, you may want to begin by organising them into pairs to make a 3-d model from a net, or from other plastic shapes you have in the classroom, such as Polydron.

The main challenge is to match the information from one diagram onto the other. Once the children realise how the two diagrams are connected, the arithmetic is relatively trivial. Labelling the vertices and then opening up the net of the solid can help make the connections too.

Key questions

There are $9$s on both diagrams. Does that help? How?

What about the $25$?
What number does the F represent? How does the name of the shape help you to know?

Possible extension

You may wish to offer some other model-making activities which serve to consolidate 2d representation of 3d objects. There is a paper folding example of a dodecahedron here, which dextrous children may be able to do, perhaps with a little support. You might want to try making one yourself first!

Possible support

Support children who find visualising difficult with the latest in digital technology - use a digital camera to take photographs of the front and back of different 3d shapes including polyhedra and ask them to match them up, identifying common edges or vertices.