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### Number and algebra

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# Air Nets

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Age 7 to 18

Challenge Level

- Problem
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The video clips below show nets made from Polydron.

You will see our mathematician attempt to assemble the net into a solid 3D shape, sometimes successfully and sometimes unsuccessfully.

Before watching the mathematician fold each net, consider these questions:

You will see our mathematician attempt to assemble the net into a solid 3D shape, sometimes successfully and sometimes unsuccessfully.

Before watching the mathematician fold each net, consider these questions:

- Can you imagine folding the net up into a solid shape?
- Do you think that the net will fold into a shape with all sides clicked together?
- Can you imagine the shape of the final solid if the net does indeed correctly fold together?

As you watch the mathematician fold each net, consider these questions:

- Were you correct? Was the result a surprise in any way?
- Try again to imagine how the shape folded together.
- Draw an accurate drawing of the net. Can you see which sides joined together? Can you indicate this clearly on your diagram?
- If you have access to Polydron, try building each net and replicating the final solid, where one was created. Could you make a solid shape from the net in the cases where our mathematician failed, or is it actually impossible to make the net into a solid shape?

Finally, consider the mathematical properties of the nets:

- How might you be able to look at a net and be
*certain*that the net*will not*fold up into a solid? - How might you be able to be
*certain*that the net*will*fold up into a solid? - In what cases might you be unsure as to whether or not a net will fold up correctly? Can you give a good set of conditions for a net being a
*good possible candidate*for folding up into a solid?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

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.