### Christmas Presents

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

### Platonic Planet

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?

### 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.

# Air Nets

##### Age 7 to 18 Challenge Level:

The video clip below shows a net made from Polydron. You can change the net using the drop down menu on the right hand side.

When you press play, you will see our mathematician attempt to assemble the net into a solid 3D shape. Sometimes he will succeed, sometimes he will fail.

Before watching each video clip, consider these questions:

1. Can you imagine folding the net up into a solid shape?
2. Do you think that the net will fold into a shape with all sides clicked together?
3. Can you imagine the shape of the final solid if the net does indeed correctly fold together?
Now watch the videos and consider some of these questions:
1. Were you correct? Was the result a surprise in any way?
2. Try again to imagine how the shape folded together.
3. Draw an accurate drawing of the net. Can you see which sides joined together? Can you indicate this clearly on your diagram?
4. 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:
1. How might you be able to look at a net and be certain that the net will not fold up into a solid?
2. How might you be able to be certain that the net will fold up into a solid?
3. 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?

If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.