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Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Cubic Conundrum

Which of the following cubes can be made from these nets?

Instant Insanity

We have a set of four very innocent-looking cubes - each face coloured red, blue, green or white - and they have to be arranged in a row so that all of the four colours appear on each of the four long sides of the resulting cuboid.

Platonic Planet

Age 14 to 16
Challenge Level


Glarsynost the alien lives on a platonic planet whose shape is that of a perfect regular dodecahedron.

Every day, she likes to go for a walk to have a look at her planet and see if she has any visitors.

What fraction of the planet's surface can she see from the middle of a face? From an edge? From a vertex?

Her walk needs to start and end at the same place, and she needs to be able to see every part of the planet's surface at some stage during her walk. Investigate the possible paths she could take. The challenge is to find the shortest path you can!

One way of investigating and recording this could be to create a net of a dodecahedron, and draw the path on the net, being careful to consider which faces will join when the net is folded up.

Here are two nets you could use, but you may find it easier to visualise an efficient path using a different one.

Net 1

Net 2