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All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

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Three Cubes

Can you work out the dimensions of the three cubes?

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Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

Twelve Cubed

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

In total there are $12\times12\times12$ or $12^3$ cubes with edge length $1$cm. Each cube has twelve edges, so the sum of the lengths of the edges on one cube is $12$cm. Therefore the total length of all the edges of all the centimeter cubes is $12^3\times12 = 12^4 = 20,736$cm

This problem is taken from the UKMT Mathematical Challenges.