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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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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

Extended Cube

Stage: 3 and 4 Short Challenge Level: Challenge Level:1
Each pyramid has five faces, but the square base is glued onto the cube, and so is not a face of the star. There are 6 pyramids, each contributing 4 faces, so the star has 24 faces.

There is one height of pyramid which will make the two faces that meet at each of the original edges become one face, giving a solid with 12 rhomboidal faces.

This problem is taken from the UKMT Mathematical Challenges.
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