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?

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Daniel's Star

Stage: 3 Short Challenge Level:
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.