### Proximity

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

### 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?

### Three Cubes

Can you work out the dimensions of the three cubes?

# Sliced

##### Age 14 to 16 Challenge Level:

Slice the tetrahedron in two with a cut through the line $b$ and one of the equal edges. The perpendiculars are very important because of what you need to find the volume of a pyramid. You now have two identical pyramids like this: