### 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:

An irregular tetrahedron has two opposite sides the same length ($a$ say) and the line joining their midpoints is perpendicular to these two edges and is of length $b$. What is the volume of the tetrahedron?