Polyhedra

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Can you work out the dimensions of the three cubes?

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?

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