We received this solution from someone who didn't give their name:

In a tetrahedron any two faces have a common edge so no two faces can be the same colour. A tetrahedron needs 4 colours. If we start by colouring one face, then the 3 faces adjoining it need 3 more colours.

A cube needs at least 3 colours because 3 faces meet at a point. Three colours are sufficient because each pair of opposite faces can be painted in one of the 3 colours.

An octahedron needs 2 colours. At each vertex 4 faces meet and they can be painted in alternate colours.

A dodecahedron needs at least 4 colours because if we start by colouring one face then we have to use 3 more colours to paint the faces around it. The net shows how the dodecahedron can be painted with 3 faces of each colour so 4 colours are sufficient.

An icosahedron needs at least 3 colours because we have to use 3 colours to paint the 5 faces around each vertex. Three colours are sufficient as shown in the net.