First I asked whether it was possible to colour the edges with 3 colours so that at every vertex 3 different colours met. Initially we thought it would be impossible and set out to find a proof of this. However, our investigations showed that it was actually possible! The colours around a vertex can be ordered in two ways: blue-red-green or blue-green-red. It turns out that out of the 20 vertices, 16 of them will have the same colour ordering and 4 of them with be the other one. And these 4 vertices (circled in the top picture) form the vertices of a regular tetrahedron! Magic.
After this challenge, the next obvious question was whether we could colour the edges with 5 colours so that around every face of the dodecahedron there are 5 different colours. This was a much easier task but the results were, in many ways, much more beautiful.
Since there are 30 edges to be coloured in 5 colours, there must be 6 edges of each colour. Toby noticed that there are 5 different cubes that can be embedded in a dodecahedron, and wondered if there would be a relation between the cubes and the colouring. The photo below shows where the 5 differently coloured cubes sit inside the dodecahedron. A colouring of the faces is therefore obtained by looking at the pentagram formed inside a pentagonal face, and colouring the opposite edges in the colours given.