  We consider vertex colourings of the dodecahedral graph with five colours,
such that on each face the vertices are coloured with all the five colours. We
show that the total number of these colourings is 240. All such colourings can
be obtained from any given such colouring, by permuting the colours, and
possibly applying central symmetry with respect to the centre of the regular
dodecahedron. For any such colouring, the colour classes form the vertex sets
of five regular tetrahedra. These tetrahedra together form one of the two
compounds of five tetrahedra, inscribed in the regular dodecahedron. We give
two proofs: a combinatorial one, and a geometrical one. Our result is related
to the result in W. W. Rouse Ball -- H. S. M. Coxeter, stating that there are
four such colourings, as follows. There are four such colourings, up to
applying an arbitrary orientation-preserving congruence of the regular
dodecahedron.
