Toying Around II

Here are all the possibilities

{1, 1, 1, 1}
{2, 2, 2, 2}
{3, 3, 3, 3}
{4, 4, 4, 4}

{2, 1, 1, 1}
{2, 2, 1, 1}
{2, 2, 2, 1}

{3, 1, 1, 1}
{3, 3, 1, 1}
{3, 3, 3, 1}

{4, 1, 1, 1}
{4, 4, 1, 1}
{4, 4, 4, 1}

{3, 2, 2, 2}
{3, 3, 2, 2}
{3, 3, 3, 2}

{4, 2, 2, 2}
{4, 4, 2, 2}
{4, 4, 4, 2}

{4, 3, 3, 3}
{4, 4, 3, 3}
{4, 4, 4, 3}

{3, 2, 1, 1}
{3, 2, 2, 1}
{3, 3, 2, 1}

{4, 2, 1, 1}
{4, 2, 2, 1}
{4, 4, 2, 1}

{4, 3, 1, 1}
{4, 3, 3, 1}
{4, 4, 3, 1}

{4, 3, 2, 2}
{4, 3, 3, 2}
{4, 4, 3, 2}

{4, 3, 2, 1}
{4, 2, 3, 1}

Only in the case of 4 different colors can we have "chirality", i.e, where order of colors matter.

Let the rotation group of the tetrahedron (isomorphic to $A_4$ ) act on the set $S$ of all $4^4$ colourings of a tetrahedron. We want to know the number of distinct colourings (to within rotation), namely the number of orbits of this group action. By standard group action bookwork, this is equal to $\frac{1}{|A_4|}\sum_{g \in A_4} X(g)$ where $X(g)$ is the number of colourings in $S$ which are fixed by $g$ .

• If $g=1$ is the identity, then $X(g) = 4^4$ .
• If $g$ is a rotation through $\tfrac23\pi$ about an axis through one of the vertices, the colourings which are fixed by $g$ are those for which the three faces touching at vertex are all the same colour. Thus $X(g) = 4^2$ .
• If $g$ is a rotation through $\pi$ about an axis through the midpoints of a pair $e_1,e_2$ of opposite edges, then the colourings fixed by $g$ are those for which the two faces touching $e_1$ are the same colour, and the two faces touching $e_2$ are also the same colour. Thus $X(g) = 4^2$ in this case as well.

Thus the number of orbits is $\frac{1}{12}\big[4^4 + 8\times4^2 + 3\times4^2] \; = \; \boxed{36}$