Solution
Painting a Tetrahedron
Show the problem again
Each face of a regular tetrahedron is painted one of 4 available colours; colours may repeat and need not all be used. Two painted tetrahedra are the same if one can be rotated to match the other (reflections are not allowed). How many different painted tetrahedra are there? Give an integer.
Worked solution
The answer is 36. The rotation group has 12 elements: the identity, 8 rotations by ±120° about an axis through a vertex and the opposite face centre, and 3 half-turns about axes through midpoints of opposite edges. Fixed colourings: identity $4^4 = 256$; a 120° rotation fixes one face and cycles three, so $4^2 = 16$ (8 of these give 128); a half-turn swaps faces in two pairs, again $4^2 = 16$ (3 give 48). Burnside: $(256 + 128 + 48)/12 = 432/12 = 36$. A direct count agrees: 4 one-colour, 12 with a 3+1 split, 6 with 2+2, 12 with 2+1+1 and 2 mirror-image tetrahedra using all four colours.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.