Solution
Painting a Dodecahedron
Show the problem again
Each of the 12 faces of a regular dodecahedron is painted one of 3 colours (repeats allowed, not all colours need be used). Two painted dodecahedra are the same if a rotation carries one to the other. How many different painted dodecahedra are there? Give an integer.
Worked solution
The answer is 9099. The rotation group has 60 elements. Classify them by their action on the 12 faces. The identity has 12 face cycles. There are 6 axes through opposite face centres, each with 4 nontrivial rotations (by multiples of 72°): each fixes 2 faces and cycles the other 10 in two 5-cycles, so 4 cycles (24 rotations). There are 10 axes through opposite vertices, each with 2 nontrivial rotations: four 3-cycles (20 rotations). There are 15 axes through opposite edge midpoints, each with a half-turn: six 2-cycles (15 rotations). Burnside: $\frac{1}{60}\left(3^{12} + 24 \cdot 3^4 + 20 \cdot 3^4 + 15 \cdot 3^6\right) = \frac{531441 + 1944 + 1620 + 10935}{60} = \frac{545940}{60} = 9099$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.