Solution
Colouring the Edges of a Cube
Show the problem again
Each of the 12 edges of a cube is coloured red or blue. Two colourings are the same if some rotation of the cube carries one to the other. How many different colourings are there? Give an integer.
Worked solution
The answer is 218. Apply Burnside with $2^{c}$ fixed colourings, where $c$ is the number of edge cycles of each rotation. Identity: $c = 12$, giving 4096. Face quarter-turns (6 of them): the 12 edges fall into three 4-cycles (top ring, bottom ring, verticals), $2^3 = 8$ each, total 48. Face half-turns (3): six 2-cycles, $2^6 = 64$ each, total 192. Vertex-axis turns (8): four 3-cycles, $2^4 = 16$ each, total 128. Edge half-turns (6): the axis edge and its opposite are fixed and the other 10 edges form five 2-cycles, so $c = 7$, $2^7 = 128$ each, total 768. Sum: $4096 + 48 + 192 + 128 + 768 = 5232$, and $5232/24 = 218$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.