Solution
Painting a Cube in Three Colours
Show the problem again
Each face of a cube is painted red, white or blue (colours may repeat, and not all need be used). Two painted cubes are the same if one can be rotated to look exactly like the other. How many different cubes are there? Give an integer.
Worked solution
The answer is 57. The rotation group of the cube has 24 elements. By Burnside, count colourings fixed by each, i.e. $3^{c}$ where $c$ is the number of face cycles. Identity: $3^6 = 729$. Six quarter-turns about face axes (2 fixed faces + one 4-cycle, $c = 3$): $6 \times 27$. Three half-turns about face axes ($c = 4$): $3 \times 81$. Eight turns about vertex diagonals (two 3-cycles, $c = 2$): $8 \times 9$. Six half-turns about edge axes (three 2-cycles): $6 \times 27$. Total $729 + 162 + 243 + 72 + 162 = 1368$, and $1368/24 = 57$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.