Solution
Black and White Corners
Show the problem again
Each of the 8 vertices of a cube is coloured black or white. Two colourings are the same if a rotation of the cube carries one to the other. How many different colourings are there? Give an integer.
Worked solution
The answer is 23. By Burnside over the 24 rotations, count $2^{c}$ with $c$ the number of vertex cycles. Identity: $2^8 = 256$. Six face quarter-turns (two 4-cycles): $6 \times 4 = 24$. Three face half-turns (four 2-cycles): $3 \times 16 = 48$. Eight vertex-axis turns (2 fixed vertices + two 3-cycles): $8 \times 16 = 128$. Six edge half-turns (four 2-cycles): $6 \times 16 = 96$. The sum is $256 + 24 + 48 + 128 + 96 = 552$, and $552/24 = 23$. As a sanity check, by number of black vertices the counts are 1, 1, 3, 3, 7, 3, 3, 1, 1 for 0 through 8 blacks, which also total 23.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.