Question:

We use Burnside's lemma to count rotationally distinct colorings: the number of distinct colorings equals the average number of colorings fixed by each rotation in the rotation group. The rotation group of a cube has 24 elements: 1 identity, 6 face rotations (90°, 270°), 3 face rotations (180°), 8 vertex rotations (120°, 240°), and 6 edge rotations (180°). For each rotation, we count fixed colorings (where colors remain unchanged). The identity fixes all $2^8 = 256$ colorings. Face 90° rotations fix $2^3 = 8$ colorings (3 independent cycles of 4 vertices each). Using this analysis for all 24 rotations and applying Burnside's lemma: $\frac{1}{24}(256 + 6\cdot8 + 3\cdot16 + 8\cdot2 + 6\cdot4) = \frac{552}{24} = 23$.