How many distinct ways are there to colour the six faces of a cube with two colours, counting two colourings as the same when one rotates into the other?

How many distinct ways are there to colour the six faces of a cube with two colours, counting two colourings as the same when one rotates into the other?

Approach: Classify the 24 rotations of the cube by axis type, count the cycles each one induces on the six faces, then average the number of colourings each rotation fixes.

10. Burnside's lemma says the number of orbits is the average number of colourings fixed by each element of the rotation group, and a rotation fixes a colouring exactly when every cycle of its action on the faces is one colour, giving 2^{number of face cycles}. The 24 rotations split into the identity with 6 face cycles, 6 quarter turns about face axes with 3 cycles, 3 half turns about face axes with 4 cycles, 8 rotations of order 3 about diagonals with 2 cycles, and 6 half turns about edge axes with 3 cycles. The fixed colourings total 2^6 + 6 * 2^3 + 3 * 2^4 + 8 * 2^2 + 6 * 2^3 = 64 + 48 + 48 + 32 + 48 = 240, and 240/24 = 10. Listing them by the count of one colour gives 1, 1, 2, 2, 2, 1, 1, which sums to 10 and confirms the cycle structure bookkeeping.

Follow-up: How many colourings are there with three colours, and how does the answer change if reflections are also allowed?

Key concepts: burnside lemma, rotation group, cycle structure, fixed colourings.