Solution
Three Pairs on a Cube
Show the problem again
A cube is painted so that exactly two faces are red, two are white and two are blue. Two painted cubes are the same if one can be rotated to match the other. How many different painted cubes are there? Give an integer.
Worked solution
The answer is 6. Classify by which colour pairs sit on opposite faces. If all three pairs are opposite: 1 cube. If exactly one colour's faces are opposite (3 choices of that colour), the other four faces form a ring, and the two remaining colours must each occupy two adjacent faces of the ring, which is unique up to rotation: 3 cubes. Exactly two opposite pairs is impossible, since the third pair would then be forced opposite as well. If no pair is opposite, each colour sits on two adjacent faces, and the three pairs wrap around the cube; these come in two mirror-image forms that no rotation interchanges: 2 cubes. Total $1 + 3 + 2 = 6$. Burnside agrees: the identity fixes $6!/(2!)^3 = 90$, each of the 3 face half-turns fixes 6, each of the 6 edge half-turns fixes 6, all other rotations fix none, and $(90 + 18 + 36)/24 = 6$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.