Solution
Six Colours, Six Faces
Show the problem again
Six different colours are available, and each face of a cube is painted a different colour (so each colour is used exactly once). Two cubes are the same if one can be rotated to look exactly like the other. How many distinguishable cubes are there? Give an integer.
Worked solution
The answer is 30. There are $6! = 720$ ways to paint the faces of a fixed cube, and since all colours differ, no nontrivial rotation leaves any painting unchanged. Each cube therefore corresponds to exactly 24 paintings (the size of the rotation group), giving $720/24 = 30$. Directly: rotate so colour 1 is on the bottom. The top can be any of the other 5 colours. The remaining 4 colours go around the sides, and rotating about the vertical axis leaves $3! = 6$ distinct circular orders. That is $5 \times 6 = 30$.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.