Problem 12 · Painting the faces of a cube
Colourings up to rotation: $6$, $30$ and $6!/24 = 30$.
Integer answer, at most 4 digitsMònica has six colours to paint the faces of a cube. In how many ways can she do it if:
- All faces are the same colour?
- Five faces are the same colour and one face is a different colour?
- All faces are different colours?
Two paintings are considered different if one cannot be obtained from the other by rotating the cube. As your answer, give the product of the three answers.
Reasoned solution
Key idea: the cube's rotation group has $24$ elements; in each part we look at what is genuinely different.
a) Just choose the colour: $6$ ways.
b) Choose the majority colour ($6$) and the odd face's colour ($5$): the position of the odd face is irrelevant, since any face can be taken to any other by a rotation. Total: $6 \cdot 5 = 30$.
c) There are $6! = 720$ assignments of colours to faces, and no rotation (except the identity) fixes a colouring with six distinct colours: each class has exactly $24$ representatives. Total: $720 / 24 = 30$.