- 07 Jun '06 21:08My rather artistic friend is painting the faces of a cube: on each

face she uses one of her two favourite colours: either red or green.

How many different ways can she paint the cube?

Suppose now we declare that two paintings are in the same "rotation class" if there exists a rotation of the cube sending one

painting to the other. (So for example, the painting which has the

top face red, and the rest green, is in the same rotation class as the

painting which has the bottom face red, and the rest green.)

How many different rotation classes are there?

Now repeat all the above using the three colours red, green, blue

to paint the cube, instead of two. - 07 Jun '06 22:59

I'm thinking 2^6 and 3^6, 2=32 ways and 3 colors=729 ways.*Originally posted by SPMars***My rather artistic friend is painting the faces of a cube: on each**

face she uses one of her two favourite colours: either red or green.

How many different ways can she paint the cube?

Suppose now we declare that two paintings are in the same "rotation class" if there exists a rotation of the cube sending one

painting to the other. (So for example, the ...[text shortened]... eat all the above using the three colours red, green, blue

to paint the cube, instead of two. - 08 Jun '06 11:43

I think 55 rotation classes:*Originally posted by SPMars***Agingblitzer is correct -- there are 10 rotation classes with two colours.**

But his answer is a little too low on the case of 3 colours. The tricky case (I think) is when you have two of each colour.

(Of course, the number of paintings are 2^6 and 3^6 respectively.)

6 -> 3 ways

5,1 -> 6

4,2 -> 12

4,1,1 -> 6

3,3 -> 6

3,2,1 -> 18

2,2,2 -> 4