cube

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 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.

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.

I'm thinking 2^6 and 3^6, 2=32 ways and 3 colors=729 ways.

And I'm wondering why your post is shorter than his!

Originally posted by Bowmann
And I'm wondering why your post is shorter than his!

Yours is shorter again!

for 2 colours I would say 10 rotation classes

and for 3, errrrm, 50

42 combinations for 2 if rotation didnt matter..

and for 3..... i owuld have to use paper for that.

I don't really know, 18??

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.)

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.)

I think 55 rotation classes:

6 -> 3 ways
5,1 -> 6
4,2 -> 12
4,1,1 -> 6
3,3 -> 6
3,2,1 -> 18
2,2,2 -> 4

Originally posted by XanthosNZ
Yours is shorter again!

So?

Almost howardbradley!

There are 57 rotation classes when we use 3 colours (all your figures are correct, except for the 2-2-2 case when there are 6 possibilities.)

Originally posted by SPMars
Almost howardbradley!

There are 57 rotation classes when we use 3 colours (all your figures are correct, except for the 2-2-2 case when there are 6 possibilities.)

Rats! I forgot about chirality 😳