I think you want only the DISTINGUISHABE ways to paint a cube. There are only 2
DISTINGUISHABLE ways.
1. Use one color to paint three mutually adjacent faces (which have one vertex
in common) and use the other color to paint the other three mutually
adjacent faces (which also have one vertex in common).
2. Use one color to paint three faces which are not mutually adjacent, i.e., the
middle one is adjacent to the other two, No vertex is in common to all three.
Then use the other color to paint the other three faces, which are the same way,
not mutually adjacent, i.e., the middle one is adjacent to the other two, No
vertex is in common to all three.
If it seems to you that that there are other ways besides these two, remember
that any other way is simply a 90, or 180 degree rotation left, right, forward
or backwards of one of the above two.
Edwin
