In how many ways can the 6 faces of a cube be colored black or white? Two colorings are the same if one can be obtained from the other by a rotation.
Let's observe that it suffices to count how many colorings exist that have exactly 0, 1, 2, and 3 black faces, because the other cases are symmetric. With one or no black face, there is a unique coloring for each case. When we have two black faces, we have two possible colorings: when these faces are opposite and when they are not. Lastly, with three black faces, we also have two cases: when two of these black faces are opposite and when there are no opposite black faces. Thus, in total, we have $1+1+2+2+2+1+1=10$ possible colorings.

## Related Problem

In how many ways can the 12 edges of a cube be colored white or black? Two colorings are considered the same if one can be obtained from the other by a rotation.

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