# In how many ways can the faces of a rectangular box can be painted so that the color changes occur only at each corner?

With three differently colored paints, in how many ways can the faces of a rectangular box can be painted so that the color changes occur only at each corner?

I was trying to solve this by principle of inclusion and exclusion, but I am unable to enumerate the number of ways to color the rectangular box( without any restrictions) because there are some distinct faces(adjacent faces) and some are non distinct faces(opposite faces).
Edit: The colorings which differ by rotation/s are considered to be same.

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Exactly what do you mean when you say that color changes occur only at each corner? – Brian M. Scott Dec 10 '12 at 6:26
Do you mean, at each edge? – Did Dec 10 '12 at 6:28
Opposite faces could be distinct, I think. Are mirror-symmetric colorings considered identical? – Gyu Eun Lee Dec 10 '12 at 6:29
Is there a rule that the colour must change where two walls meet? Also, I expect you can figure out the answer, the number is small. To simplify calculations, imagine that the West wall is painted red, and then multiply whatever answer you get by $3$. – André Nicolas Dec 10 '12 at 6:29
@did yes it means edges – Saurabh Dec 10 '12 at 6:30