If you paint the six sides of a 3x3x3 cube and then slice it into 27 unit cubes, how many will have paint on 0, 1, 2, 3 of their square sides?

Generalize to any dimension:
If you paint the sides of a n dimensional cube with sides length 3 and slice it into 3ⁿ unit cubes, how many will have paint on 0, 1, 2, ..., n of their (n-1) dimensional sides?

The math certainly checks out. 

Just to be clear (for my own sake), the term "paint", as it applies to a n-dimensional cube, refers to a covering of n-1 dimension, correct?  So painting a 4-cube implies covering it with some 3-dimensional coating that makes it appear completely covered to a 4-dimensional observer?

Posted by Bryan on 2015-03-13 11:30:34