I have a large cube-shaped block made of smaller cubes with sides of length 1 unit. The smallest such block is, of course, 1 cube big. Now I want to categorize the various blocks I make based on how many small cubes make up a side. So a 2-Block would be a block made up of 8 smaller cubes (2 x 2 x2). Sound good?
If I disassemble the block: how many cubes in my 2-Block are painted on a given number of sides? For this solution, I can see that every corner has cube with paint on three sides. Next, I noticed that I have exactly 8 corners on my block and my block is made of 8 cubes. So in a 2-Block, there are 8 cubes painted on 3 sides and that is it. What if I added a cube to every dimension. ie: what about a 3-Block? Do I now have cubes that are only painted on 2 sides or 1 side? How big do I have to get before I have a cube painted on zero sides? Is there a general formula?