## Painted cube puzzle

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?

## About Samantha from SocialMath

Applied Mathematician and writer of socialmathematics.net.
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### 1 Response to Painted cube puzzle

1. Algebra and Painted Cubes says:

For n=0, a no little cube cube, no sides are painted.

For n=1, a one little cube cube, all 6 sides are painted.

For n>1 the following is true.

No cubes are ever painted on 4 or more sides.

Corner cubes:
8 little cubes are paints on 3 sides.

Edge cubes:
12*(n-2) little cubes are painted on 2 sides.

Face cubes:
6* (n-2)^2 little cubes are painted on 1 side.

Embedded cubes:
(n-2)^3 little cubes have no paint.

Check:
If you add these four algebraic expressions the result is n^3, the number of little cubes in all.