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?

Advertisements

About Samantha from SocialMath

Applied Mathematician and writer of socialmathematics.net.
This entry was posted in Games. Bookmark the permalink.

One 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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s