Math Forum - Project of the Month, November 1996
#### A Math Forum Project

# November POM - Winner

## Michael Mandel

Germantown Academy, Fort Washington, Pennsylvania

**From: Mandelin@aol.com**

In this cube where all sides are painted and then is divided into unit cubes
all of my calculations are all for cubes more than one unit on each side.
Every cube has exactly eight corners, and each corner is painted on three
sides because it is the intersection of three of the sides of the cube. To
find out the number of unit cubes painted on two sides in a cube of sides of
length "x", you must multiply (x - 2) by 12. It must be x - 2 because on
each edge the two ends are both corners and therefore painted on three sides
and don't count towards the two sided cubes. It must be multiplied by 12
because there are four edges on the upper square, four on the bottom square
and four connecting the two. It is only the edges because they are the
intersection of two sides. The number of cubes painted on one side can be
found by multiplying (x - 2) by (x - 2) by 6. It must be (x - 2) squared
because each side is x long and the sides on the edges have paint on more
than one side. The edges are on both sides of the face so it must be (x - 2)
squared. It will be multiplied by 6 because there are 6 faces of the cube.
The number of sides with no paint on them can be found by cubing (x - 2).
This is because on both ends of the unpainted cube there are painted unit
cubes, so you cannot count them in the unpainted number. This holds true for
all three dimensions and therefore it is (x - 2) cubed. In a cube measuring
1 unit cube on each side there is one cube painted on six sides and no other
cubes painted on any different number of sides.

### My Comments

Michael did a nice job of explaining where the equations come from, and
does it from a very visual perspective. He also points out that the
equations only work if n is greater than 1. The one thing he might have
done to improve his solution is to break it up visually - it is hard to
read a huge blocks of text, and it's a good idea to at least break it into
paragraphs.
For his effort, Michael won himself a t-shirt. The back of it looks like
this:

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