Math Forum - Project of the Month, November 1996

A Math Forum Project

November POM - Winner

Michael Mandel
Germantown Academy, Fort Washington, Pennsylvania


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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