Associated Topics || Dr. Math Home || Search Dr. Math

### Painted Cube

```
Date: 06/21/2001 at 14:35:06
From: Ste
Subject: Painted cube

We've been given the painted cube topic in preperation for gcse's
next year. I've done loads of diagrams and tables, but just can't see
or find a formula for the problem (it must be in algebra). Would you
please send me a formula with a brief explenation?

Thanks, Dr. Math.
STE
```

```
Date: 06/21/2001 at 15:42:17
From: Doctor Greenie
Subject: Re: Painted cube

Hello, Ste -

You will learn a lot more about mathematics if you figure out the
formulas on your own than if we give them to you. I will try to help
you find a way to develop the formulas; if you are able to develop the
formulas with the help I give you, then you will be able to provide
explanations of the formulas without my help.

We have a large cube made up of smaller cubes, with each dimension of
the large cube being equivalent to n of the smaller cubes. We paint
the large cube, and then we want to know how many of the small cubes
have paint on 0, 1, 2, 3, 4, 5, or 6 faces.

To do this, you want to visualize the large cube and determine where
the small cubes are that have 0 faces painted, where the small cubes
are that have 1 face painted, and so on.

Let's try to build equations (functions of n) that describe the
numbers of small cubes with different numbers of faces painted. In the
discussion that follows, I will use f0(n) to denote the function of n
defining the number of small cubes with 0 faces painted, f1(n) to
denote the function of n defining the number of small cubes with 1
face painted, ..., and f6(n) be the function of n defining the number
of small cubes with 6 faces painted.

If the "large" cube is a single small cube, then there is one small
cube, and it is painted on all 6 faces. That is a special case, since
in all the larger cubes there are no small cubes with 6 faces painted.
In fact, in all the larger cubes there are no small cubes with more
than three painted faces.

So we have (for n > 1)

f6(n) = 0
f5(n) = 0
f4(n) = 0

Now let's visualize the picture to determine f3(n), f2(n), f1(n), and
f0(n).

f3(n)...

The small cubes that have three faces painted are on the corners of
the cube. How many corners are there on an n by n by n cube?  The

f2(n)...

The small cubes that have two faces painted are on the edges of the
cube - except for the small corner cubes, which have three faces
painted. How many edges does a cube have? And on an n by n by n cube,
how many small cubes are there on each of those edges, not counting
the small corner cubes? So how many small edge cubes (not counting
corner cubes) does that make on all the edges together? The answer is
f2(n).

f1(n)...

The small cubes that have one face painted are on the faces of the
cube - except for the small edge cubes, which are painted on two
faces. How many faces does a cube have? And on an n by n by n cube,
how many small cubes are there on each of those faces, not counting
the edge cubes? So how many small "face" cubes (not counting edge
cubes) does that make on all the faces together?  The answer is f1(n).

f0(n)...

The small cubes that have no faces painted are on the interior of the
large cube; these small cubes with no painted faces together form a
smaller cube inside the large cube. If the large cube is n by n by n,
what are the dimensions of the cube inside the large cube formed by
all of the small cubes with no painted faces? So how many small cubes
are there in that "interior" cube? The answer is f0(n).

Now, all together, f0(n), f1(n), ..., and f6(n) must account for all
of the small cubes in the large cube. So you can perform a confidence
check of your answers for f0(n), f1(n), f2(n), and f3(n) by seeing if
together they add to the total number of small cubes in the large
cube, which is n^3. If they do, then you probably did the reasoning
correctly and have found the correct formulas for f0(n), f1(n), f2(n),
and f3(n). If they don't, go back and check your work - at least one
of the four formulas you found for f0(n), f1(n), f2(n), and f3(n) is
incorrect.

See what you can do with this problem on your own now.  Write back if
you would like more help with this.

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search