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

### Hidden Faces of Cubes

```
Date: 06/20/2001 at 05:21:34
From: Nigel Vowles
Subject: Equation to show hidden faces of cubes

Hi,

My daughter has to produce an equation to show the number of hidden
faces when three rows of cubes are placed together on a flat surface.
The data look like this

Number of Cubes(n)		 Hidden Faces(h)
3                      7
6                     20
9                     33
12                     46
15                     59
18                     72
21                     85
24                     98
27                    111
30                    124

I've worked out the difference between n squared and h, and the
differences between these results. The difference between the
differences I calculate as 18 constantly, so the next set of
differences will obviously be 0. I can't see how to get a formula
from this, though, and am not sure if I'm heading in the right
direction. I haven't done algebra for 20 years and used to hate it,
but I must say I'm quite enjoying this problem!

Thanks in anticipation of your help.
```

```
Date: 06/20/2001 at 12:41:28
From: Doctor Peterson
Subject: Re: Equation to show hidden faces of cubes

Hi, Nigel.

Before I get started, I should ask whether you are doing this, or your
daughter. I hope you are giving her opportunities to learn from this
as you work together; learning together is great!

You haven't told me how old she is; I'll assume she is learning
algebra, so what you are doing is relevant to her.

The approach you are taking, analyzing the data after the fact to find
a formula that fits it, uses a method called finite differences, which
is explained in our archives; here is one such page:

Method of Finite Differences
http://mathforum.org/dr.math/problems/gillett.10.12.00.html

But I don't like this method for a problem like yours. Why? Because
when you get a formula, all you will know is that it fits the
particular data you used. It doesn't tell you whether you made a
mistake in your numbers, or whether the pattern will continue when the
numbers get larger. And in math, I like to KNOW, not just ASSUME.

If she was told to use this method (and was told how), then it is fine
to use it; and you will certainly enjoy learning the method anyway.
But here's how I would prefer to do the problem: rather than looking
at the data you gathered, I would look at HOW you gathered it, and
find the pattern BEHIND the numbers.

So what causes hidden faces, and how do they grow?

I'm going to start out talking not about the number of cubes, but the
number of columns of three, so that all natural numbers will work, not
just multiples of three. With one column

c = 1

X
X
X

there are three hidden faces on the bottom (one per cube), and four
between cubes (two per pair of adjacent cubes). This gives seven in
all.

When I add a second column

c = 2

X X
X X
X X

I have added another seven under and between the new cubes; I have
also added six more, between the old and new columns (two per pair of
adjacent cubes). So my new total is 7+7+6 = 20.

Now each time I add another column, the same thing will happen, and I
will add 7+6 more hidden faces. (In other similar problems, it will be
a little more complicated.)

So for c = 1, I have 7, and for each increase of 1, I add 13. This is
a linear equation:

H = 7 + 13(c-1)

since with c columns I have added c-1 columns to the first.

Since my c is 1/3 of your n, your formula (for n a multiple of 3) will
be
H = 13c - 6 = 13n/3 - 6

Go ahead and try the other method, and verify that it works. Then try
some more complicated patterns, such as making layers of cubes in a
three-dimensional block.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra

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