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/