How Many Hidden Faces?

Date: 07/07/2002 at 11:54:54
From: Amy
Subject: Hidden Faces, Sequences

I have to do an investigation at school on hidden faces. When you 
look at a cube you cannot see one face - this face is hidden. So if 
you have one cube then two cubes all in a straight line up to eight 
you will work out a pattern 

  1-1 2-4 3-7 4-10 5-13 6-16 ...

Can you please find a formula to find out the number of hidden 
faces, eg. 3n-2?  I also need proof.  Can you please help me? Ive 
spent 5 hours on it already!

Date: 07/07/2002 at 22:58:41
From: Doctor Peterson
Subject: Re: Hidden Faces, Sequences

Hi, Amy.

Actually, if I look at a cube I can see only three of its faces at 
one time; I presume you mean that it is sitting on a table and you 
are allowed to look all around; only the face in contact with the 
table (or with another cube) is "hidden".

One way to approach this kind of problem is to think about how 
things change from one step to the next. If I have a row of cubes, 
say your three in a row with 7 hidden, and add another to the end, 
how many new hidden faces are there? There's the one on the end of 
the row you had, which is now covered and becomes hidden; and there 
are two on the new cube that will be hidden, one on the bottom and 
one that touches the row. So each time you add a cube, you add three 
more hidden faces. Sounds like an arithmetic sequence to me.

Another approach is to think about one whole row and break the 
hidden faces down into groups. There are the faces on the bottom 
(one per cube) and the faces between cubes (two wherever a pair of 
cubes meet). That easily gives you a formula.

If you need more help, please write back and show me how far you got.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/