Visible/Hidden Sides on Stacked Cubes

Date: 12/12/2001 at 15:33:08
From: robin_spruell
Subject: Number Patterns

I have been set the task of finding a formula for the number of sides 
hidden/visible on cubes when put in different arrangements -

1. a line
2. a double line
3. stacked in a 3D way

I am having trouble finding out the numbers of sides and finding a 
formula.

Please help!

Date: 12/12/2001 at 22:15:46
From: Doctor Peterson
Subject: Re: Number Patterns

Hi, Robin.

Start with the simple case of a single row; here's a top view:

    +---+---+---+---+---+
    |   |   |   |   |   |
    +---+---+---+---+---+

How many sides are there in n cubes? Since each cube has 6 faces, 
there will be 6n faces in all.

How many sides are hidden? There are n bottoms that you can't see 
(assuming you are supposed to include those); and there are (n-1) 
places where two cubes touch (5-1 = 4 in my example), each of which 
hides two faces, one on each cube. So there are n + 2(n-1) hidden 
faces. You can simplify the expression.

That's how you think through this sort of problem: take an example, 
break it up into parts (bottom and touching faces), write an 
expression for the number of each kind, and add them up.

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