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### 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.

```

```
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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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