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### 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
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/
```
Associated Topics:
High School Puzzles
High School Sequences, Series
Middle School Puzzles

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