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### Ratios of Ripped Paper

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Date: 11/19/95 at 20:48:32
From: Anonymous
Subject: Brain Teaser ques.

Ok Dr. Math, I'm really stuck with this question.

Grab a piece of 8 by 11 paper. Rip it in half, then put that half over
the other, then rip it again.  For example I cut it in half, so now I
have 2 pieces.  Now I place them over each other, and rip it again.
Now there are 4.  So what is the length (or how high, like 5 miles or
something) when the paper is ripped 30 times?

Thanx, doc.
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Date: 11/19/95 at 14:22:58
From: Doctor Ethan
Subject: Re: Brain Teaser ques.

Hello,

That is a neat question. I would like to mention one thing though. You
actually could never rip a piece of paper that many times in half. But we
can imagine. Let's figure it out.

Let's call w the original width of the paper.

After one rip the width of the two halves combined is 2w.

Then after one more rip we have four pieces so the total width is 4w.
After three rips we have 8 pieces so the total width is 8w.

Let us make a chart.  Check my answers by actually ripping paper.

# of rips     Thickness
1             2w
2             4w
3             8w
4             16w
5             32w
6             64w

Do you see a pattern?

I hope so.  These numbers (2,4,8,16,32,64) are the powers of 2.

Does that make sense to you.

That means that

2=2
2*2=4
2*2*2=8
2*2*2*2=16
2*2*2*2*2=32

So after 30 rips you would have a stack 2^30 sheets high.
(That symbol 2^30 means 2*2*2*2... thirty times.)

That is   1,073,741,824w  That is really big.

Okay now let's assume that a sheet of paper is 1/500 of an inch
(it is actually a little thicker)

Then w = .002 inches

so   1,073,741,824w  = 2147483.648  inches

To convert to miles we divide by 63360 to get 33.89 miles.

Hope this helps,

-Doctor Ethan,  The Geometry Forum

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