Paper-folding

Date: 30 Dec 1994 15:22:42 -0500
From: Marlys J. Brimmer
Subject: I have a question

Hello,

My students and I have done the paper-folding thing several times, and
we thought it was only possible to fold a single sheet of paper 7 or
8 times.  A couple days ago on TV they said 10 times!  Do you agree?

Please reply asap.  Thank you very much.

Mrs. Brimmer, Greenfield Jr. High, Bakersfield, CA

Date: 31 Dec 1994 16:21:35 -0500
From: Dr. Ken
Subject: Re: I have a question

Hello there!  

An interesting question!  I tried this myself several times with different
kinds of paper and stuff, and here are my results.  With an ordinary sheet
of paper, I could only get seven folds-in-half out of it.  With a Kleenex, I
got eight.  With a piece of tissue paper, I got nine folds, and with a big
bed blanket, I got six folds.

So what does this all mean?  How does it relate to math?  Well, here's the
deal.  See, every time you fold the paper in half, you're making a new
structure whose thickness is twice the thickness of the previous structure.
So you can see that the thickness is going to get REALLY big, REALLY 
FAST. That's the important thing here; when it gets too thick, you can't 
fold it in half anymore.  

This is an example of what we mathematicians call a Geometric Sequence.
Each term in the sequence is twice as big as the term before it.  So we call
this a Geometric Sequence with common ratio 2.  That just means that if 
you take any term in the thickness sequence and divide it by the previous 
term, you'll get 2.  

Have you ever heard of the chessboard-rice problem?  If you put one grain 
of rice in the first square on a chessboard, and then two grains on the next
one, four on the next, eight on the next, then sixteen, etc., how many
grains of rice will there be on the last square?  Or even on the fifteenth
square?  As it turns out, there will be A LOT OF RICE!  A way big huge
amount.  And I'm not kidding.  Geometric growth is fast.

Another interesting thing about this problem is that you'll get basically
the same number of folds no matter what kind of sheet you use.  I mean, 
I got 6, 7, 8, and 9 folds when I used vastly different materials.  It's not
like we were getting twenty or thirty folds, or only two or three; they 
were all around seven or eight.  Which tells you something: the starting
thickness really doesn't affect things very much.  The mathematician 
would say that the first term of a Geometric Sequence doesn't affect its 
growth rate very much.  For instance, if you started with a piece of paper 
that was twice as thick, you should be able to fold it one fewer time.  
Not half as many times, but only one fewer.  That's not much difference.

So that's what I have to say about paper folding.  Actually, that's not ALL
I have to say; I'm kind of an origami nut.  But that'll have to do for now.
You might think about the following questions:  How does the size 
(length and width) of the paper affect how many times you can fold it?  
How many times could you fold it in thirds?  In fifths?

Anyway, thanks for the question.  Write back if you have more!

-Ken "Dr." Math

From: Adrian M. Whatley
Date: 3/17/98 at 13:39

On 31st December 1994 you mentioned the chessboard rice problem in a
"Math Forum" answer. Do you happen to know to whom this problem is 
first due?

Any leads would be most welcome,

Thanks,

Adrian

-- 
Adrian M. Whatley
Institut fuer Neuroinformatik der Universitaet/ETH Zuerich,
WWW: http://www.ini.unizh.ch/~amw   

From: Dr. Ken
Date: 3/17/98 at 12:46

Hi Adrian,

I didn't know the origins of this problem, but I did find one helpful page
by searching the Web:

   http://www.richmond.edu/~educate/stohr/chess/chess.html   

Hope this gives you something to go on.


-Dr. Ken, The Math Forum