Algebra Paper-Folding Problem

Date: 02/10/99 at 00:34:39
From: Lauren
Subject: Algebra Paper Folding Problem

Start with a standard piece of paper (8 1/2 by 11). Fold it in half 
so that the fold line is perpendicular to the longest edge. Then fold 
again so that the new fold line is perpendicular to the new longest 
edge. Continue this process... How many crossing points (intersections) 
will there be after 20 folds?

I found that after 20 folds there will be 1,046,529 intersections. The 
next problem was to find an equation for the number of intersections 
after "x" folds.  

I have been unable to find a pattern for this one. I found that for all 
the even numbers of folds, the equation is x^2 ("x" being the previous 
number of fold lines) and for all the odd numbers of folds, it is 2x+1, 
but that is as far as I can get. I have been unable to even consider an 
equation.  

Date: 02/10/99 at 13:19:57
From: Doctor Peterson
Subject: Re: Algebra Paper Folding Problem

I think the key is to list not just the number of intersections, but 
also the number of folds in each direction at each step. Here's a 
start:

    folds  |  long   |  short  | intersections
           | creases | creases |
    -------+---------+---------+--------------
       0   |    0    |    0    |      0
    -------+---------+---------+--------------
       1   |    0    |    1    |      0
    -------+---------+---------+--------------
       2   |    1    |    1    |      1
    -------+---------+---------+--------------
       3   |    1    |    3    |      3
    -------+---------+---------+--------------
       4   |    3    |    3    |      9
    -------+---------+---------+--------------
       5   |    3    |    7    |      21

Now look not only for a pattern in each column, but for a reason why 
each should grow in a certain way. You'll find that the number of 
intersections is the product of the two crease counts, and that each 
crease count is one less than a certain familiar sequence. Your idea 
of separating the answer into odd and even cases is good, but your 
formulae are wrong. Your number is about a quarter of what I got, but I 
may have missed something in my quick calculation.

On the other hand, the real answer is that it's impossible to fold a 
standard sheet of paper 20 times!

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