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Paper Folding 

Problem Statement
If a piece of paper could be folded in half fifty times, what would be the thickness of the folded paper? A ream (500 sheets) of paper is 2 inches thick.

Problem setup

If you fold a piece of paper fifty times, how thick would the paper be?   Five hundred sheets of paper compose a ream and a ream is two inches thick.  Determine how many sheets thick the paper would be and then convert from sheets to reams, and then from reams to inches (or feet or miles).  


Plans to Solve/Investigate the Problem

I knew from testing something I had been told as a youth, that one cannot fold any piece of paper more than seven times.  It just gets too thick.  So, I tried to imagine folding a piece of paper fifty times, knowing it is impossible.  I decided I would try to fold a piece of paper several times and see if I could determine a formula or pattern when doing so and hope to determine how thick the paper would be. 


Investigation/Exploration of the Problem

I saw quickly that when you fold a piece of paper once you have 2 sheets (half sheets), when you fold it twice, you have 4 sheets, and when you fold it 3 times you have 8 sheets.  Using this data, I saw that if you take the number 2 and you raise it to the power of two and then three, you come up with the thickness of your paper in sheets.  Example:  2 raised to second power is 2x2 which is 4.  Two to the third power is 8 (just like when you fold the paper 3 times).  When you fold the paper 4 times it is the same as taking the number 2 and raising it to the fourth power.  Your answer is 16.  There are sixteen layers of paper.  To know how many layers of paper there would be if you folded it 50 times, all you must do is take 2 and raise it to the 50th power.  I attempted to work on this problem using an excel spreadsheet, rather than try to continually do the problem manually.  If you plug the number of folds into the A column and then plug the formula "2 ^A" for the next column (2 raised to the power of A with a equaling the number of folds), then you come up with the correct potential sequence.  After a while you get some pretty big numbers.  The answer is in billions of layers of paper and after you convert the pieces of paper into inches, you still have a huge number ( that is in scientific notation due to size).  All from folding a piece of paper fifty times!


Extensions of the Problem

I decided I wanted to get an answer that would be in a more appropriate unit of measure than inches or reams since the answer was in billions of inches.  When you convert from inches to feet, the answer is still a long number, so I converted to miles by taking my first answer and dividing it by 12 ( to get feet) and then dividing it by 5280 to get miles.  The paper that is folded fifty times is over 71,000,000 miles thick!  Over half the distance from the earth to the sun (93,000,000 miles).

Author & Contact
Kevin Smith
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