Title
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
Email me!
