# How many times can a piece of paper be folded?

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Its a common misconseption that a piece of paper can only be folded in half approximately 7 times, at most 8 times. The guys over at Mythbusters thought they would get a piece of paper, roughly the sixe of a football field, a roller, and a forklist. And they folded the paper more times than eight times. They actually folded the piece of paper 11 times.

Source: Mythbusters – trying to fold 11 times

Following this feat, their explanation of being unable to accomplish a twelfth fold, was that it would have produced something resembling a semi-circle and not an actual fold. But what if they obtained a piece of paper the size of two football fields or even three for that matter? Would they still be able to fold it even more?

A student named Brittany Gallivan was able to successfully fold an extremely long piece of toilet paper in half twelve times. Later, students at the St. Marks school in Southborough, together with their teacher, James Tanton, stated they were able to fold an even longer piece of toilet paper a total of thirteen times, breaking the previous record set by Gallivan.

Britney Gallivan: Singlehandedly folder paper 12 times.

But according to Gallavan, she claimed that both the Mythbusters and Southborough students had “broken the rules.” Brittany explained in a press release that Mythbusters and the students from St. Marks had both admittedly used paper that needed to be taped together, rather than the single sheet that was required by this paper folding challenge.

When Gallavan made her record-breaking twelve folds, she supported it with a book entitled “How to Fold Paper in Half Twelve Times: An Impossible Challenge Solved and Explained,” which is what inspired the Southborough students to break her record. In her book, she noted exactly how she was able to perform this stunt and also backed it up with some mathematical equations that precisely explained her victory and the limitations that this challenge presented.

## Oh boy – Here Comes The Math!

When she was originally presented with this problem by one of her high school math teachers, to reach a dozen creases, she came up with an equation that addressed folding the paper in half the traditional way, top to bottom and side to side. Where the width (W) results from the thickness (t) of the material and number (n) of folds.

She then came up with a second equation where she believed she could conquer this challenge by folding the paper in half lengthwise instead and continue to fold it in this way. Here length (L) was derived from the thickness (t) of the paper and number (n) of folds.

Her conclusion was that with each lengthwise fold, it would require roughly four times more paper to arrive at a dozen.  In order to reach her goal of twelve, her computations showed that she would need a piece of very thin paper that was 1.2 kilometers in length, or almost 4,000 feet long.

What this mathematician was ultimately able to prove with her equations is that the smaller and thicker the piece of paper, the fewer of times you can fold it and vice versa. As we continue to fold, the smaller square footage of the paper doesn’t allow for the thickness of the folds to continue, which makes logical sense. The bottom line is that as the height of the folds continue to rise, as the amount of paper also increases, the thickness will grow exponentially.

More simply put, for ‘n’ number of folds, you will end up with 2^n pieces of paper in thickness.  This adds up quick.  3 folds is easy:  2 x 2 x 2  = 8 pieces of paper thick.  When you get to 7 (my personal record and seen below), you’re at 2^7 or 2 x 2 x 2 x 2 x 2 x 2 x 2  =  128 pieces thick.  I could not make the 8th fold because 256 pieces of paper this is like trying to fold a piece of lumber, and I am weak from sitting at a computer all day.

7 folds… and it’s already looking rough. 8 would be physically impossible for a piece of paper this size (8.5×11).

## It gets exponentially thick…

The thickness grows so exponentially quickly, in fact, according to another mathematician, these amounts can be staggering, stretching the limits of the known universe. His figures show that as the folded pile grows…

• At 23 folds, you’ll reach one kilometer or 3,280 feet
• 30 folds will get you into outer space at 100 kilometers or just over 62 miles
• Over 40 will land you on the moon
• A little over 50 gets you to the sun
• More than 80 folds is near the thickness of the Andromeda galaxy, estimated to be 141,000 light years wide
• Near 90 folds and it will be bigger than Virgo supercluster at around 130 million light years in length
• Finally, at 103 folds, you’re now outside of the observable universe which is estimated to be 92 billion light years in diameter.

While these mathematicians and students can clearly see through Brittany’s numbers to accomplish even more folds, to the ends of the universe and back, but at what price? To what lengths, literally, are we willing to go to be the paper folding champion? How much time are we willing to spend?  How many trees must die to accomplish our paper folding dreams? Who will rise as our next Paper Folding Hero?

In Brittany’s case, the cost for her extra-long roll of toilet paper was \$85 and the length was almost 4,000 feet as mentioned previously and 13,000 feet for the students at Southborough. It took Brittany over seven hours to get to the eleventh fold and snap her record breaking picture and subsequently make the twelfth and final fold. All of this was to simply get some extra credit in her high school math class.

These paper-folding pros keep talking about “breaking a record,” but is it really a world record if it isn’t even recognized by Guinness? According to Mr. Tanton, the teacher from St. Marks, the people from The Guinness Book of World Records don’t even have a category for paper folder and doesn’t keep track of these kinds of records, which is a damn shame.

And as for Gallivan’s claim that both the Mythbusters team and the students didn’t follow the rules of it being one continuous piece of paper, who made up those rules in the first place? Was it her teacher? Guess you’ll have to buy her book to find out.

As far as notoriety goes, obviously Ms. Gallivan has gotten plenty of online recognition for her achievement and is still selling copies of her book. Her equation was also mentioned on the CBS prime time series “Numb3rs,” where a young mathematician helps his older brother solve crimes for the FBI by using various equations.

## … And In Conclusion

Today, we dispelled the myth that paper can only be folded in half 7 or 8 times.  We tried ourselves, but failed in our attempt.  Others have succeeded however – Mythbusters, Brittany, and the students from Southborough – all the way up to 13 folds.  We also learned a new way to get famous:  fold a single piece of paper in half 14 times… But good luck on the final fold, it’s 16,384 sheets of paper thick.