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How Many Times Can You Fold a Piece of Paper?

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YouTube / astroboy0969

When I was a kid, I learned that there was a limit to the number of times a piece of paper could be folded. It was a lesson in exponential growth, the idea being that each fold doubles the paper's thickness, and even with something as thin as paper, quickly you'll end up with an unmanageable mess, too thick to fold further.

But the big question was always: Okay, so how many times can a given piece of paper be folded? In a brief third-grade science lesson we tried this experiment with various kid-sized pieces of paper, and often got to around six folds—and I just did it now with a large sticky note, and again got to six folds easily. Somebody (I can't recall whether it was our teacher or a fellow student) imparted the sage wisdom: seven folds is the most. This seemed plausible, because it seemed to hold up to all the testing a room full of savvy eight-year-olds could manage. Case closed: The universe only allowed for seven paper-folds on a given sheet. Oh, our minds would be blown in a few decades.

In January 2002, Britney Gallivan, then a junior in high school, folded a 4,000-foot-long roll of toilet paper to prove that 12 folds were possible (note that she used single-direction folding, given the long, narrow nature of her paper; my class had been using multi-directional folding, but still—wow). What's more, she did this after deriving a paper folding theorem (yes, it involves pi) that allows calculation of maximum folds based on paper thickness, length, and/or direction of folding, and accounts for the loss of usable paper at the edges due to the rounding that comes with extreme folding. That is some math magic right there, with empirical proof to boot.

Since Gallivan's proof, people have gotten up to quite a bit of fun with this. In 2007, the MythBusters tried the experiment and got nearly as far—but needed heavy machinery and used multi-directional folding, requiring a truly gigantic piece of paper to start with. Take a look:

Then in 2012, students at St. Mark's School in Southborough, Massachusetts visited MIT to attempt 13 single-direction folds. They didn't actually use Gallivan's single-sheet method, instead choosing to layer the first 64 sheets (equivalent to six folds) on top of each other and then begin the folding, but this is still a lot of fun:

For more on Gallivan's achievement (and the math), read this page from The Historical Society of Pomona Valley.

See also: Folding Space-Time Using a Music Box

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Can You Figure Out How Many Triangles Are in This Picture?

Time for another brain teaser. How many triangles do you see here? A Quora user posted the image above (which we spotted on MSN) for fellow brainiacs to chew on. See if you can figure it out. We’ll wait.

Ready?

So, as you can see, all the smaller triangles can combine to become bigger triangles, which is where the trick lies. If you count up every different triangle formed by the lines, you should get 24. (Don’t forget the big triangle!)

Some pedantic Quora users thought it through and realized there are even more triangles, if you really want to go there. There’s a triangle formed by the “A” in the signature in the right-hand corner, and if we’re counting the concept of triangles, the word “triangle” counts, too.

As math expert Martin Silvertant writes on Quora, “A triangle is a mathematical idea rather than something real; physical triangles are by definition not geometrically perfect, but approximations of triangles. In other words, both the pictorial triangles and the words referring to triangles are referents to the concept of a triangle.” So yes, you could technically count the word “triangle.”  (Silvertant also includes a useful graphic explaining how to find all the pictorial triangles.)

Check out the whole Quora discussion for in-depth explainers from users about their methods of figuring it out.

[h/t MSN]

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This Puzzling Math Brain Teaser Has a Simple Solution
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Fans of number-based brainteasers might find themselves pleasantly stumped by the following question, posed by TED-Ed’s Alex Gendler: Which sequence of integers comes next?

1, 11, 21, 1211, 111221, ?

Mathematicians may recognize this pattern as a specific type of number sequence—called a “look-and-say sequence"—that yields a distinct pattern. As for those who aren't number enthusiasts, they should try reading the numbers they see aloud (so that 1 becomes "one one," 11 is "two ones," 21 is "one two, one one,” and so on) to figure the answer.

Still can’t crack the code? Learn the surprisingly simple secret to solving the sequence by watching the video below.

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