This past week I attended the Math Circle Summer Teacher Training Institute. The idea of a Math Circle is that students learn math best by constructing it themselves. In Math Circles, you pose interesting problems that could probably lead to deep mathematical insight, and then let students discover those insights through conversation and collaboration. The leader is there to ask questions, moderate and guide, but should not steal the opportunity from any child to discover something for themselves. It’s a very cool way of learning math, which I personally thoroughly enjoyed. Is it the future of education in schools? I’m not 100% sure, but that’s a topic for a future post.

My favorite part of the week was struggling through some awesome problems with other math teachers in the training. I wanted to share a few of these problems because I think they are really fascinating, and could be used in traditional classroom environments too. For the next few posts, I’ll pose the problem, and then discuss some of the solutions a little lower in a “read more” tab (so if you don’t want any spoilers before you solve it, don’t click on that!).

Folding Paper and the Dragon Curve

The first problem is courtesy of James Tanton, who poses a ton of rich mathematical problems on both Twitter (@jamestanton) and his website (jamestanton.com). Here’s the problem:

Take a strip of paper and fold the right side over to the left. Unfold. Notice that if you hold it as you originally held it and then look at it from the side, the fold makes a little valley instead of a peak. We are going to number valleys with 1s and peaks with 0s. So to convert the fold pattern into a sequence, after the first fold the sequence would be just 1.

Now take the same paper, and always holding it in the same orientation, fold your original fold again from right to left, and then fold one more time. So you are bringing the fold from the right to the edge on the left. After you crease and unfold, and then hold it in its original orientation, you notice that there are two valleys and then one peak, so the sequence for this fold is 110. If you fold one more time (again being careful to always fold right to left), the next sequence should be 1101100.

So what’s going on with the pattern?
What can you say about the pattern created by the 100th fold?
What is the 112th digit in the 100th iteration of the pattern?

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So this is a really cool problem, mostly because it’s a great pattern that is really accessible for really anyone who can notice things. There is no math level barrier for this problem. Below are the keys we found when working on it, but here is James Tanton’s write-up of solution.

KEYS:

• Notice that the pattern is symmetrical in a way. Because it was a fold, the sequence from the front will be the inverse of the sequence from the back. So for example, if it starts with 111101, it must end in 010000.
• Every successive step includes the entire pattern from the step before at the beginning.
• There is always a 1 in the middle.
• Since the old folds don’t go away, that 1 in the middle comes from your original fold. So all the even folds actually come from the step before (try lining up the tables with the same folds above each other like I did in the picture above). This means that all new folds are inserted in every other step (all the odd positions).
• Magically enough, the new folds alternate 10101010101010…! (try it)
• The nth pattern has 2^n – 1 digits, 2^(n-1) 1’s, and 2^(n-1) – 1 0’s.

“ANSWERS”:

EXTENSIONS:

After you fold a few times, unfold it and make all of the folds go at right angles (see picture below). This is called the Dragon Curve and it’s an interesting fractal that will never fold back around onto itself (this make sense thinking about the folding pattern). The person who posed this problem posited that perhaps this curve could be related to complicated protein folding. So how exactly is the pattern discussed above related to the dragon curve and to protein folding??