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Math Circle Problem: Folding & Dragons

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 ( 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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