This problem was originally posed by Sue on her blog Math Mama Writes (and was presented by her at the Match Circle Institute). There’s a kids’ game called Spot It, where there are cards with pictures all over them in a pile. If you have a match with the pile in the middle, you call out the name of that icon and grab the card. The person who collects the most cards win.
But here’s the interesting part – despite there being 57 different pictures and 55 different cards, every card has one and only one match with every single other card.
How did they make this game?
Would it work for every number of pictures?
Is there an algorithm for every number of pictures?
The best way to see why this is such an interesting question is by trying to make your own deck with 3, 4, 5, 6 etc different pictures on each card. If you do that, my “solution” below might make some sense!
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.