Math Circle Session 7: Combinatorics, Pascal’s Triangle, and the Sierpinski Triangle Again

Today saw Session 7 of our math circle. No blog on Session 6, as I was away for work – my wife explored The Towers of Hanoi in my absence.

Today’s session was based on a combination of two sessions in Rozhkovskaya’s book. The idea is as follows. Imagine a maze at a tourist attraction. The maze is such that there is one entrance, but seven exits labelled 0 to 6. Imagine that you are standing facing north at the entrance. At this entrance, and at each intersection, you can move ahead-left on a bearing of 315 degrees by one unit or ahead-right on a bearing of 45 degrees by one unit until you meet the next intersection or exit. Thus the maze decomposes into 7 levels (including the entrance and exits). Now further imagine that you must roll a die at each decision point, taking the left route if you roll even and the right route if you roll odd.

The first question: If I were an ice-cream vendor, where should I place my van to capture most people coming out? Children had an intuitive feeling that most people would come out in the middle. Only one child (the oldest) offered a reasoned explanation for this: “we expect as many odd throws as even throws, which means the most likely exit would be the middle one”.

We got the children to experiment by rolling routes through the maze, and collected a tally of how many trials ended up at each exit, confirming this suspicion. Children naturally noticed that exits zero and seven were unlikely, as they would require rolling “all odd” or “all even”.

This naturally led onto the next part of the exercise, to annotate each decision point with the number of ways in which it could be reached. Children found this hard, and made many mistakes, because they were trying to count all the ways to reach a given decision point from the maze entrance from scratch for each decision point. I explained that there are only at most two ways to reach any decision point, so it is sufficient to sum the possibilities found to these two preceding points.

Thus, Pascal’s triangle was revealed. We asked the kids to sum the rows of Pascal’s triangle, and they discovered they summed to successive powers of two.

Finally, we had the kids colour all the even numbers in a large version of Pascal’s triangle (pre-printed). One of the kids immediately recognised a fractal-like shape. Once complete, we pointed out the resemblance to Sierpinski’s triangle, created in Session 7. Oddly (to me) this wasn’t immediately apparent, even to the kids who were there.

Overall, I was a little disappointed with today’s session. I think the material was good, but it was a sunny day, and two of the kids showed no sign of wanting to be there, inside after school. This meant I had to spend a lot more time than I ever imagined trying to get these kids on task, meaning a lot less time to discuss the mathematics of what we were seeing, which was a shame. My one rule for Math Circle was that the kids should want to be there and, for whatever reason, I don’t think this was true for 1/3 of them this time. I hope it’s a temporary blip, but I need to ask for views from primary school teachers; I was again reminded today how different kids are to the adults I’m used to!

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