Earlier this week Steven Strogatz posted this question on twitter:
The problem received lots of attention (for a math problem on Twitter!) and I thought it would be fun to talk through the problem with the boys today. This type of problem is obviously beyond their ability to solve, but I thought they would have some interesting ideas about it anyway . . . and they did!
We started by just talking through the problem and getting a little clarity on what was going on with the dogs. One of the challenges in talking about a problem like this with kids is that elementary school kids don’t usually see problems involving motion, so it takes them a while to find the language and the ideas to describe what’s going on. They got there eventually, though, and by the end of this video the kids think that the dogs will move inside of the original polygon and eventually collide in the center.
With the introductions and initial thoughts out of the way, we tried to talk in more detail about how you would even approach solving this problem.
One idea my younger son had was to look at a triangle rather than trying to solve the problem about the n-gon. The interesting reason for this is that he thought maybe the dogs would run the same distance in all of the n-gon’s so why not look at a simple polygon first.
We tried to draw the shape that the dogs ran in the triangle. Their approach involved approximating the path with straight lines. Also, their drawing showed the idea of the dogs spiraling towards the center of the triangle.
At the end of this video my younger son has the idea to look at infinitely many dogs on a circle!
So the idea of looking at infinitely many dogs basically stopped me in my tracks. It seemed so fun to try to see what was going on in this situation. Here’s what they came up with:
Finally we moved to looking at one of the programs that Dan Anderson made for this problem. The program we looked at is from this tweet:
Sorry for the technical difficulties on my computer while we looked at Dan’s program. Despite Firefox crashing, the kids thought the curves were really cool:
So, despite being a pretty advanced problem, it is still a fun idea for kids to think through. From a “watching kids learn math” perspective, it is neat to hear the ideas that they have about how to approach this complicated problem, too.
Thanks to Steven Strogatz and Dan Anderson for sharing their ideas about this problem.