Sharing the “lonely runner” problem with kids

I saw a really neat unsolved problem shared on twitter this morning:

An Unsolved Problem

N runners on a circular track of length L start from the same spot with constant, distinct speeds. A runner is "lonely" if its distance from everyone else is at least L/N

Lonely Runner Conjecture: Each runner is lonely at some time

It's only proved for N≤7 pic.twitter.com/G7a3jHkwYD

— Tamás Görbe (@TamasGorbe) August 4, 2019

I thought it would be a really fun problem to talk through with the boys – especially since kids can definitely say something about the n = 2 and n = 3 case.

Here’s the introduction to the problem:

After the introduction we talked about the n = 2 case.

Now we moved on to the n = 3 case. The boys had an interesting idea on this one that caught me a little off guard. The neat thing is that they were able to come up with a pretty good hand waving argument that in the general case the three runners would eventually form an equilateral triangle.

After that neat hand waving argument from the last video, we tried to find a more precise argument for solving the n = 3 case. Solving it in general was just a bit out of reach, but they did find an argument for why at least one runner would become lonely.

Since we ended up pretty close to the proof of the general case for n = 3, I explained the last step after we finished with the last video. I think this is a really nice problem for kids to play around with and I think that lots of young kids will find the ideas in this problem to be really fascinating.

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