Sharing Tamás Görbe’s circle packing problem with my younger son

Yesterday I saw a really neat tweet from Tamás Görbe:

It turns out that we looked at this problem a few years ago, but for reasons I don’t remember my younger son wasn’t part of that project:

https://mikesmathpage.wordpress.com/2018/04/03/a-terrific-problem-to-share-with-calculus-and-geometry-students-from-the-iowa-state-problem-collection/

Today I thought it would be really fun to tackle the problem with my younger son. We started with a quick introduction to the problem and then a discussion of how to approach the solution:

The first thing my son tried was finding the radius of a single inscribed circle inside of an equilateral triangle:

Next he tried to find the radius of the circles when there were 3 inscribed circles. This part was pretty challenging for him, but his work really shows what a kid struggling through a math problem can look like:

Now we got to the heart of the problem – what is the radius when there are “n” inscribed circles:

Finally, we looked what the area covered by the circles would be in the limit as n goes to infinity. We also talked a bit about the surprise – why isn’t all of the area of the triangle covered?

I really think this problem is a great one to share with kids who have see geometry. It is great to see how they approach the problem, and also really nice to see how they thinking about the area in the limit.

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