Story left to the reader as an exercise

Puzzle for the week of 4/20

Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 4/27/25). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem! The solution and submitted responses will be posted around Wednesday at 10 am.

I make no guarantees my solutions are correct! You are all smart people so please comment if you think I made a mistake!

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I usually like to wrap my puzzles in a story or somewhat realistic situation but I couldn’t figure out a good one for this week’s problem ¯\_(ツ)_/¯

You start with a regular triangle. At the midpoint of each side you draw a circle on the inside of the triangle where the circle is tangent to the side. Each circle is the same size and its radius is maximized such that the interior circles touch each other. Below is a diagram to illustrate.

If we consider a regular triangle, what is the ratio of the area of the 3 interior circles to the area of the triangle? How about a square? How about a regular n-gon?

Please submit your answers here. Please ask any questions in the comments.