Call a “ring” of circles a collection of six circles of equal radius, say r, whose centers lie on the six vertices of a regular hexagon with side length 2r. This makes each circle tangent to its two neighbors, and we can call the center of the regular hexagon the “center” of the ring of circles. If we are given a circle C, what is the maximum proportion of the area of that circle we can cover with rings of circles entirely contained within C that all are mutually disjoint and share the same center?

When submitting an answer, you can either send in a closed-form solution, or your answer out to 6 decimal places.

Note: This puzzle was inspired by the math problems posted at https://www.janestreet.com/bonus-problems/. We are publishing our video series Real Numbers, about problem-solving for high school students with a passion for math, to YouTube over the next few weeks. These problems were released to celebrate the occasion. Enjoy!