I was looking at this problem:

https://en.wikipedia.org/wiki/Circle_packing_in_a_circle which roughly asks what is the optimal packing of n circles of the same area $q$ into a larger circle. For simplicity I let the area of the larger circle be 1. Moreover what is the value of $nq$.

I was focusing $n=3,4$ (The case of 4 appeared in a Math GRE exam) but after chewing on it for some time it was not clear to me how to prove optimality,

It's easy to reason for $n=2$ that each circle must touch the boundary (if it didn't you can always shift and expand it) and from there the diameter argument follows.

For $n=3$ We have one circle touching the boundary, but now it's not obvious to me how to solve this without resorting to some non-linear optimization problem. I think I'm missing some geometric intuition on how to prove that they need to be symmetrically distributed.


For $n=3$ the triangles must touch each other or you could expand one. Draw the triangle formed by the centers, which is equilateral if the inner circles are the same size. Draw the radii of the small circles to the big circle and you can compute the radius of the small circles.

For $n=4$ a similar calculation gives you the radius. Draw the square formed by the centers, then the radii out to the big circle.


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