Circles packing

In this kind of problems, we have to pack the maximum number of unit circles in a given area,
or equivallently, to find the smallest area for a given number of circles.
We can then pack 9 unit circles in a larger circle with radius R in that way.
Value of radius R ? Solution

But we can do better ! We can pack ten unit circles in this same circle.
How ? proof ? Solution

Can we do even better ? 10 circles into a smaller circle ? Solution

Details

 

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