Suppose C has radius 1. Then the radius for each of the circles in a ring inscribed in C can be shown to be 1/3.
For the next ring, the largest radius one can achieve is if we rotate the ring by 30 degrees, in which case we get a maximum radius1 of (2 + 4 Sqrt[3] − 4 Sqrt[1 + Sqrt[3]])/18.
The total area covered by the infinite set of nested rings can then be calculated using a geometric series. The fraction that is covered comes to
6/(9 - (1 + 2 Sqrt[3] − 2 Sqrt[1 + Sqrt[3]])2),
which to 6 decimal places is .783464.
Congrats to this month’s solvers!
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Corresponding to the positive solution to the equation r2 + (2/3 − r Sqrt[3])2 = (1/3 + r)2. ↩