# A question on circles

Given that a lamp-post can light a surrounding circle of radius 100 m, what is the minimum number of such lamp-posts required to light a circular ground of radius 1000 m.

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## 1 Answer

I doubt that anybody has ever found this precisely: for example this seems to give the best results known for using a given number of gircles, but only goes as far as 12 circles at which point the radius of the bigger disk is 2.769... times the smaller disk.

In your problem there is an obvious lower bound of 100 lampposts, since the radius of the bigger circle is 10 times that of the smaller circles, and so the area 100 times.

This bound can be improved, since the optimal coverage of a plane involves hegaxonal packing, which reduces the effective area of each small circle by a factor of $\frac{3 \sqrt{3}}{2 \pi} \approx 0.82699$ so you would need at least 121 lamposts, and probably a few more.

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+1 interesting link! – wxu Jun 6 '11 at 12:18