There are $11$ coins with $1$ unit radius and we are trying to put them inside a big disc with some radius. So

What is the minimum radius of this disc?

  • If this question was asked for $2$ coins, the answer would be obviously $2$ unit radius,
  • If was asked for $3$ coins, the answer would be $1+\frac{2}{\sqrt{3}}\cong 2.16$ unit radius as shown below:

enter image description here


The answer is


which can be done like this (image courtesy of Wikipedia):


but it is highly non-trivial to demonstrate that this is optimal. It was first proved by Hans Melissen in "Densest packings of eleven congruent circles in a circle", Geometriae Dedicata 50(1) (1994), pp. 15–25, by a proof sufficiently complex that I don't want to reproduce it here.

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  • $\begingroup$ lol I had to idea there was such a paper -.- $\endgroup$ – Oray Nov 14 '17 at 19:36
  • 3
    $\begingroup$ When a seemingly simple geometry problem that the ancient Greeks could have understood isn't proven until the 90s, you know that it has to be complicated :D $\endgroup$ – DqwertyC Nov 14 '17 at 19:44
  • $\begingroup$ en.wikipedia.org/wiki/Circle_packing_in_a_circle $\endgroup$ – Seyed Nov 14 '17 at 22:00

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