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


1 Answer 1


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.

  • $\begingroup$ lol I had to idea there was such a paper -.- $\endgroup$
    – Oray
    Commented Nov 14, 2017 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
    Commented Nov 14, 2017 at 19:44
  • $\begingroup$ en.wikipedia.org/wiki/Circle_packing_in_a_circle $\endgroup$
    – Seyed
    Commented Nov 14, 2017 at 22:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.