# The smallest disc which could contain 11 coins

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: ## 1 Answer

The answer is

$1+\frac{1}{\sin\frac{\pi}{9}}\approx3.923$

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.

• lol I had to idea there was such a paper -.- – Oray Nov 14 '17 at 19:36
• 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 – DqwertyC Nov 14 '17 at 19:44
• en.wikipedia.org/wiki/Circle_packing_in_a_circle – Seyed Nov 14 '17 at 22:00