# Adding coins inside a ring of coins

17 identical coins with diameter 1 are lying flat on a table, such that their midpoints build the vertices of a regular 17-gon (regular heptadecagon) and adjacent coins touch each other.

What is the maximum number of coins, which can be added inside the 17-gon, none of them overlap and lying flat on the table?

Note: the coins to add are of the same size as the coins building the 17-gon.

• If it's the pool table I used to play on at a dive by the beach, then fifty easy. That thing was all kind of warped Apr 16 at 15:58
• Are you, by any chance, in the process of making pepperoni pizza? :-)
– Bass
Apr 16 at 20:23

I'll get things started with

14

• I got the same, filling concentric rings of width 1. It likely cannot be improved by 2 coins, assuming this best-known 33-coin packing is indeed optimal, but it is hard to tell whether maybe one extra coin can be squeezed in. Apr 17 at 6:07
• loopy walt pointed out a way to prove that it is optimal at the current state of research. Apr 22 at 6:03

UPDATE: @Rob Pratt's is optimal at current state of research:

Here is the best known packing of 32 circles into a regular heptadekagon.

Coordinates taken from http://hydra.nat.uni-magdeburg.de/packing/ced/ced.html If we could pack 15 circles into a regular 17-ring we would beat that. Therefore if we believe the experts it is not possible.

Mind you, it's pretty darn close. All the red lines indicate touches. Those which are not marked but look as if they ought to be are less than 6 permille (1/10), 3 permille (1/10) or less than 1 permille (8/10) of a radius apart.

original post: