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        riddles >> easy >> coin in a circle
        
(Message started by: jollytall on Dec 14^th, 2012, 10:23am)

Title: coin in a circle
Post by jollytall on Dec 14^th, 2012, 10:23am

Is it true that if you have identical coins with a radius of 1, then for every N there is an R, where in a circle with radius R N, but also maximum N coins can be placed (no overlapping, smashing, standing on the edge, etc.).
With other words, if you start with a small circle (R<1) and gradually increase the R, then always one extra coin can be added (coins can be rearranged) at the same time (i.e. it does not happen that because of e.g. symmetry when you reach a certain R, then suddenly 2 or more coins can be added at the same time).
I do not know if it is easy or hard, I have no solution.

If too easy: what about coins in other shapes like squares? What about small balls in a large ball?

Title: Re: coin in a circle
Post by Grimbal on Dec 15^th, 2012, 7:13am

It seems the optimal packing of 18 circles allows for a free 19th circle.
http://hydra.nat.uni-magdeburg.de/packing/cci/cci18.html

Status: debunked

Title: Re: coin in a circle
Post by jollytall on Dec 15^th, 2012, 1:47pm

Well, thanks. It seems someone has spent a lot of time on it already. Good find.

Btw. 19 in proven as per the description, but 18 is not (though seems very likely). Anyway I accept it as a likely enough solution. :-)

Title: Re: coin in a circle
Post by towr on Dec 16^th, 2012, 7:17am

Why pick the 18 and 19, and not 6 and 7?

Title: Re: coin in a circle
Post by Grimbal on Dec 16^th, 2012, 12:41pm

Er... good question.

I searched "circle packing in a circle", found a list of optimal packings with the ratio between outer and inner circle. I searched a bit randomly for 2 equal values and I found it at 18 and 19.

Indeed, 6 and 7 is much more convincing.