This question comes from a discussion with my wife about the more efficient way for cooking biscuits. Here the problem:
We have a circular pan with a diameter of $30 cm$ and we have two round stamps for biscuits with diameters of $5$ and $3$ cm and we want produce a number $n$ of biscuits in a proportion that is about $1/3$ of $5 cm$ biscuits and $2/3$ of $3 cm$. What is the more efficient way to put the biscuits on the pan?
My wife say that usually she cooks separately the two measures, but I say that it's better to mix the two in a way the better use the space. So she asked to me what is this better way. But, after a bit of thinking, I've not a good answer.
I know that the circles packing is a difficult problem, with a lot of literature produced about it, but my knowledge about it is very poor. I've seen the wikipedia page where are listed some optimal configurations, but all with the little circle with the same diameter.
So I ask if there is some way to solve the problem that is not by trial and error.
( sorry for my bad english).
Perhaps this paper would be of interest. It's about compactly packing two different sized circles in the plane.