I need determine the maximum number of squares of the given size that can be packed into a circle of the given radius. Squares can be rotated. I'm not sure how complex this problem is and i can find little about it on the other sites.
1$\begingroup$ This sounds like a hard problem. But while I understand the need to pack circular/spherical things (e.g. oranges) into square things (e.g. boxes), why would one want to pack square things in a circle?? $\endgroup$– TMMMar 2, 2012 at 18:21
2$\begingroup$ @TMM: I think that there is a thin connection with getting as many silicon wafers (squares?) as possible out of a circular piece. $\endgroup$– André NicolasMar 2, 2012 at 18:39
1$\begingroup$ see Open Access paper published: 03 October 2018 : Rigorous packing of unit squares into a circle, T Montanher et al, Journal of Global Optimization volume 73, pages547–565(2019) $\endgroup$– sammy gerbilApr 24, 2020 at 22:45
This is a hard case-by-case problem. This beautiful page shows the records for the smallest circle packed with $n$ unit squares for $n$ from 1 to 35. You can see that there's nothing obvious about most of the solutions. Of course, as you pack more and more squares into a circle, there's less and less to be gained by finding a clever arrangement.