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.
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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.