The problem of finding the optimal placement of $N$ identical, non overlapping, circles with maximum radius in the unit square is a well known challenge both in classical geometry and in optimization. A database of putative optima is currently maintained at \url{www.packomania.com}. Recently, through clever use of an extremely simple global optimization method, we succeeded in finding improved configurations for several instances. The improved configurations are in the range $N \leq 90$, i.e., they improve even over relatively small instances (even $N=53$), an event that some researchers did not believe to be possible. We also improved larger instances using a simpler strategy initialized at the previously known putative optimum.
Citation
DSI report 01/2005 Dipartimento di Sistemi e Informatica Universita' di Firenze