TITLE: How can we find n points on a plane so that as many pairs of points as possible have the same distance?
QUESTION [10 upvotes]: There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.

REPLY [14 votes]: The number is tabulated at OEIS. It seems that it's only known up to $n=14$ (and some scattered larger values). Links are given there to some papers on the topic. Evidently, no one knows how to do it for general $n$. 
Also discussed on math.stackexchange.