It's always reassuring to know that a problem that seemed elementary and stumped me also stumped Erdos. However it's also humbling to see how much farther he (and others) got in their work on this problem than I have been able to. I see how the square lattice establishes the upper bound and I can also see how to experiment a bit for smaller n with other lattice paterns like equilateral triangles or other subsets of the square lattice. But how did they establish the lower bound? Thanks to everyone for your help and all the great references. One thing that I haven't seen addressed so far: for small n, like 6 or 7 even, how many distinct arrangments of points are there that attain the fewest possible distinct distances? For n = 4, as I said, I found 6 arrangements with only 2 distances and I *think* I got them all ... for n=5 i think there's only the regular pentagon? --Joshua Zucker