TITLE: Erdos distance problem n=12
QUESTION [15 upvotes]: The recent paper On the Erdos distinct distance problem in the plane
Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example.  Here is what I cannot find:  an example of 12 distinct points in the plane with only 5 different distances between points.  The regular 12-polygon has 6 different lengths but I cannot do better. http://oeis.org/A186704 implies that there is one>

REPLY [5 votes]: Thought it might be nice to show the set placed on the hexagonal lattice. 

Here’s a Desmos link if you wish to play around with such point configurations.