A friend asked me a question to ponder over:
You got $14$ dots which you need to place on a plane in such a way so that you get the maximum amount of similar distances between each $2$ points. I managed to get $31$ ($12$ first hexagon $+ 12$ second hexagon$ + 7$ distance between each point of $2$ hexagons) by drawing $2$ hexagons one below the other with the distance between the centers equal to the side length
The answer is incorrect though according to him.
Any insight towards the solution would be helpful.
PS: don't mind the black-red, I made such distinction just for the sake of pointing out the position of $14$ dots ($2$ hexagons + $2$ centers)
Best Answer
Apparently the maximum is $33$, as was worked out in C. Schade: Exakte Maximalzahlen gleicher Abstände, Diploma thesis directed by H. Harborth, Techn. Univ. Braunschweig 1993. Sadly, I couldn't find a description of the proof. This maximum can be realized in two ways, up to graph isomorphism:
Image source: Jean-Paul Delahaye, Les graphes-allumettes, (in French), Pour la Science no. 445, November 2014, pages 108-113. http://www.lifl.fr/~jdelahay/pls/2014/252.pdf
More resources are listed at https://oeis.org/A186705.