My previous puzzle asked for the maximum number of 4-point circles attainable from a configuration of $n=10$ points drawn on a plane. I am now interested in generalizations of this puzzle to arbitrary $n$.
I wrote a hill-climbing program that searches for configurations with integer coordinates. Here are the best solutions it has found so far:
Here are the questions I want answered:
- Can you improve any of these solutions? You can use either integer or non-integer coordinates.
- Can we construct any upper/lower bounds on the maximum number of circles possible for an arbitrary $n$?
- The solutions for $n$=8, 10 and 12 use two concentric polygons. Can we conjecture that for even $n \geq 8$ the best solution will use two concentric $(n/2)$-polygons?
- For $n=13$ my best solution uses 43 circles, which is exactly like the $n=12$ case. Surely that extra point must be useful for a few more circles?