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arithmetic – General orchard planting downside for circles


My earlier puzzle requested for the utmost variety of 4-point circles attainable from a configuration of $n=10$ factors drawn on a airplane. I’m now fascinated about generalizations of this puzzle to arbitrary $n$.

I wrote a hill-climbing program that searches for configurations with integer coordinates. Here are the very best options it has discovered up to now:

$n=8$, 12 circles: (2,16) (10,20) (7,26) (12,6) (7,16) (12,11) (22,11) (16,14)
enter image description here

$n=9$, 14 circles: (9,17) (8,18) (5,17) (6,16) (7,19) (7,15) (6,18) (7,17) (8,16)
enter image description here

$n=11$, 30 circles: (5,27) (41,29) (29,37) (44,40) (35,33) (19,35) (35,7) (23,28) (35,37) (30,32) (17,19)
enter image description here

$n=12$, 43 circles: (27,7) (33,5) (37,17) (27,47) (21,5) (32,7) (12,17) (27,22) (42,17) (27,2) (17,17) (22,7)
enter image description here

Here are the questions I need answered:

  1. Can you enhance any of those options? You can use both integer or non-integer coordinates.
  2. Can we assemble any higher/decrease bounds on the utmost variety of circles doable for an arbitrary $n$?
  3. The options for $n$=8, 10 and 12 use two concentric polygons. Can we conjecture that for even $n geq 8$ the very best resolution will use two concentric $(n/2)$-polygons?
  4. For $n=13$ my finest resolution makes use of 43 circles, which is strictly just like the $n=12$ case. Surely that additional level should be helpful for a couple of extra circles?
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