Tuesday, November 15, 2022
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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)

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

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

$$n=12$$, 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)

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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