Orchard planting problem of 5 points per circle [closed]

In General orchard planting problem for circles , the problem of 4 points per circle has been studied.

The problem here is that what is the maximum number of 5-point circles for a configuration of n points drawn on a plane?

It is easy to show that we need 8 points to get two 5-point circles and 9 points to get three 5-point circles

10 points could reach five 5-point circles:

11 points to reach seven 5-point circles and 12 points to reach nine 5-point circles:

In all pictures above, one point is at infinite point and circle-inversion transformation (turn line into circle) could be used to transform it to normal point.

• What's the red circle in your second picture point to? Commented Jan 26, 2023 at 2:51
• great idea for a problem Commented Jan 26, 2023 at 4:14
• @justhalf Ignore it. It shows that we could not add a third circle to the picture. Commented Jan 26, 2023 at 14:16
• With 12 points you can make 12 5-point circles. Make a stereographic projection of an icosahedron. Or project a dodecahedron and you get 24 circles using 20 points. Commented Jan 26, 2023 at 23:29
• astonishing result. Searching by computer and we could not find any combination with 13 5-point circles so that 12 5-point circles are best result of 12 points Commented Jan 27, 2023 at 3:38

Here is the 12 points, 12 5-circles solution I mentioned in a comment.

And adding a single point in the center gives you 3 more circles.

Add an equivalent result of Florian F. (one infinity point) for 13 points with 15 5-point circles (where black point and black dash circles are the extra point and 3 extra 5-point circles) And similarly we could add one more extra point and 3 extra 5-point circles to reach 14 points with 18 5-point circles and one more point and 2 extra 5-point circles to reach 15 points with 20 5-point circles.