The classic Orchard planting problem asks for the maximum number of 3-point straight lines attainable from a configuration of $n$ points drawn on a plane.

Here we are interested in a variant of this problem. What is the maximum number of squares attainable from a configuration of 10 points drawn on a plane? Each corner of an attained square must contain a point.

Here is a similar puzzle for circles: Orchard planting problem for circles

  • $\begingroup$ Just to clarify, are all the sides of a square the same length and at a 90 degree angle? Or are trapezium's allowed? $\endgroup$ – LiefdeWen Sep 1 '20 at 7:47
  • $\begingroup$ squares must have sides of the same length and 90 degree angles. $\endgroup$ – Dmitry Kamenetsky Sep 1 '20 at 8:34

May not be optimal but the best I can seem to get is

$7$ squares

With the following arrangement

enter image description here
That is, four of side length $1$, one of side length $2$ and two of side length $\sqrt{2}$.

  • 2
    $\begingroup$ Yep that's the solution I found. Not sure if it can be improved. $\endgroup$ – Dmitry Kamenetsky Sep 1 '20 at 11:06
  • $\begingroup$ I checked that this answer is optimal with a computer program. $\endgroup$ – Dmitry Kamenetsky Sep 5 '20 at 13:52

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