Orchard planting problem for squares

Question

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

Just to clarify, are all the sides of a square the same length and at a 90 degree angle? Or are trapezium's allowed? — LiefdeWen, Sep 01 '20 at 07:47
squares must have sides of the same length and 90 degree angles. — Dmitry Kamenetsky, Sep 01 '20 at 08:34

score 3 · Accepted Answer · answered Sep 01 '20 at 10:15

3

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

$7$ squares

With the following arrangement

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

answered Sep 01 '20 at 10:15

hexomino

135,910
10
384
563

2

Yep that's the solution I found. Not sure if it can be improved. – Dmitry Kamenetsky Sep 01 '20 at 11:06
I checked that this answer is optimal with a computer program. – Dmitry Kamenetsky Sep 05 '20 at 13:52

Orchard planting problem for squares

1 Answers1