[seqfan] Re: How many squares can you make from n points in the plane?

[Peter Munn]

On Thu, September 30, 2021 6:38 am, Neil Sloane wrote:
> This is just about a(n), A051602, where the n points are restricted to the
> integer lattice Z x Z.
[...]
>
> Is there a clever argument that says that if n = m^2, the square mXm grid
> arrangement  is optimal?

On the contrary, it isn't for m > 6 (probably also m = 6, I need to double
check). De-select a grid point from a corner of the square and select
instead a point just beyond the middle of an opposite side. For m = 7, I
reckon this increases the square count by 3, and I think another 2 is
possible by repeating this for another corner. For m = 9, I suspect the
optimal arrangement is achieved when this adjustment is enacted for all 4
corners.

Best regards,

Peter