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

[Neil Sloane]

Benoit,  Thanks for the lovely example for n=49 !
Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
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On Fri, Oct 1, 2021 at 1:54 PM Benoît Jubin <benoit.jubin at gmail.com> wrote:

> > I saw that Peter, Sascha, ... observed that moving corner points to just
> > outside the middle of the edges did better than the square array.  When
> > does this beat the square array?  And does it produce a case when b(n) >
> > a(n)?  That now seems very likely!
>
>
> For the moment, all points are on the grid, so there is no witness
> that a(n) and b(n) may differ.  What they mean, I think, is for
> instance for m = 7:
> ....x....
> ..xxxxx..
> .xxxxxxx.
> .xxxxxxx.
> xxxxxxxxx
> .xxxxxxx.
> .xxxxxxx.
> ..xxxxx..
> ....x....
> This wins over the square, because a corner was in 6 squares (so we
> lose 4*6 - 1 squares), whereas the added midpoints are in 7 squares
> each, plus 1 common square through the four midpoints. So it should be
> a +6 win.
>
> Benoît
>
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