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

Benoît Jubin benoit.jubin at gmail.com
Fri Oct 1 22:41:53 CEST 2021

```> Benoit,  Thanks for the lovely example for n=49 !

That shows that a(49)>=202 if I counted correctly, but Sascha has an
example with 205 apparently.

Peter, Sascha: can you give me your optimal configurations ? (best
would be: in place of an "x" to denote a vertex, put the number of
squares it belongs to)

Neil: since Sascha is on a 3-day trip, I may wait Monday or Tuesday to
update the entry, if that's ok with you.

Benoît

> 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.
> Email: njasloane at gmail.com
>
>
>
> 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
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/

```