[seqfan] Re: How many squares can you make from n points in the plane?
Benoît Jubin
benoit.jubin at gmail.com
Fri Oct 1 19:54:21 CEST 2021
> 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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