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

[hv at crypt.org]

I should have said: I worked below _without_ the assumption that the
points had integer coordinates, so if accurate this proves something
slightly stronger. _With_ that assumption, it would not have been
safe to assume we can scale any arrangement to make the smallest
square be a unit square.

Hugo

I wrote:
:Neil Sloane <njasloane at gmail.com> wrote:
::It seems that surprisingly little is known - see A051602. Even a(7) is an
::open question, although it is easy to see that a(7) >= 3.
:
:Unless I'm missing something, the 7-point case doesn't seem too hard.
:
:Fix 4 points on the standard unit square, and wlog assume this is the
:smallest square in the final combination. There are then four
:possibilities.
:
:A) There's another square of the same size, sharing two points with the
:first. Then the 7th point can only give us a third square in two
:symmetrically placed ways, of side sqrt(2). No fourth is possible.
:
:B) There's another square of the same size, sharing one point with the
:first. Then all 7 points are accounted for. Any third square must use
:two points from each of the existing two squares, so again can only
:have a side of sqrt(2); that is possible, but there can't be a fourth.
:
:C) There's another square of a different size, sharing two points with
:the first; this can only have side sqrt(2). The only ways to select
:three of those points such that the 7th point can be added to give a
:square in each case reduce us to case A or B.
:
:D) There's another square of a different size, sharing one point with
:the first. All points are accounted for, and any additional square
:must share two points with each of the first two squares; but then
:any such third square must reduce us to case A or C.
:
:So the possibilities are:
:
:.XX
:XXX
:XX.
:
:and
:
:.X.
:XXX
:XXX
:
:Hugo