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

[hv at crypt.org]

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