  A homothetic packing of squares is any set of various-size squares with the
same orientation where no two squares have overlapping interiors. If all $n$
squares have the same size then we can have up to roughly $4n$ contacts by
arranging the squares in a grid formation. The maximum possible number of
contacts for a set of $n$ squares will drop drastically, however, if the size
of each square is chosen more-or-less randomly. In the following paper we
describe a necessary and sufficient condition for determining if a set of $n$
squares with fixed sizes can be arranged into a homothetic square packing with
more than $2n-2$ contacts. Using this, we then prove that any (possibly not
homothetic) packing of $n$ squares will have at most $2n-2$ face-to-face
contacts if the various widths of the squares do not satisfy a finite set of
linear equations.
