How many contacts can exist between oriented squares of various sizes?

Abstract

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.