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

TL;DR

This paper investigates the maximum number of face-to-face contacts in arrangements of squares of various sizes, providing conditions for when more than 2n-2 contacts are possible and establishing bounds for general packings.

Contribution

It introduces a necessary and sufficient condition for homothetic square packings to exceed 2n-2 contacts and proves bounds for non-homothetic packings with arbitrary sizes.

Findings

Maximum contacts for homothetic packings can be characterized by linear conditions.

Non-homothetic packings have at most 2n-2 contacts unless sizes satisfy specific linear equations.

Conditions for exceeding 2n-2 contacts are explicitly described.

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.