Axis-parallel square packing with an exact formula for optimal packing of seven squares

Abstract

How large a square is needed to hold axis-parallel squares of prescribed side lengths? We give exact formulas for the optimal side through seven squares, with explicit constructions and geometric lower-bound proofs. For every $n \ge 6$, the optimum for $n$ squares is a sum of at most $n-3$ of the prescribed side lengths. This bound is sharp on an open set of side lengths and yields the optimal Euclidean Lipschitz constant $\sqrt{n-3}$ of the optimum as a function of the $n$ side lengths.