Packing Unit Squares in Squares: A Survey and New Results

Abstract

Let $s(n)$ be the side of the smallest square into which we can pack n unit squares. We present a history of this problem, and give the best known upper and lower bounds for $s(n)$ for $n\le100$, including the best known packings. We also give relatively simple proofs for the values of $s(n)$ when $n = 2$, 3, 5, 8, 15, 24, and 35, and more complicated proofs for $n=7$ and 14. We also prove many other lower bounds for various $s(n)$.