Efficient packing of unit squares in a square

Abstract

Let s(N) denote the edge length of the smallest square in which one can pack N unit squares. A duality method is introduced to prove that s(6) = s(7) = 3. Let nr be the smallest integer n such that s(n2 + 1) ≤ n + 1/r. We use an explicit construction to show that nr ≤ 27r3/2+O(r2), and also that n2 ≤ 43.