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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 n r be the smallest integer n such that s(n 2 + 1) ≤ n + 1/r. We use an explicit construction to show that n r ≤ 27r 3 /2 + O(r 2), and also that n 2 ≤ 43. 1. Introduction. Erd˝ os and Graham [1]… (More)

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