Abstract

If c>5 \) and if \(x\) is sufficiently large, then any collection of rectangles of sides of length not greater than \(1\) with total area smaller than \( x^2-cx^{5/6} \) can be packed into a square of side length \(x\).

Highlights

  • First publications related to packing of rectangles or squares appeared over fifty years ago

  • In 1957 Kosinski [6] proved, among others, that any sequence of rectangles of total area V and with sides of length not greater than D can be packed into a rectangle of side lengths 3D and (V + D2)/D

  • For example, “Can every set of rectangles of total area 1 and maximal side 1 be accommodated in a square of area 2?” or “What is the smallest number A such that any set of squares of total area 1 can be packed into some rectangle of area A?”

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Summary

Introduction

First publications related to packing of rectangles or squares appeared over fifty years ago. In 1957 Kosinski [6] proved, among others, that any sequence of rectangles of total area V and with sides of length not greater than D can be packed into a rectangle of side lengths 3D and (V + D2)/D. This result was improved in [4,7,8]. Denote by s(x) the greatest number such that any collection of rectangles of sides of length not greater than 1 with total area smaller than s(x) can be packed into Ix. Groemer [4] proved that s(x) ≥ (x −1) provided x ≥ 3.

Preliminaries
Packing into a large square
Full Text
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