Perfect packing of squares

Joós, Antal (2022) Perfect packing of squares. arXiv e-prints. ISSN 2331-8422

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Abstract

It is known that ∑i=1∞1/i2=π2/6. Meir and Moser asked what is the smallest ϵ such that all the squares of sides of length 1, 1/2, 1/3, … can be packed into a rectangle of area π2/6+ϵ. A packing into a rectangle of the right area is called perfect packing. Chalcraft packed the squares of sides of length 1, 2−t, 3−t, … and he found perfect packing for 1/2<t≤3/5. We will show based on an algorithm by Chalcraft that there are perfect packings if 1/2<t≤2/3. Moreover we show that there is a perfect packing for all t in the range log32≤t≤2/3.

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