Perfect packing of squares

Abstract

It is known that $\sum _{i=1}^{\infty }1/i^2={\pi }^2/6$. Meir and Moser asked what is the smallest $\epsilon$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\dots$ can be packed into a rectangle of area ${\pi }^2/6+\epsilon$. A packing into a rectangle of the right area is called perfect packing. Chalcraft packed the squares of sides of length $1$, $2^{-t}$, $3^{-t}$, $\dots$ and he found perfect packing for $1/2<t\le 3/5$. We will show based on an algorithm by Chalcraft that there are perfect packings if $1/2<t\le 2/3$. Moreover we show that there is a perfect packing for all $t$ in the range ${\log }_{3}2\le t\le 2/3$.

9 pages, 1 figure, Math. Rep. accepted (2019)