Perfect packing of squares

Abstract

It is known that $\sum\limits_{i =1}^\infty {1/ i^2}={\pi^2/6}$. Meir and Moser asked what is the smallest $\epsilon$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\ldots$ 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 $1$, $2^{-t}$, $3^{-t}$, $\ldots$ and he found perfect packing for $1/2<t\le3/5$. We will show based on an algorithm by Chalcraft that there are perfect packings if $1/2<t\le2/3$. Moreover we show that there is a perfect packing for all $t$ in the range $\log_32\le t\le2/3$.