Packing $1.35\cdot 10^{11}$ rectangles into a unit square

Abstract

It is known that $\sum\limits_{i=1}^{\infty} \frac{1}{i (i+1)} = 1$. In 1968, Meir and Moser asked for finding the smallest $\epsilon$ such that all the rectangles of sizes $1/i \times 1/(i + 1)$ for $i = 1, 2, \ldots$, can be packed into a unit square or a rectangle of area $1 + \epsilon$. In this paper, we show that we can pack the first $1.35\cdot10^{11}$ rectangles into the unit square and give an estimate for $\epsilon$ from this packing.