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

Abstract

It is known that $\sum _{i=1}^{\infty }\frac{1}{i(i+1)}=1$. 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 $i=1,2,\dots$, 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 $1.35\cdot 10^{11}$ rectangles into the unit square and give an estimate for $\epsilon$ from this packing.

7 pages, 4 figures