The sequence $\frac{1}{2}, \frac{1}{2\cdot 2}, \frac{1}{3\cdot 4}, \frac{1}{4\cdot 6}, \frac{1}{5\cdot 8}, \frac{1}{6\cdot 10},\ldots$ has a curious property, as follows:

a) the series with these terms sums to 1;

b) no process of sequentially packing open intervals with these lengths into the unit interval $[0,1]$ can ever come to an impasse.

Many other sequences also enjoy this property.

Question: has this type of phenomena ever appeared in the literature?

In particular I wonder about possible decompositions (up to a set of measure zero) of, say, the unit square (or unit sphere) into open sets which enjoy the corresponding property.

arbitrarycollection ofcirclesexcept for a bound on total area; I ask about a carefully contrived ordered collection of open sets. – David Feldman Mar 6 '12 at 7:13