Just-in-time scheduling is a central industrial paradigm that means manufacuring parts only when they are needed, thus eliminating inventories and streamlining assembly lines. Just-in-time requires a partition of time into a given number $m$ of non-overlapping time-frames of decreasing density. That is, for every $i=1,...,m$, elements of the $i$th part should appear with constant frequency, but with frequency that is lower than that of the $i-1$st part. The partition should be made ``in real time'' meaning that at time $t$ one should decide quickly to which part $k$ belongs.
In 1973 an Institute scientist solved the above partition problem, while being driven by a different motivation (and while being unaware of the Just-in-time application). In his solution the $i$th subset is the set of all integers of the form $[k\cdot(2^m-1)/2^(m-i)]-2^(i-1)+1$, where $k=1,2,...$ and $[x]$ means rounding down $x$ to the nearest integer. He conjectured that the method is unique; that is, for $m\geq 3$, the sequence of numbers $a_i=(2^m-1)/2^(m-i)$, for $i=1...,m$, is essentially the only sequence that can partition the integers into $m$ subsets.
Mathematicians have since only been able to prove partial results of the conjecture; and industrial scientists have used and perfected it to design optimal Just-in-time procedures.