There are four lists, each with $100$ numbers in $[0,1]$. You want to perform as few swaps between pairs of numbers as possible, so that the difference between the sums of numbers in any two lists becomes at most $1$. What is the largest number of swaps that you may have to perform?
A worst case seems to be when two lists are all $1$ and the other two lists are all $0$, which needs $100$ swaps. A method for achieving the goal (difference $\le 1$) is that, as long as the goal is not met, we swap a largest number in a largest-sum list with a smallest number in a smallest-sum list. This always terminates because after the swap, both lists have a larger sum than the previous sum of the smaller-sum list. But does it always terminate after at most $100$ swaps?
