  This paper attacks the following problem. We are given a large number $N$ of
rectangles in the plane, each with horizontal and vertical sides, and also a
number $r<N$. The given list of $N$ rectangles may contain duplicates. The
problem is to find $r$ of these rectangles, such that, if they are discarded,
then the intersection of the remaining $(N-r)$ rectangles has an intersection
with as large an area as possible. We will find an upper bound, depending only
on $N$ and $r$, and not on the particular data presented, for the number of
steps needed to run the algorithm on (a mathematical model of) a computer. In
fact our algorithm is able to determine, for each $s\le r$, $s$ rectangles from
the given list of $N$ rectangles, such that the remaining $(N-s)$ rectangles
have as large an area as possible, and this takes hardly any more time than
taking care only of the case $s=r$. Our algorithm extends to $d$-dimensional
rectangles. Our method is to exhaustively examine all possible
intersections---this is much faster than it sounds, because we do not need to
examine all $\binom Ns$ subsets in order to find all possible intersection
rectangles. For an extreme example, suppose the rectangles are nested, for
example concentric squares of distinct sizes, then the only intersections
examined are the smallest $s+1$ rectangles.
