Let $P_{ n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each rectangle coincides with a point in $P_{ n}$ and the total area covered by the rectangles is maximized \cite{ ibmpuzzle}, \cite{ Winkler2007}, \cite{ Winkler2010a}, \cite{ Winkler2010b}. The longstanding conjecture has been that at least half of $U$ can be covered when such rectangles are properly placed. In this paper, we give an existential proof of the conjecture.
Categories: cs.DM math.CO