On Packing Almost Half of a Square with Anchored Rectangles: A Constructive Approach

In this paper, we consider the following geometric puzzle whose origin was traced to Allan Freedman \cite{croft91,tutte69} in the 1960s by Dumitrescu and T{\'o}th \cite{adriancasaba2011}. The puzzle has been popularized of late by Peter Winkler \cite{Winkler2007}. Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. The problem is to construct $n$ 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. We would term the above rectangles as \emph{anchored rectangles}. The longstanding conjecture has been that at least half of $U$ can be covered when anchored rectangles are properly placed. Dumitrescu and T{\'o}th \cite{Dumitrescu2012} have shown a construction method that can cover at least $0.09121$, i.e., roughly $9\%$ of the area.