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.

An Existential Proof of the Conjecture on Packing Anchored Rectangles

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.

Absolute Calibration and Characterization of the Multiband Imaging Photometer for Spitzer. III. An Asteroid-based Calibration of MIPS at 160 microns

We describe the absolute calibration of the Multiband Imaging Photometer for Spitzer (MIPS) 160 micron channel. After the on-orbit discovery of a near-IR ghost image that dominates the signal for sources hotter than about 2000 K, we adopted a strategy utilizing asteroids to transfer the absolute calibrations of the MIPS 24 and 70 micron channels to the 160 micron channel. Near-simultaneous observations at all three wavelengths are taken, and photometry at the two shorter wavelengths is fit using the Standard Thermal Model. The 160 micron flux density is predicted from those fits and compared with the observed 160 micron signal to derive the conversion from instrumental units to surface brightness. The calibration factor we derive is 41.7 MJy/sr/MIPS160 (MIPS160 being the instrumental units). The scatter in the individual measurements of the calibration factor, as well as an assesment of the external uncertainties inherent in the calibration, lead us to adopt an uncertainty of 5.0 MJy/sr/MIPS160 (12%) for the absolute uncertainty on the 160 micron flux density of a particular source as determined from a single measurement. For sources brighter than about 2 Jy, non-linearity in the response of the 160 micron detectors produces an under-estimate of the flux density: for objects as bright as 4 Jy, measured flux densities are likely to be ~20% too low. This calibration has been checked against that of ISO (using ULIRGS) and IRAS (using IRAS-derived diameters), and is consistent with those at the 5% level.