For 25 years, Macalester College mathematician Joe Konhauser offered a “problem of the week” to his students. Here’s a sample, from the collection Which Way Did the Bicycle Go? (1996):
Fifteen sheets of paper of various sizes and shapes lie on a desktop, covering it completely. The sheets may overlap one another and may even hang over the edge of the desktop. Prove that five of the sheets can be removed so that the remaining ten sheets cover at least two-thirds of the desktop.
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“Imagine that the desktop was painted just before the paper was put on it. Then some of the sheets of paper will have picked up some paint when they were put on the desktop. More precisely, each sheet of paper will have picked up paint on it from any part of the desk that it covers that is not covered by any sheet lower in the pile. Since the desktop is completely covered by the paper, the total area of the paint on all the sheets of paper is exactly equal to the area of the desktop. If we remove the five sheets with the least paint on them, then clearly the total area of paint on the sheets remaining is at least two-thirds of the area of the desktop, and therefore at least two-thirds of the desktop is covered by the remaining sheets.”
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