PigeonHole - Notes for class 7 ITI:111 Paul Kantor 2009 W...

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Notes for class 7. ITI:111 © Paul Kantor 2009 We’re going to discuss three different topics today: the pigeon-hole principle, the principle of mathematical induction, and the concept of triangular numbers. 1. The Pigeon-hole principle A pigeon-hole desk from: www.bostonminiatures.com Actual pigeon holes (probably what inspired the desk and the principle) from commons.wikimedia.org I asked the class and determined that everyone in the class is either a freshman, a sophomore, a junior or a senior, a total of four different years. I then randomly picked five people and asked them to stand up and I asked this question “can I be absolutely certain that some two of these five people belong to the same class?” It was pretty obvious to everyone in class that the answer is “yes”. There are more people than there are classes and so there must be two people in the same class. This is the first example of the famous pigeon-hole principle. The name “pigeon-hole” comes from the fact that old fashioned writing desks, which a few of you may still have seen somewhere, have little boxes that are about two inches by three inches at the opening to put letters and bills and stuff like that into. These are called pigeon holes, because they have the same shape as the holes in ancient caves where pigeons used to be kept. In this case the things in the pigeon holes aren’t real things, they are abstractions such as a particular year at Rutgers. We show try the same idea by considering the months in which people are born. I asked thirteen people to stand up and again it’s perfectly clear that there must be at least two of
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them who are born in the same month. If we start assigning months to the people, we will run out of months before we run out of people, and someone is going to have to be added to a month that has already been used.
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PigeonHole - Notes for class 7 ITI:111 Paul Kantor 2009 W...

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