May 2016

After looking at the potential topics for this paper I was initially drawn to “Mathematics and Music” because my wife is very musical and likes the topic but as I read about it, it didn’t really sing to me. As I read the rest, I thought the pigeonhole principle seemed very interesting and I decided to write about that. In chapter three in the book “To Infinity and Beyond” the pigeonhole principle states: “Suppose we have four pigeons but only three pigeonholes. No matter how we assign a hole to each pigeon, at least two pigeons will have to share the same hole.” We can apply this principle to conclude some surprising results with very little work.
Before I give some other examples this principle is based on certainty and not chance. So each example will happen but the specifics may differ from case to case. One example I found outside of the book was “if you pick five cards from a standard deck of 52 cards, then at least two will be of the same suit.” (16 fun applications of the pigeonhole principle) The reason that I liked this example is that I could easily test it and see for myself at home. While this principle made logical sense, it was good to be able to test it and see for myself that it worked. I then scaled this principle up. I found that when I had nine cards, three of them were of the same suit. While many of the conclusions of the principle are difficult to confirm I found this to be fun to scale and confirm. [Calculation 9 = 4 x 2 +1]
You might wonder to yourself at this point about how this is a nice trick but what use is this principle to mathematics or science? As I was reading I learned the pigeonhole principle is called an “Existence theorem” because it confirms that a solution exists but doesn’t give a method for finding it. So when working on a difficult problem we can use an Existence theorem like the pigeonhole principle, to confirm ahead of time that a solution exists before working on it. This would be preferable to wasting time on a problem where a solution doesn’t exist and for which that could have been confirmed previously.
Another principle spoken of in the chapter is “Uniqueness of a solution.” The Uniqueness theorem is essentially if there is a solution, it is the only solution. While the pigeonhole principle itself is not a unique solution with two pigeons that can be random among the group, this theorem is an extension of that principle. This theorem can be applied to many things in life where there is a presumption of one solution. The book uses an example of finding the one true church. An example popular in the world would be finding your one true love. After you found that person continuing to search would be illogical as you have already found the person.
The pigeonhole principle was fascinating to learn about and test myself with its predictive ability. While I’m not the sort of hard mathematics person to use this in my working life, I think it will be fun to share this sort of thing with friends as a conversation piece they can see and test in person with a few examples.