The abstract theory of the Pigeonhole Principle can be very useful in the life of statistics. Knowing that if there’s a limited number of input and that it has to go through a limited number (smaller than the input) of hoops and every hoop has to be filled before the extras can go through, can be helpful, especially when thinking of it in terms of Chemistry.
In Chemistry, after hypothetically combining the atoms with bonds, the student has to distribute the electrons evening so that every atom has eight electrons, including those atoms that share a certain number of electrons. Pigeonholing is used here. The total number of electrons are the pigeons and that every atom has to have eight electrons are the holes. We know that each atom has to be connected by a covalent bond, thus completing the principle that in some order there are atoms sharing electrons. Also within Chemistry the principle of Pigeonholing is used. Within electron configurations, for each energy level, the electrons have to fill up one row spinning one way (i.e. down or up) then fill in what’s left spinning the opposite way. When putting together the electron configuration I don’t know for certain which portion of the configuration has a partner or not but I know that at least one does, especially if the number of electrons for a certain element is in the middle of the periodic table.
The PDF book that I read, Infinity and Beyond by Dr. Kent A Bessey mentioned other theories that go along with the Pigeonhole Principle, such as the Existence and Uniqueness Theorems. I find that these three go hand in hand very well. The Existence Theorem explains that there is a solution to every Pigeonhole problem no matter what the numbers and situations are. The Uniqueness Theorem explains that there is one solution to the situation.
At first I thought the theorems above were annoying and useless. What is the purpose of having a possible solution if you don’t know how to get it or find it? It was not clear until I read a portion of chapter three in Infinity and Beyond. It states, “knowing that a problem has a solution is wonderful, but equally satisfying is knowing there is only one solution” (Bessey 46-53). Strangely enough, this is a thought process that I’ve known all my life. From grade school to the present, knowing that there is a solution to the problem I am solving is wonderful, because I know that I’m not wasting my time. My teacher used it a lot in AP Calculus AB. He would give us the problem and the answer so we could check our work.
The uses of Pigeonholing, and Existence and Uniqueness theorems are used throughout our lives whether it is within Chemistry, some sort of statistics or simply trying to figure out how many people per seat you can fit into your car.