It’s a principle that provides the basic and the essential part in the mathematics of sorting and even counting. This principle was introduced by Johann Peter Gustav Lejeune Dirichlet in 1834. He called it the shelf principle or drawer principle. However, many people appreciate his insight and call it the Dirichlet drawer principle. The name pigeonhole was introduced in 1940 by Raphael Robinson. In the literal sense, pigeonholes refer to shelves or holes used to place pigeons. Thus, the principle roughly states that if one has a given number of boxes and a given number of items to put in the boxes, with the number of items being more than the boxes, then at least one box will contain more than one item. At first, the principle was used in finite sets. However, over the years mathematicians have used it with infinite sets (Francis, 2010).
Historical application of the principle in finite sets
Johann Peter Gustav Lejeune Dirichlet used the principle as a basic counting argument in an attempt to prove a result about Diophantine approximation. In the approximation, he found that if one puts kn+1 pigeons into n pigeon holes definitely one pigeon hole will contain K+ 1 pigeons. This use was extended to various problems. For example, the birthday paradox, in this problem one is challenged to determine the probability that in a given number of randomly chosen people some of them will have the same birthday. Applying the principle, it was realized that, there is a 50% probability that a pair of people will have same birthday in a group of 23 randomly selected people. However, the probability increases as the number of persons considered is increased as shown in the graph below. This paradox was introduced and discussed by Harold Davenport. Dirichlet also used the principle to approximate irrational numbers using rational numbers (Bogomolny, 1999).
(Bogomolny, 1999).
Modern Application of Pigeonhole Principle
The theory is currently used in hash tables. To increase the efficiency of hash tables one needs to use a large enough hash value so as to avoid the repeat stated by the pigeonhole principle. However, this uniqueness may be unattainable in a situation where one is required to make an excessively high number of unique data entries. For example, the global population is around six billion. Thus, it is not possible to have enough keys to represent the DNA of every person in a 32-bit hash function because the possible values are 232 ~= 4.3 billion only. There is a possibility that the number of keys falls below that of indices in the array. This shows that collisions are unavoidable in hashing algorithm (Francis, 2010).
The principle is also used in printing unique codes used to identify objects. The principle can calculate the number of codes that can be generated without duplicating any code. This number increases as the number of the digits in the code increase. Moreover, it can be used to determine the number of identical codes that are printed once the unique threshold is exceeded (Bogomolny, 1999).
The ability to rely on the pigeonhole principle to generate codes is used to generate identification numbers or even passport numbers. This principle makes sure that there is no time two people will share one identification number. Thus, the government can associate all the information of the person to her unique number. Moreover, the concept is applied in generating unique numbers used by telecommunication companies in manufacturing airtime cards. The program set ensures that it is difficult for people to guess the code and the code is never repeated. A telecom company could lose a lot of money if it were possible for customers to guess airtime top up codes. The non-repeat aspect ensures a top up will not fail because the code has been used by another person. This is attainable by making the code as long as possible. Many companies use a sixteen digit code (Bogomolny, 1999).
The application of the pigeonhole principle has been extended into the judiciary particularly in the law of tort. A plaintiff requests a judge to order the defendant to compensate him as a result of an injury suffered by an action of the defendant. However, if all the torts are placed in pigeon holes, the plaintiff should successfully place the wrong committed in one pigeon hole (Niloybagchi, 2014).
David Speyer was able to use pigeonhole principle in solving pell’s equation. He realized that x2−Dy2=1 is a unique equation that has excessively many solutions. In the equation, D is taken to be square-free. This proof is a simplification of the complex Lagrange's complex proof. The pigeonhole principle was introduced in 1934 by Johann Peter Gustav Lejeune Dirichlet. This concept did not evolve; however, mathematicians have been able to the extent its applicability to various areas. For example, in computer science, it is applied in the concept of harsh theory while in basic counting it is used to determine the probability of occurrence of a given event in Diophantine approximation (Francis, 2010).
References
Bogomolny, N. (1999). Pigeonhole Principle. Retrieved January 20, 2017, from http://www.cut-the-
knot.org/do_you_know/pigeon.shtml
Niloybagchi, M. (2014, October 26). Pigeon Hole Theory: Aspects of Criticism. Retrieved January 20,
2017, from http://www.legalservicesindia.com/article/article/pigeon-hole-theory-aspects-of-criticism-1716-1.html
Francis, S. (2010). Pigeonhole Principle. Retrieved January 20, 2017, from
https://www.math.hmc.edu/funfacts/ffiles/10001.4.shtml
