Pigeonhole Principle: Theorem, Statement & Examples

The Pigeonhole Principle in Discrete Mathematics | Comprehensive Guide

In this guide, we’ll explore the Pigeonhole Principle, a simple but powerful concept in discrete mathematics. The Pigeonhole Principle states that if more items are distributed into fewer containers than there are items, then at least one container must hold more than one item. This principle is widely applied in combinatorics, number theory, and computer science to solve various counting and distribution problems. By the end of this guide, you will understand the Pigeonhole Principle, how to apply it, and its real-world applications.

What is the Pigeonhole Principle?

The Pigeonhole Principle can be stated as:

"If nnn items are put into mmm containers, and n>mn > mn>m, then at least one container must contain more than one item."

In other words, when there are more items than containers, it's impossible to distribute the items such that each container has exactly one item. Some containers must hold more than one item.

Formal Statement:

Let nnn and mmm be positive integers. If n>mn > mn>m, and nnn objects are placed into mmm containers, then at least one container must hold more than ⌊nm⌋\left\lfloor \frac{n}{m} \right\rfloor⌊mn​⌋ items.

Example of the Pigeonhole Principle:

Simple Example:

Consider 11 pigeons and 10 pigeonholes. If we place each pigeon into a pigeonhole, the Pigeonhole Principle guarantees that at least one pigeonhole must contain more than one pigeon. This is because there are more pigeons than there are pigeonholes.

Real-World Example:

Suppose there are 13 people in a room, and we want to know if any two people share the same birth month. Since there are only 12 months in a year, the Pigeonhole Principle tells us that at least two people must have been born in the same month, as there are more people than months.

Generalized Pigeonhole Principle:

The generalized Pigeonhole Principle is an extension of the basic principle. It states that if nnn items are placed into mmm containers, then at least one container must hold at least ⌈nm⌉\left\lceil \frac{n}{m} \right\rceil⌈mn​⌉ items.

Formal Statement:

If nnn items are placed into mmm containers, then at least one container must hold at least ⌈nm⌉\left\lceil \frac{n}{m} \right\rceil⌈mn​⌉ items, where ⌈x⌉\lceil x \rceil⌈x⌉ denotes the ceiling function (the smallest integer greater than or equal to xxx).

Example:

If you have 100 apples and 9 baskets, the generalized Pigeonhole Principle guarantees that at least one basket will contain at least ⌈1009⌉=12\left\lceil \frac{100}{9} \right\rceil = 12⌈9100​⌉=12 apples.

Applications of the Pigeonhole Principle:

Number Theory:

• The Pigeonhole Principle is used in number theory to prove results about divisibility and modular arithmetic. For example, if you choose 5 integers from the set {1,2,3,4,5,6,7,8}\{1, 2, 3, 4, 5, 6, 7, 8\}{1,2,3,4,5,6,7,8}, the principle guarantees that at least two of the chosen integers will have the same remainder when divided by 4.

Graph Theory:

• In graph theory, the Pigeonhole Principle can be used to prove that certain properties must hold for graphs. For instance, in a group of people where each person shakes hands with others, if there are 6 people, the principle guarantees that at least two people must have shaken hands with the same number of people.

Computer Science:

• The Pigeonhole Principle is used in hashing algorithms. If more items are inserted into a hash table with fewer available slots, the principle ensures that at least one slot will contain multiple items (causing a collision).

Combinatorics:

• In combinatorics, the Pigeonhole Principle helps in counting problems where items must be distributed into categories. It guarantees that certain conditions must be met, such as some category receiving more than the average number of items.

Passwords and Security:

• In cryptography, the principle is used to analyze security vulnerabilities. For example, if a password space has fewer possible combinations than the number of users, the Pigeonhole Principle guarantees that at least two users will have the same password, making the system less secure.

Example Problem Using the Generalized Pigeonhole Principle:

Problem: In a group of 20 students, at least how many students share the same birth month?

Solution: There are 12 months in a year, so we need to distribute 20 students into 12 months (pigeonholes). Applying the generalized Pigeonhole Principle:

⌈2012⌉=⌈1.67⌉=2\left\lceil \frac{20}{12} \right\rceil = \left\lceil 1.67 \right\rceil = 2⌈1220​⌉=⌈1.67⌉=2

Therefore, at least 2 students must share the same birth month.

Why Learn About the Pigeonhole Principle?

The Pigeonhole Principle is a simple yet powerful concept that helps solve a variety of problems in mathematics, computer science, and real-world scenarios. It provides an intuitive way to reason about distributions and proves that certain conditions must be true in any situation where there are more objects than containers.

Topics Covered:

Basic Pigeonhole Principle: The concept of distributing more items into fewer containers.

Generalized Pigeonhole Principle: How to determine the minimum number of items in one container when items are distributed unevenly.

Applications: Use cases in number theory, graph theory, computer science, and real-world problems.

For more details and further examples, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/discrete-mathematics-the-pigeonhole-principle/.