What is the Pigeonhole Principle?

What is the Pigeonhole Principle?

If you have more pigeons than pigeonholes into which you want to distribute those pigeons, at least one pigeonhole must contain more than one pigeon. In other words, if you have n objects and m containers, where n > m, then at least one of the m containers must contain more than one object. This principle is named after the analogy of pigeons (objects) being placed in pigeonholes (containers)....

Key Points of Pigeonhole Principle:

It provides a simple way to prove the existence of certain arrangements or occurrences without explicitly finding them. It’s often used to establish the existence of duplicates, repetitions, or patterns within a set of objects or elements. The principle is a useful tool in various areas of mathematics, computer science, and problem-solving, including combinatorics, graph theory, and competitive programming....

How to Identify Pigeonhole Principle Problems ?

Observe the problem statement carefully: First, read the problem description carefully. Look for keywords that may indicate the presence of the Pigeonhole Principle, such as “divisible”, “distribution,” “buckets,” “categories,” “objects,” or “elements.” Determine the distribution requirement: Distribution is a common indicator of the pigeonhole problem. Any problem requiring to distribution of elements(pigeons) into buckets (pigeonhole) may involve the pigeonhole principle. Analyze Constraints: Analyze the constraints of the task, including the number of objects and classes (or pigeons). If the number of objects exceeds the number of categories(pigeonholes) , this is a strong indication that the Pigeonhole Principle can be applied. Look for Repetitions: The pigeonhole principle often deals with patterns, repetitions, or duplicates between objects or elements. Compare with known pigeonhole problems: If you have solved or encountered similar problems related to the Dovecote principle, please compare them. Some famous pigeon problems include: Birthday Paradox: In a group of 23, there is an even greater chance that at least two people have the same birthday. This intuitive result is an application of the pigeon principle. Socks in drawers: If the dresser has more socks than drawers, at least one drawer should have more than one sock....

How to solve problems of Pigeonhole Principle?

1. Determine the Pigeons and Pigeonholes: After Identifying the problem statement carefully, you need to determine the pigeons and pigeonholes in the problem. Pigeons are the objects which need to be distributed and pigeonhole corresponds to buckets into which pigeons will be distributed. As in the above-solved problem, the remainders were the pigeons in the problem and the pigeonholes corresponded to the distinct remainders that can occur when dividing by n. 2. Apply the Pigeonhole Principle: Once the pigeons and pigeonholes have been identified, the pigeonhole principle can be applied. The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must have more than one pigeon. Apply this to your problem by figuring out if there are more holes than holes. If your problem setup matches this, it means you can apply the pigeonhole principle....

Practice Problems of Pigeonhole Principle:

Find a non empty subset in an array of N integers such that sum of elements of subset is divisible by N Maximum adjacent difference in an array in its sorted form Construct String with given frequency and minimum continuous occurrence of a letter Find a triplet (X, Y, Z) such that all are divisible by A, exactly one is divisible by both A and B, and X + Y = Z Find Binary String of size at most 3N containing at least 2 given strings of size 2N as subsequences Minimum number of socks required to picked to have at least K pairs of the same color Count of subarrays of size K having at least one pair with absolute difference divisible by K-1 Largest subset with sum of every pair as prime...