# Applications of pigeonhole principle with example. soft question 2022-12-24

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The pigeonhole principle, also known as the "boxes and pigeons" principle, is a simple but powerful concept in mathematics that states that if there are more objects than available spaces (or "pigeonholes"), then at least one space must contain more than one object. This principle has many applications in various fields, including computer science, economics, and even daily life.

One of the most common applications of the pigeonhole principle is in computer science, specifically in the field of data compression. In data compression, the goal is to represent a large amount of data using a smaller number of bits. One way to do this is by using a technique called "lossless compression," where the original data can be recovered exactly from the compressed version. The pigeonhole principle can be used to prove that certain lossless compression schemes are optimal, meaning that no other scheme can compress the data more efficiently. For example, suppose we have a set of data consisting of the letters A, B, C, and D. If we want to represent this data using only 2 bits per letter, we can use the pigeonhole principle to prove that at least one of the letters must be represented by two different combinations of 2 bits. This means that the data cannot be losslessly compressed using 2 bits per letter, and we must use a different method or a higher number of bits to achieve optimal compression.

Another application of the pigeonhole principle is in economics, specifically in the study of market equilibrium. Market equilibrium occurs when the quantity of a good or service that is being supplied is equal to the quantity that is being demanded. The pigeonhole principle can be used to prove that under certain conditions, market equilibrium is always possible. For example, suppose we have a market for a certain type of good, and there are three sellers who each have a certain number of units of the good to sell. The pigeonhole principle states that if the sellers have a total of more than three units of the good, then at least one of them must have more than one unit to sell. This means that there must be at least one buyer who is willing to purchase more than one unit of the good, which is necessary for the market to reach equilibrium.

In daily life, the pigeonhole principle can also be used to solve practical problems. For example, suppose you have a group of friends who are going on a road trip, and you need to decide which car to take. You have three cars to choose from, each with a different number of seats. The pigeonhole principle states that if you have more friends than the total number of seats in the three cars, then at least one of the cars must have more than one person in it. This can help you decide which car to take, and also serve as a reminder to carpool to save space and reduce environmental impact.

In conclusion, the pigeonhole principle is a simple but powerful concept that has many applications in various fields, including computer science, economics, and daily life. Its versatility and simplicity make it a valuable tool for solving a wide range of problems.

## Applications of Pigeonhole Principle

Otherwise, that means none of these three points are connected and hence they are mutual strangers. Then by the Pigeonhole Principle, at least one of these small squares so call pigeonholes should contain at least 3 points ie the pigeons. Every infinite decimal expansion has the following property. It is easy to see that for any two cards A and B either distance A,B or distance B,A must always be less than or equal to 6. Repeat this process, dividing that half in half and so on. Solution: Let the sequence be a 1, a 2, â€¦.

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## big list

Example 1 If you are required to pick five cards from a standard deck of 52 cards, at least two will be of the same suit. A pigeon is delivering 4 mails and has to place all its mails into the available pigeonholes. Can we provide an answer to the problem by applying the Pigeonhole Principle? So nice example, but not really practical I think. Next, we will like to proceed to a more creative aspect of the application of Pigeonhole Principle by showing how it can be used to design interesting games: 7. He just has to take 4 socks from the drawer! It will be worthwhile to note the above technique can be useful in analyzing accuracy of weapons in shooting practices and tests. READ ALSO: What is Illusory correlation in psychology: Definition and examples As reported by Legit.

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## 16 fun applications of the pigeonhole principle

In comparison there are millions of people in New York City. Or there will be degenerate cases of collinearity which can also be easily treated. Introduction of Pigeonhole Principle 2003 Singapore Maths Project Festival Senior Section Applications of the Pigeonhole Principle River Valley High School Team members: Edwin Kwek Swee Hee Huang Meiizhuo Koh Chan Swee Heng Wee KuanContents Page 1. The principle can also be used to prove that any lossless compression algorithm, provided it makes some inputs smaller as the name compression suggests , will also make some other inputs larger. As we all know, there are 12 months in a year, thus, even if the first 12 people have their birthday from the month of January to the month of December, the 13th person has to have his birthday in any of the month of January to December as well.

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## (PDF) Pigeonhole Principle: Real Life Applications and Mathematical Investigation

The participant does not show these cards to the magician, but does show them to the magicians accomplice. Otherwise, that means none of these three points are connected and hence they are mutual strangers. Application of pigeonhole principle in card games We like to introduce the application of pigeonhole principle in two exciting card tricks: a. Hence by the Pigeonhole Principle, two members of the sequence, say a and b, are associated with the same ordered pair s, t. In such questions, a given number of darts are thrown onto a dartboard, the general shape and size of which are known.

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## proofs

The magician immediately identifies the fifth hidden card. If GG is a complete graph KnKn, the numbers f Kn,H f Kn,H and rb Kn,H rb Kn,H are called anti-Ramsey numbers and rainbow numbers, respectively. Let how many balls each box contains. Lets assume that there is one person in the room that has no acquaintance at all, then the others in the room will have either 1, 2, 3, 4, , 48 acquaintance, or do not have acquaintance at all. Hope you'll find them useful: 1-Given five lattice points on the plane, we connect any two of them by drawing a line between them. We first note that by applying the Pigeonhole Principle, we can show that in any permutation of 10 distinct numbers there exists an increasing subsequence of at least 4 numbers or a decreasing subsequence of at least 4 numbers.

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## Discrete Mathematics

At first blush, it might seem like these questions require very different approaches to answer. This would be a set of three points without any edges. Case 1: everyone has at least one friend If everyone has at least one friend, then each person has between 1 to nÃ‚Â­-1 friends. The magician now knows that the suit of the mystery card is hearts. The accomplice's next task is thus to let the magician know that he must add the value 4 to the number 3 to obtain the final value of 7 for the hidden card! While this version sounds different, it is mathematically the same as the one stated with pigeons and pigeonholes. This means that two or more of the cards picked will belong to the same suit.

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## Art of Problem Solving

But we are picking randomly so we include after we apply pigeon principle. There are great applications of pigeonhole principle PHP in some olympiad problems and some theorems, both in finite and infinite structures. Granted, this is semantics, not mathematics. If some f i is 0, it means that the ith person does not have any friends in the group. Can we show that there will always be two darts which are at most 10 units apart? Consequently, two of the patrons on a sold out show must share the same first and last initials. There are five other points it could possibly connect to.

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## Mathematics

As you know, mathematics develops through a process of generalization. Here are 16 of my favorite applications, categorized by difficultly:. Actually, it's not obvious to me how you could even prove it using pigeonhole principle though I see how you could prove it by contradiction, misguided as it seems to me. Pigeonhole principle examples Here are some examples that will help you get a better understanding of the pigeon hole theory. The first card that the accomplice shows to the magician is one of these two cards.

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## Pigeonhole Principle (Defined w/ 11 Step

Of course, by the Pigeonhole Principle, there will be at least one pigeonhole with 2 or more pigeons! To prove the result, we may divide the square into 25 equal smaller squares of side units each. Combinatorial Card Trick 10 b. Now, given any two cards A and B, define distance A,B as the clockwise distance from A to B. In a packed Carnegie Hall performance, there will be two people who have the same first and last initials. Example 07 Show that in any group of 30 people, we can always find 5 people who were born on the same day of the week.

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