PIGEONHOLE PRINCIPLE. Student redefine this as common sense behind this basic idea of this mathematical basic principle; if there are n objects to be situated in m receptacles (with m < n), at least two of the things must go in to the same field. Whereas the idea is commonsensical, in the hands of an able mathematician it can be designed to do extraordinary things. There is one of the very most famous applications of Pigeonhole Theory which there's at least two people in New York City with the same amount of hairs on their head.

The concept itself is attributed to Dirichlet in 1834, although he in fact used the word Schubfachprinzip. The exact same maxim is often known as in honour of Dirichlet who used it in resolving Pell's equation. The pigeon seems to be a fresh addition, as Jeff Miller's website on the first use of some math words provides,

"Pigeon-hole principle occurs in English in Paul Erd¶s and R. Rado, A partition calculus in collection theory, Bull. Am. Math. Soc. 62 (Sept. 1956)".

In a recent debate on a brief history group Julio Cabillon added that there are a number of names in various countries for the theory. His list contained,

Le principe des tiroirs de Dirichlet, France for the basic principle of the drawers of Dirichlet

Principio da casa dos pombos in Portuguese for the home of pigeons principle

Das gavetas de Dirichlet for the drawers of Dirichlet.

Dirichlet's principle

The Package principle

Zasada szufladkowa Dirichleta which imply the concept of the drawers of Dirichlet in Polish

Schubfach Prinzip which signify drawer theory in German

## INTRODUCTION

Let's get this to thing easier by imagine some typically common daily awkward point in time which related to Pigeonhole Basic principle. Sometimes, I wake up and get ready for classes early in the morning. But then, the room still dark and my room-mate still in sleep. Let see, I've socks of three different colors in my drawer and to be within sloppy order. So, how do i pick a matching pair of same coloured socks in most convenient way without disturbing my associates (which indicate turning on the light)? A simple math will defeat this problem. I just have to get only 4 socks from the drawer! Naturally it is the Pigeonhole Concept applied in the real life.

So, what's Pigeonhole Concept then? Let put a good example to demonstrate this principle. For example, there are 3 pigeonholes around. You can find 4 pigeon and each of them holds one mail. The pigeons are providing the mails and have to place most of its mails into available pigeonholes. With only 3 pigeonholes around, there clear to be 1 pigeonhole with at least 2 mails!

Thus, the overall rule states when there are k pigeonholes and there are k+1 mail, they will be 1 pigeonhole with at least 2 mails. A more complex version of the theory would be the following:

If mn + 1 pigeons are positioned in n pigeonholes, then you will see at least one pigeonhole with m + 1 or even more pigeons in it. However, this Pigeonhole Basic principle tells us little or nothing about how to locate the pigeonhole that contains two or more pigeons. It only asserts the presence of a pigeonhole formulated with two or more pigeons.

The Pigeonhole Principle seems trifling but its uses are deceiving astonishing! Thus, in our project, we plan to learn and find out more about the Pigeonhole Principle and illustrate its numerous interesting applications inside our daily life.

## RESULTS OF RESEARCH AND REAL WORLD EXAMPLES

## CASE 1 : LOSSLESS DATA COMPRESSION

Lossless data compression algorithms cannot assure compression for everyone input data sets. Frankly says, for any (lossless) data compression algorithm, you will see an suggestions data established that didn't get reduced in size when prepared by the algorithm. This is naturally proven with elementary arithmetic using a counting argument, as follows:

Assume each particular data file is symbolized as a string of pieces (in count up of arbitrary size)

We inference that there surely is a compression algorithm that changes everything of the file into another type of file that your size is reduced than the initial file, and this regardless one document will be compressed into something that is shorter than itself.

Let M be the least number in a way that there is a file F with size M parts that compresses to something shorter. Let N be the length (in bits) of the compressed version of F.

F = Document with duration M

M = Least number that compressed into something shorter

N = length (in bits) in compressed version of F

Since N < M, each record of size N helps to keep its size throughout the compression. There are 2N such data. Mutually with F, this makes 2N + 1 files which all compress into one of the 2N data files of length N.

2N < 2N + 1

But 2N is smaller than 2N + 1, consequently from the pigeonhole concept there must be some document of period N which is at the same time, the productivity of the compression function on two different inputs. That data file cannot be decompressed dependably (which of the two originals suppose to be yield?), which contradicts the assumption that the algorithm was lossless.

Hence, we can finalize our original hypothesis (that the compression function makes no data file much longer) is actually fallacious.

For any lossless compression algorithm that changes some files shorter, must automatically make some data files longer, but it is not necessary that those data files become very much much longer. Most functional compression algorithms provide an "escape" service that can turn off the normal coding for documents that would become longer by being encoded. Then the only increase in size is a few parts to let know the decoder that the normal coding has been switched off for your source. In example, for each 65, 535 bytes of input, DEFLATE compressed documents never need expansion by more than 5 bytes.

In reality, for just about any lossless compression that reduces how big is some record, the expected length of a compressed data file (averaged over all possible files of period N) must actually be higher than N if we consider data files of span N, if all data were equally apparent. So if we don't have any idea about the properties of the info we are considering for a compressing, we probably not compress the document at all. A lossless compression algorithm is merely come in useful when we are choose to compress a specific types of data than others; after that the algorithm could be designed to compress those types of data in a far greater way.

Whenever opting for an algorithm always means implicitly to select a subset of most files that can be usefully shorter. This is the theoretical reason we assume to consider different kind of compression algorithms for different types of files: there are almost impossible for an algorithm that ideal for all types of data. Algorithms are usually quite specifically tuned to a specific type of file such like this example; lossless audio compression programs do not work very well on text data, and vice versa.

Above all, documents of random data cannot be regularly compressed by any likely lossless data compression algorithm: undeniably, this effect can be used to determine the idea of randomness in algorithmic complexity theory.

## CASE 2 : DARTBOARD

Another kind of problem demanding the pigeonhole principle to resolve is those which require the dartboard. In such questions, the overall shape and size of Dartboard that are known, confirmed variety of darts are tossed onto it. Then we determine the distance between two convinced darts is. The hardest part is to establish and identify its pigeons and pigeonholes.

## EXAMPLE 1

On a round dartboard of radius 10 units, seven darts are thrown. Can we verify that there will be two darts which are at most 10 units apart?

To demonstrate that the ultimate proclamation will usually true, we first have to separate the group into six comparable areas as shown;

Therefore, we allowing each one of the sectors to be always a pigeonhole and each dart to be a pigeon, we have seven pigeons to be transferred into six pigeonholes. By pigeonhole rule, there will be at least one sector made up of a minimum number of two darts. The assertion is shown to be true in any case since the most significant distance relating two points lying in a sector would be 10 products.

In actual fact, it is also possible to establish the circumstance with only six darts. In such a case, the circle this time is redefined into five divided industries and all else follows. But, put attention that is not necessarily true to any further level if we use five darts or less.

## EXAMPLE 2

On a dartboard which is produced as a regular hexagon of area length 1 product, nineteen darts are then thrown. How would we confirm that you will see two darts within products each other?

All once more, we must identify our pigeonholes by dividing the hexagon into six equilateral triangles as illustrated below.

While the 19 darts as pigeons and with the six triangles as the pigeonholes, we uncover that there has to be in any case one triangle with a minimum of 4 darts in it.

Now, considering another scenario, we will have to endeavour an equilateral triangle of aspect 1 device within 4 details inside.

If locate all the points as far aside from each other as is feasible, we should come to conclusion of conveying each of the first three things to be at the vertices of the triangle. The fourth or the previous point will then be exactly at the centre of the triangle. Since we realize that the length from the centre of the triangle to each vertex is of the altitude because of this triangle, that is, systems, we will get that it is unquestionable potential to find two darts that happen to be units aside within the equilateral triangle.

## CONCLUSIONS

In conclusion, however the Pigeonhole Principle appears to be simple, but, this matter is very useful in helping you to definitely devise and smooth the progress of computation and showing steps for various important numerical problems. This rule is very helpful inside our life though it seem so simple. This Basic principle also can be applied in our daily life, whether we realizes it or not. It really is fun when the condition can be resolved in a manner that we know, employing this principle.

## RECOMMENDATIONS

We wish to provide you some recommendation on making the Pigeonhole Rule far more interesting like:

Using variety of leaning materials and variety of examples to help scholar to get more understand the Pigeonhole Theory.

Create a proper imagination of what exactly are the real reasons for having the Pigeonhole Process.

Search more information from the web about the Pigeonhole Theory.

Make a lot of exercise that is related about the Concept.

Make an organization discussion and talked about about the topic.