# The pigeon hole principle

## The pigeon hole principle

## Pigeonhole Principle

- The
**Pigeonhole Principle**is a simple yet powerful mathematical concept. It states if you have more items than containers to put them in, at least one container must hold more than one item. - It is a form of
**combinatorial logic**, used to draw certain conclusions from given information. - It is particularly useful in proving the existence of certain outcomes without identifying them specifically.

## Explicit and General Form

- The
**explicit form**states that if`n`

items are put into`m`

containers, and`n > m`

, then at least one container must contain more than one item. - The
**general form**extends this by stating that if`n`

items are distributed in`m`

containers, then at least one container must hold at least`⌈n/m⌉`

items. Here`⌈n/m⌉`

denotes the ceiling function, which rounds a number up to the next largest integer.

## Application

- The principle is used in different branches like discrete mathematics, coding theory, and somewhat unexpectedly, in some areas of computer science – proving its versatility.
- It can help to prove broad statements about the distribution of sets of numbers, useful in certain algebra and number theory problems.
- It doesn’t require a deep understanding of complex mathematical concepts, but often requires
**creative thinking**to see how it can be applied.

## Examples

- A simple realization of the Pigeonhole Principle is the statement: “In any group of six people, at least two must share the same birthday month.”
- Another common example deals with spatial organization. If five socks are chosen at random from a drawer containing just ten socks (five pairs), there must be at least one pair among the chosen socks.

## Limitations

- Remember that while the Pigeonhole Principle proves the existence of certain outcomes, it doesn’t usually provide a method for finding the outcomes.
- Like any mathematical theorem, it applies only under the conditions that its assumptions hold – in this case, that the items and containers are distributed in a certain way.

## Learning Tip

- Practice Pigeonhole Principle problems to become familiar with the way it’s used. Start with simple examples and gradually move on to more complex scenarios.
- Always look for a way to apply the Pigeonhole Principle when presented with problems that involve distributing items across containers in some way.