Pigeonhole principle

Simply put, the principle states that if you have N pigeons and M pigeon holes, where N is greater than M, at least one of the holes will have more than one pigeons in it. For example, 10 pigeons and 9 holes - one hole must have 2 pigeons. Which is obvious, right? There more...

There more...

Using the pigeonhole principle, we can show that if you have a party with at least two people, and everyone attending knows at least one person, then there will be two people who know the same number of people at the party

Or, we can show that there are two (non-bald) people in London with the same number of hairs on their head.

Or if you draw 5 dots on the surface of a ball, 4 dots will lie on the same half of the ball.

Or, in a group of six people, there will be either 3 mutual friends or 3 mutual strangers

And, if a lossless compression algorithm makes any file smaller, it must make at least one file larger.

Finally, a handy one as the winter months draw in - if you have N different colours of socks in a draw, picking out N+1 socks will guarantee you a pair.