Examples of the Pigeonhole Principle

From the Wikipedia article:

• You have a drawer with 10 black socks and 12 blue socks. Without looking in the drawer, how many socks do you need to pull out to get at least one pair of socks the same color?
• Show that there must be at least two people in London with the same number of hairs on their head.
• If there are N people (N >= 2) who can arbitrarily shake hands with one another, show that there are at least two people who shake the same number of hands.
• Show that there cannot be a lossless compression algorithm that will compress any file to a strictly smaller size.

Ties into probability theory:

• If 2 pigeons are randomly assigned to 4 pigeonholes, there is a 25% chance that at least one pigeonhole will hold more than one pigeon. For 5 pigeons and 10 holes, it's 69.76%. For 10 pigeons and 20 holes, it's 93.45%.
  
• Given a group of 23 randomly chosen people, there is more than a 50% chance that at least two of them will have the same birthday. For 60 or more people, the probability is greater than 99%. (Known as the Birthday Paradox)

Puzzles whose solutions use the pigeonhole pricniple:

• You have a chessboard with two diagonally opposite corners removed. Is it possible to cover the remaining board with domino pieces that are each two board squares in size?
• There are 500 boxes with apples, each containing no more than x apples. Find the maximal possible value of x , such that one can for sure find 3 boxes containing equal number of apples.
  
• Several math proofs using the pigeonhole principle.
• And some other interesting math problems on the PHP.