Introduction

The Pigeon hole principle asserts: If we must put N+1 or more pigeons in N holes, then some pigeon hole must contain two or more pigeons. Although it might look nothing much than common sense, this principle is very useful in various types of problems. It allows us to sometimes draw quite unexpected conclusions in situations, when it even seems that we do not seem to have enough information.

Problem 1. A bag contains beads of two colors: black and white. What is the smallest number of beads which must be drawn from the bag, without looking, so that among these beads there are two of the same color.

Solution. We can draw 3 beads from the bag. If there are no more than one of each color among these, then there would be no more than 2 beads altogether. This, obviously, contradicts to the fact that we have chosen 3 beads.On the other hand, it is clear that choosing 2 beads is not enough, since they may happen to be of the same color. In this problem the beads are the pigeons, and the colors play the role of pigeon holes.

Methodological Remark. Notice, that in this problem, has in itself certain vagueness, which is, in a sense, characteristic for all problems related to the pigeon hole principle. This comes from the fact that the very principle is an assertion of existence without having an explicit way to single out the solution. At the same time, the solutions of these problems include some of the rigidity of an axiomatic proof. Students have often problems with this. They should first solve a couple of simple problems as the one before, tracking carefully how we go from what is given to what is needed to prove.

Problem 2. Given 12 integers, show that two of them can be chosen whose difference is divisible by 11.

Hint. Look at the twelve integers as pigeons and let the eleven possible remainders when dividing by 11 to be the holes.

Problem 3. The city of Sofia has 1,300,000 citizens. Show that two of these must have the same number of hairs on their heads, it it is known that no person has more than one million hairs on his or her head.