The Pigeonhole Principle is one of the simplest principles of mathematics. However, its clever application has led to some extraordinarily powerful results. In its simplest form the pigeonhole principle states that placing N+1 objects into N slots requires that at least one slot contain more than one object. An expanded version of the pigeonhole principle states that placing N objects into K slots requires that at least one slot contains at least éN/Kù objects.

Example: Show that at least two people in the City of Charlottesville or Albemarle County (who are not bald) have the same number of hairs on their head.

We guestimate that the number of hairs on a person's head does not exceed 50,000. Charlottesville and Albemarle Co. combined have in excess of 60,000 (nonbald) persons. Therefore, by the Pigeonhole Principle, at least two persons have the same number of hairs on their head.

Numerous clever (and more useful) examples are given in the text.