ABSTRACT
The Pigeonhole Principle states the obvious: if there are more pigeons than holes, and every pigeon is in a hole, then some hole contains more than one pigeon. Yet, this simple idea proves to be amazingly powerful. In this chapter, 11 puzzles and a theorem are employed to demonstrate this power. In one puzzle, the principle is used to show that in any ten numbers chosen from 1 up to 100, there are disjoint non-empty sets with the same sum. In another, that 42 positive integers in a row can always be grouped for multiplication and addition in such a way that they become an exact multiple of a million. In yet another, that if 26 baseball players line up alphabetically, that there must be six that are also in height order, either increasing or decreasing.
Finally, the theorem uses the Pigeonhole Principle to show that the fractional parts of multiples of any irrational number are dense in the unit interval.
