Learn more about the Pigeonhole Principle and other powerful techniques for combinatorics problems
in our Intermediate Counting
& Probability textbook by USA Math Olympiad winner (and MIT PhD) David Patrick.
Pigeonhole Principle
Pigeonhole Principle
The basic pigeonhole principle says that if there are 
holes, and 
pigons (k>1), then one hole MUST contain two or more pigeons. The extended version of the pigeonhole principle states that for n holes, and 
pigeons, j>1, some hole must contain k+1 pigeons. If you see a problem with the numbers n, and nk+1, think about pigeonhole.
Examples
Can users find some?
You could paste in these... (maybe, just a suggestion)
- Show that in any group of five people, there are two who have an identical number of friends within the group. (Mathematical Circles)
- Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. (Manhattan Mathematical Olympiad 2004)
- Prove that having 100 whole numbers one can choose 15 of them so that the difference of any two is divisible by 7. (Manhattan Mathematical Olympiad 2005)
- Prove that from any set of one hundred whole numbers, one can choose either one number which is divisible by 100, or several numbers whose sum is divisible by 100. (Manhattan Mathematical Olympiad 2003)
