Tutoring high school math, the pigeonhole principle might not come up very often.  Because of its applications to everyday life, the tutor often thinks of it.

Back in my May 23 post, I introduced the pigeonhole prinicple. I pointed out that, although the theory seems straightforward, it can yield some surprising results. Here is an example for you sports fans:

Example:

Let’s imagine a sports team’s season is 35 games. How many must they win to guarantee a five-game streak?

Solution: Divide the 35 games into seven sets of five:

_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|_ _ _ _ _|_ _ _ _ _

Now, imagine putting 4 Ws in each set of five games. Since there are seven sets, that makes 28 Ws. Notice that, by leaving an artful gap at the proper end of each set of games, you can place 4 Ws in each set of five so that you get no consecutive row of 5 Ws:

W W W W _|W W W W _|W W W W _|W W W W _|W W W W _|W W W W _|W W W W _ (for example)

However, if we need to insert one more W, we can’t escape getting a row of 5 consecutive Ws. Therefore, the solution is 28+1=29 wins. The team must win 29 games to guarantee a five-game winning streak.

Of course, most people might react that if you watch sports, you expect a team that wins even 25 of 35 games will likely have a 5 game streak. True, in all probability; probability, however, is a related but different branch of mathematics. (The problem above is a counting problem.) I’ll have more to say about topics from both those partitions of math in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.