From Introductory Combinatorics by Richard Brualdi
We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. To make sure that he's able to take a couple of breaks, he also decides that he wont play more than 12 games per week.
My question is, how do you prove/dis-prove that there will be a succession of (consecutive) days in which he will play k games, $k \geq 22$?
Specifically, I don't quite understand the use of the pigeonhole principle here.
Thanks in advance! :D
I'll try to clarify what I'm asking:
The goal is to prove that for every possible practice schedule that the chess master makes, there will always be a consecutive sequence of days such that he plays exactly k games.
So for example, if k = 22, how do you prove that no matter what his schedule is like, he will always play exactly k games in r days, $1 \leq r\leq 77$.
My main question is: what is the general method for proving different values of k? Are there values of k (from 1 to 132 inclusive) that are impossible to obtain?
Note that he does not necessarily have to play 132 games within 77 days! Also, keep in mind the bounds of k.