A chessmaster who has 11 weeks to prepare for a tournament decides to play at least one game every day, but in order not to tire himself, he agrees to play not more than twelve games during any one week (consider "week" as periods of 7 consecutive days, starting from the first day of preparation; that is: if the first day is Sunday, each one of the eleven weeks ends on each one of the next eleven Saturdays.)

Prove that there exists a succession of days during which the master will have played exactly twenty games.

(In reply to Trivial solution?? by ed bottemiller)

The idea of the puzzle is that no matter how the chessmaster allocates from 7 to 12 games in various predefined weeks, the chessmaster cannot avoid having a succession of some number of consecutive days in which exactly 20 games have been played.

Think of the chessmaster trying to avoid such a situation of 20 games in an integral number of consecutive days. The idea is to prove that he can't succeed in such avoidance.

Posted by Charlie on 2008-10-02 12:53:48