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.

This seems to me a rather imprecise formulation of a problem. The CM must play from 7 to 12 games in each "week".  It is not clear whether every seven-day stretch is a "week" in this sense, or only each seven-days (e.g. Monday thru Sunday) beginning with the same day of the week.

Suppose he starts on a Monday, and plays two games each weekday (Monday-Friday) and one game a day each weekend. In the first week, and each following week, he will have played 12 games.  In the first 11 days he will have played 20 games (isn't that "a succession of days"??).  This doesn't make much sense -- so what is required?

Posted by ed bottemiller on 2008-10-01 20:40:47