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.

If one takes the interpretation I just suggested (for want of a clear formulation), there would be a trivial solution: CM plays one game each day.  Then for a "succession" of 20 days, for each subset of 20 consecutive days (from the 77 total) -- i.e. for days 1 to 20 thru days 58 to 77 -- he will play "exactly twenty games."  The proposer does not make clear what is entailed by CM agreeing to play any number of games per week from seven to twelve.  Any suggestions?

 

Posted by ed bottemiller on 2008-10-02 12:27:23