Question

Concerning Application 4 attached below, my question is show that there is a succession of days during which the chess master will have played exactky k games, for each k=1,2,...,21. Is it possible to conclude that there is a succession of days which the chess master will have played exactly 22games?

Application 4. A chess master who has 11 weeks to prepare for a tournament decides to play at least one game every day but, to avoid tiring himself, he decides not to play more than 12 games during any calendar week. Show that there exists a succession of (consecutive) days during which the chess master will have played ezactly 21 games. Let a be the number of games played on the first day, a2 the total number of games played on the first and second days, as the total number of games played second, and third days, and so on. The sequence of numbers a1, 02,..,a7 is a strictly increasing sequence since at least one game is played each day. Moreover, a1 2 1, on the first, The argument actually contains a nice algorithm, whose validity relies on the pigeonhole principle, for finding the consecutive a's, which is more efficient than examining all sums of consecutive a's. Each term of the sequence is larger than the one that precedes it. 72 CHAPTER 3. THE PIGEONHOLE PRINCIPLE and since at most 12 games are played during any one week, a77 12 x 11 = 132.4 Hence, we have 1sa1< a2<- < a77 S 132. .. The sequence a1 +21, a2 +21,... ,ar7 +21 is also a strictly increasing sequence: 22 a1+21< az+21 << a77+21 132+21 = 153. Thus each of the 154 numbers a1,a2,.,a77, a1 +21, a2 +21, ... ,a7 +21 is an integer between 1 and 153. It follows that two of them are equal. Since no two of the numbers a1, a2,,a77 are equal and no two of the numbers a1 +21,a2 + a, +21. Therefore, 21,..,aT7+21 are equal, there must be an i and aj such that a = days j+1,j+2,..., i the chess master played a total of 21 games. on

Answer 1

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