Chapter 3, Problem 1E

To determine

To show: The chess master will have played exactly $k\left(=1,2,\dots ,21\right)$ games in the succession of days and find the possibility of playing 22 games.

Answer to Problem 1E

It is $\underset{\_}{\text{not possible}}$ to play exactly 22 games on the successive days

Explanation of Solution

Given:

The cumulative number of games played on the first n days is denoted by $a_n$, where $n=1,2,\dots ,77$.

The chess master must play at least one game per day, but not exceeding 12 games per week.

The maximum number of games the chess mater can play is 132 and thus, $1\le a_1\le a_2\le \cdots \le a_{77}\le 132$.

Theorem used:

If $n+1$ objects are distributed into n boxes, then at least one box contains two or more of the objects.

Description:

From the given condition, the sequence $1\le a_1\le a_2\le \cdots \le a_{77}\le 132$ is increasing, since at least one game is played each day.

Thus, the sequence $1+k\le a_1+k\le a_2+k\le \cdots \le a_{77}+k\le 132+k$ is also an increasing sequence, which has $133+k$ elements.

Thus, each of the numbers $a_1,a_2,\cdots ,a_{77},a_1+k,a_2+k,\cdots ,a_{77}+k$ must be equal to the integer between 1 to $132+k$.

From the above theorem, any of the two values in $a_1,a_2,\cdots ,a_{77},a_1+k,a_2+k,\cdots ,a_{77}+k$ must be same.

Observe that no two numbers $a_1,a_2,\cdots ,a_{77}$ are equal and also, no two numbers $a_1+k,a_2+k,\cdots ,a_{77}+k$ are equal.

Thus, there must be i and j such that $a_i=a_j+k$.

The chess master can play k games in total on the days $j+1,j+2,\dots ,i$.

Hence, the required result is proved.

Moreover, if the chess master plays 22 games on the succession of days, then the total number of games played by the master will exceed 132.

Therefore, it is $\underset{\_}{\text{not possible}}$ to play exactly 22 games on the successive days.