
To show: The chess master will have played exactly

Answer to Problem 1E
It is
Explanation of Solution
Given:
The cumulative number of games played on the first n days is denoted by
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,
Theorem used:
If
Description:
From the given condition, the sequence
Thus, the sequence
Thus, each of the numbers
From the above theorem, any of the two values in
Observe that no two numbers
Thus, there must be i and j such that
The chess master can play k games in total on the days
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
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Chapter 3 Solutions
Introductory Combinatorics
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