# Thread: Discrete math set question someone help?? thank you

1. ## Discrete math set question someone help?? thank you

In a period of 4 weeks, Joe played tennis every day. He plays at least one set everyday, and played a total of 40 sets. (In each day the nubmer of sets Joe played is an integer.) Show that there is a consecutive span of days during which Joe played exactly 15 sets of tennis.

2. Originally Posted by peiyilee
In a period of 4 weeks, Joe played tennis every day. He plays at least one set everyday, and played a total of 40 sets. Show that there is a consecutive span of days during which Joe played exactly 15 sets of tennis.
Suppose that $S_k$ is the sum total of all sets completed at the end of the kth day.
Thus $1 \le S_1 < S_2 < \cdots < S_{27} < S_{28} = 40$. That is a set of 28 different sums.
Construct a new set of 28 different sums, $S_1 + 15 < S_2 + 15 < \cdots < S_{27} + 15 < S_{28} + 15 = 55$.
Now we have 56 representations of numbers 1 to 55.
Use the pigeonhole principle to conclude.