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Chapter 5: Problem 14

During the first six weeks of his senior year in college, Brace sends out at least one resum茅 each day but no more than 60 resum茅s in total. Show that there is a period of consecutive days during which he sends out exactly 23 resum茅s.

Short Answer

Expert verified
Applying the Pigeonhole Principle, we can prove that there exists a period of consecutive days during which Brace sends out exactly 23 resumes.

Step by step solution

01

Defining Variables and Time Frame

Let's define the number of resumes sent by the nth day to be \( R_n \). So if Brace sends out one resume a day for six weeks (42 days), we know that he has 43 numbers \( R_n \) in total, starting from \( R_0 = 0 \) (0 resumes on the 0th day: the day before he started sending) to \( R_{42} = 60 \) (60 resumes by the 42nd day).
02

Applying the Pigeonhole Principle

Let's create some new terms, \( S_n \), where \( S_n = R_n - 23 \). The values of \( S_n \) will range from -23 (at \( S_0 = R_0 - 23 \)) to 37 (at \( S_{42} = R_{42} - 23 \)). We now have a total of 44 \( S_n \) numbers, and they range from -23 to 37. Here, the range gives us 61 possible 'holes', and we have 44 'pigeons'.
03

Finding the Specific Sequence

Per the Pigeonhole Principle, since we have more 'pigeons' (44) than 'holes' (61), at least two of the \( S_n \) must be the same. Therefore, if we suppose \( S_i = S_j \) and \( j > i \), it means that there is a sequence of consecutive days from \( i+1 \) to \( j \) where exactly 23 resumes are sent (since \( R_j - R_i = S_j - S_i = 23 \)). This fulfills what we were asked to prove.

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