Question

A student has 37 days to prepare for an exam. From past experience, he knows that he will need no more than 60 hours of study, To keep from forgetting the material, he wants to study for at least one hour each day. Show that there is a sequence of successive days during which he will have studied exactly 13 hours.

Short answer

Yes, it's possible due to the adjusted cumulative sequence and the pigeonhole principle.

Step-by-step solution

Define the Total Study Hours Function

Imagine each day the student accumulates study hours, creating a sequence of cumulative study hours. Let's define a function \( S(n) \) that represents the total hours studied from day 1 to day \( n \). So, \( S(n) = \sum_{i=1}^{n} \text{hours studied on day } i \).

Establish Boundaries for Total Hours

We know from the exercise that the student will study a total of 60 hours over 37 days. Each day, the student studies at least 1 hour, so by day 37: \( S(37) = 60 \), and \( S(n) \geq n \) for any day \( n \).

Create Adjusted Sequence for Analysis

Consider the sequence of adjusted cumulative study hours: \( S(n) - n \). This represents the total study hours minus the minimum hours the student intends to study by each respective day.

Analyze the Sequence Using the Pigeonhole Principle

We wish to find two days such that the difference in their adjusted cumulative hours is exactly 13. Consider \( S(n) - n \) for \( n = 1 \) to 37, resulting in the sequence \( \{S(1) - 1, S(2) - 2, \ldots, S(37) - 37\} \). The values range from 0 to 59 since \( S(37) = 60 \).

Find Two Equal Values in Adjusted Sequence

By the pigeonhole principle, with 37 adjusted numbers, ranging from 0 to 36 (since \( S(n) - n \geq 0 \)), at least one pair of these numbers must be the same. Let's denote these days as \( i \) and \( j \) where \( 1 \leq i < j \leq 37 \) such that \( S(i) - i = S(j) - j \).

Calculate the Difference in Study Hours

This means \( S(j) - S(i) = j - i \). Therefore, the actual study hours between day \( i \) and day \( j \) (inclusive) is exactly \( j - i \). We need it to be 13, so set \( j - i = 13 \).

Conclusion

The setup ensures there exists a sequence of 13 consecutive days in which the student will study exactly 13 hours because of the cumulative distribution and the pigeonhole principle.

Key concepts

Cumulative Distribution

Cumulative distribution is a concept often used to describe a sequence where each element signifies the total amount achieved up to a certain point. In the context of our exercise, this refers to the cumulative total of study hours over days.
We define a cumulative total function, represented as \( S(n) \), to denote the sum of study hours from day 1 to day \( n \). Thus, \( S(n) = \sum_{i=1}^{n} \text{hours studied on day } i \). This helps quantify progress over time and is crucial to understanding how study goals are distributed across several days.

• The cumulative distribution function gives a clear picture of the student's study timeline.
• It helps in identifying specific durations where certain study hour goals are met or need adjustment.

Visualizing study time this way allows us to pinpoint contiguous sub-segments that add up to a particular total, like the 13 exact hours needed in the problem.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values. Unlike continuous mathematics, which involves smooth curves and ranges, discrete mathematics concerns countable items.
In this exercise, the study hours each day are countable objects, and their cumulative study hours form a sequence that exemplifies discrete mathematics.

• This discrete approach allows us to use tools like functions and sequences to better analyze and find solutions to problems.
• The whole set of possible study times across different days is a collection of discrete values that are easy to calculate and compare.

The pigeonhole principle, an important element in discrete mathematics, helps us determine when conditions will yield a desired configuration, like finding two days where the adjusted cumulative study hours are the same.

Sequence Analysis

Sequence analysis involves breaking down a series of elements to find patterns or conclusions based on their properties. In the student's exercise, sequence analysis comes into play with the adjusted sequence \( S(n) - n \).
This sequence subtracts the minimum required hours from the cumulative study hours, providing an insightful tool for understanding variations in study habits.

• We create and observe sequences to reveal hidden relationships and patterns.
• By setting up the sequence \( \{S(1) - 1, S(2) - 2, \ldots, S(37) - 37\} \), we focus on the fluctuation in study hours relative to imposed minimums.

Analyzing this adjusted sequence with the pigeonhole principle allows us to find exactly 13 consecutive study hours—a clearest example of applying sequence analysis to solve a practical problem. This type of analysis breaks a larger problem into simpler, approachable parts, making it easier to understand and solve.