A student has 37 days to prepare for an examination. From...

Name: Problem 11, Chapter 3: Introductory Combinatorics
Uploaded: 2020-02-29T21:39:17-08:00
Duration: 2 min 29 s
Channel: Tomokazu Switzer
Description: A student has 37 days to prepare for an examination. From past experience she knows that she will require no more than 60 hours of study. She also wishes to study at least 1 hour per day. Show that no matter how she schedules her study time (a whole number of hours per day, however), there is a succession of days during which she will have studied exactly 13 hours.

A student has 37 days to prepare for an examination. From past experience she knows that she will require no more than 60 hours of study. She also wishes to study at least 1 hour per day. Show that no matter how she schedules her study time (a whole number of hours per day, however), there is a succession of days during which she will have studied exactly 13 hours.

Key Concepts

Partial Sum Sequence

A partial sum sequence is created by taking the cumulative totals of a sequence of numbers—in this case, the hours studied per day. By summing up the hours day by day, each term in the sequence represents the total study time up to that point. Analyzing such sequences allows for comparison between different intervals to determine the study hours accumulated over consecutive days.

Difference of Partial Sums

The sum of a sequence over an interval between two days can be obtained by taking the difference between the cumulative sums at the end and beginning of the interval. This method is crucial because if two partial sums differ by a specified target amount, it confirms that the consecutive days between them add up exactly to that target.

Pigeonhole Principle

The pigeonhole principle is a combinatorial tool that asserts if more items are distributed than there are containers, at least one container must hold more than one item. In problems like this, it is used to show that within a bounded range, certain residues or differences will repeat, leading to the existence of a subsequence of days where the hours studied sum to the desired total.

Discrete Interval Analysis

Analyzing intervals in a discrete setting involves looking at sums over finite, integer-based intervals—like hours per day in this study schedule. Given the constraints on minimum and maximum possible values, this analysis guarantees that, despite the freedom in scheduling, some consecutive block of days will necessarily have a total study time that meets the specific requirement.

Transcript

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