Question

A widget-maker makes at least one widget every day but not more than 730 widgets in a year. Given any \(n\), show that the widget-maker makes exactly \(n\) widgets in some set of consecutive days. For some \(n\), it may take more than a single year.

Short answer

A widget-maker can achieve any integer number of widgets using consecutive day sums, including numbers more than 730 by extending beyond a year.

Step-by-step solution

Calculating Total Possible Widgets

First, calculate the total possible number of widgets that the widget-maker can produce in a year. The widget-maker makes at least 1 widget per day and at most 730 in the entire year. Hence, the minimum and maximum number of widgets are 1 and 730, respectively.

Analyze Consecutive Sum Possibilities

The task is to show that, for any number of widgets less than or equal to 730, there exists a set of consecutive days in which the exact number of widgets is made. We will use the idea of differences between cumulative sums.

Understanding Cumulative Sums

Define the cumulative sum of widgets made up until day \(i\) as \(S_i\), where \(S_0 = 0\) and \(S_i = \) the total number of widgets made by the end of day \(i\). Then, \(S_1, S_2, \ldots, S_{365}\) are the cumulative sums for each day in the year.

Cumulative Difference and Consecutive Sums

For any set of consecutive days starting at day \(i\) and ending at day \(j\), the total number of widgets is \(S_j - S_{i-1}\). To find a sum of exactly \(n\) widgets, we need \(S_j - S_{i-1} = n\).

Finding the Possible Differences

Since the number of distinct sums produced by daily widgets over a year range from 1 to 730 (inclusive), and each returning to zero upon new cycles (if needed), \(S_i\) will cover every possible integer from 0 to 730, allowing for any difference thereby containing every integer \(n\) from 1 to 730 in some interval.

Addressing Possibility Beyond a Year

If an \(n\) greater than 730 is desired, the same concept holds by repeating the annual cycle. Therefore, it is always possible to achieve any \(n\) by extending the range of days to include multiple cycles, ensuring that all integers can be accounted for.

Key concepts

Cumulative Sum

The concept of a cumulative sum is essential in understanding how we can calculate the total widgets produced over a series of days. Think of it as keeping a running total of widgets. Every day, the number of widgets made is added to the sum from previous days. Suppose, on the first day, you make 3 widgets—your cumulative sum is 3. If, on the next day, you make 4 more, your cumulative sum will now be 7 (3 from the first day plus 4 from the second day).

• Let’s define this mathematically: For each day \(i\), the cumulative sum \(S_i\) is given by \(S_i = \) total widgets up to that day.
• The first day is special: \(S_0 = 0\) because no widgets have been produced before the start.
• Cumulative calculation keeps building as days proceed: \(S_1, S_2, \ldots, S_{365}\).

Therefore, the cumulative sum tells us not just how many widgets are made up to any given point, but also helps in finding out how many were made over selective periods by subtracting sums from different days.

Consecutive Days

Understanding the concept of consecutive days is crucial when calculating the widgets produced over a continuous period. Consecutive days mean days that come one after another without any gaps. If we need to show that exactly some number of widgets ( ) are produced in a set of consecutive days, we should focus on days that closely follow each other.

• For instance, if you want to know how many widgets are produced between the 5th day and the 10th day, this entire stretch is a consecutive sequence.
• To find the total for these days, you would calculate the cumulative sum up to day 10 and subtract the cumulative sum up to day 4.

Using this method allows you to capture production sums over any contiguous sequence of days, giving a clear picture of output for any time period you choose.

Widgets

Widgets are essentially the product being manufactured each day. Whether it’s machines, gadgets, or other items produced in a factory, in this context, widgets symbolize daily outputs.

• Each day, there is a minimum of one widget produced, implying consistent production activity.
• The upper limit ensures no more than 730 are produced within a year, which means each day contributes slightly towards this yearly cap.
• When watching how widgets accumulate, daily productions add up through cumulative sums, as previously mentioned.

Widgets demonstrate not just daily productivity but also help in understanding and calculating outputs over fixed periods, assisting in planning and forecasting.

Difference of Sums

The difference of sums concept is a powerful tool for determining the exact number of widgets produced over a segment of days. This involves subtracting cumulative sums from one another.

• If you want to find how many widgets were made from day \(i\) to day \(j\), calculate \(S_j\) and \(S_{i-1}\) (where \(S_j\) is the sum up to day \(j\) and \(S_{i-1}\) is the sum up to the day before \(i\)).
• The formula \(S_j - S_{i-1} = n\) tells exactly how many widgets were created over that specific period.
• This method ensures accuracy by allowing any segment of time to be analyzed for widget production.

This approach also provides for clarity in assessing any excess or shortfalls in production targets within specific consecutive day spans.