POW #1: A Timely Phone Tree
The “A Timely Phone Tree” problem covered concepts involving identifying patterns, mathematical visualization and analysis, data collection, and mathematical trends. From implementing charts to putting numbers together, I had to effectively go about finding an answer given the sequences involved with the problem description. Not only was it crucial for me to stay organized with my work and process, but I needed to ensure that each step I was taking was for a reason, and that I was being as efficient as possible with my methodology. It took quite a bit of time for me to sift through the potential ways in which I could solve the problem, but once I stuck to an approach, it was smooth sailing to the end.
Part 1: Problem Statement
In this problem, I was introduced to Leigh, whose parents were concerned with her phone usage. They limited her to making calls between 8:00 p.m. and 9:00 p.m., but Leigh wanted to find a way in which she could still spread the latest news to all of her friends. It was decided that each phone call that she or her friends made during the time period allotted would only last three minutes. When 8:00 came around, Leigh first called Mike for three minutes, and at 8:03, Leigh called Diane while Mike called Anna May. Each of those four friends would then call new friends within the next three-minute time period, and the pattern would continue until the clock struck 9. The objective was to find how many of Leigh’s friends would hear the news before an hour of calling had passed. This would also be achieved under the assumption that it always takes three minutes to complete the calling process, no one would call someone who had already been called, and no caller would receive a busy signal.
Part 2: Process (I)
To start off, I analyzed the information in order to come up with a single approach that would take me to the end in the quickest manner possible. I picked up on the fact that each call took three minutes and the chain of callers who would hear the news would spread like wildfire. Therefore, I created a chart with the time in one column, and the people who had heard the news on the other. With a simple list format instead of just a number to convey the value for the right column, I could more easily keep track of my progress. I included Leigh in the chart for the time being in order to keep track of the pattern, and planned on eliminating her at the end when I had my total. For each caller after Leigh, Mike, Diane, and Anna May, I devised mock names to differentiate the callers.
In this problem, I was introduced to Leigh, whose parents were concerned with her phone usage. They limited her to making calls between 8:00 p.m. and 9:00 p.m., but Leigh wanted to find a way in which she could still spread the latest news to all of her friends. It was decided that each phone call that she or her friends made during the time period allotted would only last three minutes. When 8:00 came around, Leigh first called Mike for three minutes, and at 8:03, Leigh called Diane while Mike called Anna May. Each of those four friends would then call new friends within the next three-minute time period, and the pattern would continue until the clock struck 9. The objective was to find how many of Leigh’s friends would hear the news before an hour of calling had passed. This would also be achieved under the assumption that it always takes three minutes to complete the calling process, no one would call someone who had already been called, and no caller would receive a busy signal.
Part 2: Process (I)
To start off, I analyzed the information in order to come up with a single approach that would take me to the end in the quickest manner possible. I picked up on the fact that each call took three minutes and the chain of callers who would hear the news would spread like wildfire. Therefore, I created a chart with the time in one column, and the people who had heard the news on the other. With a simple list format instead of just a number to convey the value for the right column, I could more easily keep track of my progress. I included Leigh in the chart for the time being in order to keep track of the pattern, and planned on eliminating her at the end when I had my total. For each caller after Leigh, Mike, Diane, and Anna May, I devised mock names to differentiate the callers.
It was at this point that I realized that I could not continue this table when considering how much the number of friends called increased after every three-minute period. As a result, I switched over to the method of using numbers to represent the number of people called as opposed to the actual names. This would prove to be easier to manage, and I was happy that I made the transition.
I had completed my table, and my answer was 1,048,575. (I had to subtract Leigh, since the question asked for the number of her friends that would hear the news).
While I had come to a conclusion and the problem was all but wrapped up, I still felt unsatisfied. There was no way for me to translate the data to equation form, and I wanted to ensure that each scenario could be applied in the same way. Because of this, I edited the leftmost column of my table to make room for a variable t, which would represent time. I designated “Friends Contacted” as the y variable in order to formulate my equation:
While I had come to a conclusion and the problem was all but wrapped up, I still felt unsatisfied. There was no way for me to translate the data to equation form, and I wanted to ensure that each scenario could be applied in the same way. Because of this, I edited the leftmost column of my table to make room for a variable t, which would represent time. I designated “Friends Contacted” as the y variable in order to formulate my equation:
After about a half hour of scanning the values in the table and each number’s correlation to the respective time, I was able to develop an equation. I noticed that the number of people that had heard the news was equal to two to the power of the time divided by three.
This came out to be:
*Equation: y = 2t/3
*The only time value in which this would not apply to would be 60, since one would need to subtract Leigh from the output, making it 1,048,575.
At this point, I was confident with both my solution and equation, and could consider the problem as completed. The graph for this problem is displayed below
(where t = x):
This came out to be:
*Equation: y = 2t/3
*The only time value in which this would not apply to would be 60, since one would need to subtract Leigh from the output, making it 1,048,575.
At this point, I was confident with both my solution and equation, and could consider the problem as completed. The graph for this problem is displayed below
(where t = x):
Part 2: Process (II)
The Honors requirement for this problem was similar in style to the initial exercise, with the exception of the limitations and requirements for how many calls each friends could make. Per the example provided, Leigh would call Mike from 8:00 p.m. until 8:03 p.m., and then both friends would make calls to Diane and Anna May from 8:03 p.m. to 8:06 p.m.. However, Leigh would be finished at 8:06 after making a maximum of two calls, while the others could call until they reached two conversations over the phone. The goal was to find out how many people would hear the news by 9:00 p.m. using the new restrictions.
To go about solving this problem, I thought it would be best if I tried my initial approach again in order to keep track of the people. The formula would be a little bit harder to figure out, so I thought that I could tackle that portion of the problem once I had recorded the data for the hour. Again, I created fake names for the friends that were not mentioned at the outset. I also changed the rightmost column to read “Friends Calling” so that I could monitor active callers. In addition, I added a 1 or a 2 next to each name, depending on which call they were on before they could not make any more calls.
The Honors requirement for this problem was similar in style to the initial exercise, with the exception of the limitations and requirements for how many calls each friends could make. Per the example provided, Leigh would call Mike from 8:00 p.m. until 8:03 p.m., and then both friends would make calls to Diane and Anna May from 8:03 p.m. to 8:06 p.m.. However, Leigh would be finished at 8:06 after making a maximum of two calls, while the others could call until they reached two conversations over the phone. The goal was to find out how many people would hear the news by 9:00 p.m. using the new restrictions.
To go about solving this problem, I thought it would be best if I tried my initial approach again in order to keep track of the people. The formula would be a little bit harder to figure out, so I thought that I could tackle that portion of the problem once I had recorded the data for the hour. Again, I created fake names for the friends that were not mentioned at the outset. I also changed the rightmost column to read “Friends Calling” so that I could monitor active callers. In addition, I added a 1 or a 2 next to each name, depending on which call they were on before they could not make any more calls.
After I filled out the rows above, I was assured that I could revert to my latter method from the previous problem and use numbers instead of names, including the variables t and y.
Following the elimination of Leigh from the total, the number of friends called under the modified rules was 28,655.
Using the information above, I recognized that the number of callers that had been reached by a specific time was equal to the previous number in addition to the sum of the differences from the two values prior to it. (For example, 143 callers had been reached by the 27th minute because 88 + [88 - 54] + [54 - 33] = 143. This is because each caller could only reach out twice, so the differences are accounting for both the callers that had already made two calls as well as the callers who were yet to make their second calls).
This would give me an equation for this problem:
**Equation: y = p1 + (p1 - p2) + (p2 - p3)
**p1 = previous number
p2 = number preceding previous number
p3 = number preceding both previous numbers
Part 4: Extensions
Reflection
The “A Timely Phone Tree” problem was crucial in my ability to understand concepts surrounding translating data into mathematical patterns, recognizing trends in a chart, formulating an equation based on recorded data, and variable application among others. I really had to take the toughest routes possible to find my answers; I was not satisfied with just a number to conclude the problem. I wanted to know why I used the methods that I did, and my persistence was ultimately what led me to success.
I believe that I deserve an “A” for each of the five categories on which I will be graded: deep mathematical thinking, breadth of exploration, documentation, habits of a mathematician, and reflection. As far as my thought process during the problem, I made sure to understand the reasoning behind each formula that I devised, and each piece of data that I recorded. With my exploration, I took multiple approaches to the same problem, and when one path led to a dead end, I did not give up. I kept fighting on until I came upon an answer that was suitable for my work, and I investigated the particulars behind that solution. As evident above, I also made sure to exhibit all of my work throughout the problem, whether that be through charts, equations, and/or explanations. I was very thorough in displaying my process, and I put a lot of time into perfecting my process. My habits of a mathematician are shown below, and I am confident that I am providing an exhaustive reflection of my steps and methods utilized over the course of the problem.
The most prominent Habits of a Mathematician shown in my work were Conjecture & Test and Stay Organized. When I formulated an equation or way that I went about solving a problem, I always made sure to prove my statements and work. In order to have an effective argument for a specific answer or value, it is essential to back it up with evidence. Many times it was difficult for me to seek reasoning behind my work, but I tried to view the problem at a variety of perspectives so that I could truly grasp it. This skill was coherent with being organized, because I needed to make sure that I kept my explanations both vivid and concise. I formatted each of my charts in a similar manner, and spaced out my work with explanations and descriptions so that the viewers could easily sort through my analysis. If one is not organized when completing work, it is not only hard for them to remember their process(es), but it is also challenging for the audience to view the problem as they had through their eyes.
Using the information above, I recognized that the number of callers that had been reached by a specific time was equal to the previous number in addition to the sum of the differences from the two values prior to it. (For example, 143 callers had been reached by the 27th minute because 88 + [88 - 54] + [54 - 33] = 143. This is because each caller could only reach out twice, so the differences are accounting for both the callers that had already made two calls as well as the callers who were yet to make their second calls).
This would give me an equation for this problem:
**Equation: y = p1 + (p1 - p2) + (p2 - p3)
**p1 = previous number
p2 = number preceding previous number
p3 = number preceding both previous numbers
Part 4: Extensions
- Simple: How many of Leigh’s friends would hear the news if her parents limited her to calling from 8:00 p.m. to 8:30 p.m.?
- Moderate: How many of Leigh’s friends would hear the news if the friend who Ana May called had a busy signal?
- Difficult: If each friend who made a call before 8:15 p.m. could call up to three people, and each friend after that time until 9:00 p.m. could only call two people, how many friends would hear the news?
Reflection
The “A Timely Phone Tree” problem was crucial in my ability to understand concepts surrounding translating data into mathematical patterns, recognizing trends in a chart, formulating an equation based on recorded data, and variable application among others. I really had to take the toughest routes possible to find my answers; I was not satisfied with just a number to conclude the problem. I wanted to know why I used the methods that I did, and my persistence was ultimately what led me to success.
I believe that I deserve an “A” for each of the five categories on which I will be graded: deep mathematical thinking, breadth of exploration, documentation, habits of a mathematician, and reflection. As far as my thought process during the problem, I made sure to understand the reasoning behind each formula that I devised, and each piece of data that I recorded. With my exploration, I took multiple approaches to the same problem, and when one path led to a dead end, I did not give up. I kept fighting on until I came upon an answer that was suitable for my work, and I investigated the particulars behind that solution. As evident above, I also made sure to exhibit all of my work throughout the problem, whether that be through charts, equations, and/or explanations. I was very thorough in displaying my process, and I put a lot of time into perfecting my process. My habits of a mathematician are shown below, and I am confident that I am providing an exhaustive reflection of my steps and methods utilized over the course of the problem.
The most prominent Habits of a Mathematician shown in my work were Conjecture & Test and Stay Organized. When I formulated an equation or way that I went about solving a problem, I always made sure to prove my statements and work. In order to have an effective argument for a specific answer or value, it is essential to back it up with evidence. Many times it was difficult for me to seek reasoning behind my work, but I tried to view the problem at a variety of perspectives so that I could truly grasp it. This skill was coherent with being organized, because I needed to make sure that I kept my explanations both vivid and concise. I formatted each of my charts in a similar manner, and spaced out my work with explanations and descriptions so that the viewers could easily sort through my analysis. If one is not organized when completing work, it is not only hard for them to remember their process(es), but it is also challenging for the audience to view the problem as they had through their eyes.