A Timely Phone Tree Pow

In this Problem of the Week, we look at A Timely Phone Tree. Leigh’s parents limit her time to call her friends to between 8 p.m. and 9 p.m. on school nights. Leigh and her friends find that 3 minutes, including the time needed to make the phone call and talk to the right person, will be enough time to tell someone the information. Once Leigh calls Michael, they will both then call another person in the next 3 minutes. Once a person is called, they will continue to call more people until 9 p.m. We have to figure out how many of Leigh’s friends will know about the situation by 9 p.m. if the pattern continues. Also, using a null hypothesis, we know that each call will take exactly 3 minutes, no one calls a person who has already been called, and the caller always reaches the person being called.

To first approach this problem, I created a short diagram to represent the first few calls that would be made. Continuing a diagram like this would be an inefficient way to chart how many people would be contacted between 8 p.m. and 9 p.m. Instead, I made a type of input-output chart. Since Leigh would be the only person to know at 8 p.m., I did not include this time in the chart. I started by calculating the various times that each set of calls would take place using intervals of three minutes. Then, after the first three minutes, two people would know Leigh’s message, including herself. After completing this chart, I found that each interval, the amount of people who knew Leigh’s message would double. This means that the pattern could be represented by 2^n (number of 3 minute intervals) to find how many people know the information by 9 p.m.

To calculate how many people would know by 9 p.m., I took sixty minutes, which is the amount of time that Leigh and her friends are allowed to be on the phone, and divided it by three to get twenty. This means that to find how many people would know Leigh’s message between 8 p.m. and 9 p.m., I would have to solve 2^20, since there could be twenty sets of calls within the hour.

Using this expression I found (2^n), I can solve for 2^20 since there are twenty three-minute intervals in the hour. I found that 2^20 = 1,048,576, which means that by 9 p.m., 1,048,576 people will known Leigh’s message including Leigh. However, since the problem asks how many of Leigh’s friends will know the message, I need to subtract Leigh from the 1,048,576 people who know the message.