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SPREADING RUMORS

Problem Statement
A rumor starts by someone telling the rumor to two people on May 1. Each of those two people are responsible for telling the rumor to two others on the next day (May 2).At this time seven people know the rumor. On May 3, the four people who heard the rumor on May 2 must each tell two more people. It is important for 8000 people to hear the rumor by May 14. Is this outcome likely to occur if the pattern for spreading the rumor continues as started?

On what day will approximately one-half of the 8000 have heard the rumor? On what day will 256 new people be told the rumor? If the rumor process continues until May 20,how many new people will hear the rumor on that day?

If the rumor process continues for n days, how many new people will be told the rumor on the nth day? What will be the total number of people who know the rumor on that day?

(Source: Mathematics Teaching in the Middle School, May 1994).

Problem setup

This problem wants me to find the pattern in the growth of the number of people involved in this NASTY rumor!  Mind you I am not a gossip – but did you hear about Brad and Angelina?  I am FLABBERGHASTED!!

 

Back to the matter at hand.  I figured at the beginning that this would grow exponentially.  It took me a few iterations to determine how the growth is related to the powers of 2. 

“How when it involves odd numbers?”  I’m glad you asked!  To remedy this I know I will have to add or subtract 1.

Plans to Solve/Investigate the Problem

Initially, I assumed that I could use 2n + 1, where n = day of the month.  This works for the first day, but thereafter?  Not so much!!

 

Investigation/Exploration of the Problem

 

In the end, I came up with the following formula: , where n is the day of the month.

This formula is utilized in the number of people column.  To get the number of people added on a given day, I simply subtracted the previous days total.

Number of people

Day of the month

Number of people added that day.

3

1

 

7

2

4

15

3

8

31

4

16

63

5

32

127

6

64

255

7

128

511

8

256

1023

9

512

2047

10

1024

4095

11

2048

8191

12

4096

16383

13

8192

32767

14

16384

65535

15

32768

131071

16

65536

262143

17

131072

524287

18

262144

1048575

19

524288

2097151

20

1048576

 

So let’s answer the questions that were asked of us.

1.      It is important for 8000 people to hear the rumor by May 14. Is this outcome likely to occur if the pattern for spreading the rumor continues as started?
I should say so! We will surpass the 8000 mark on day 12.  Pesky rumors!

2.      On what day will approximately one-half of the 8000 have heard the rumor? 

On the 11th day, close to 4100 people will know!

3.      On what day will 256 new people be told the rumor?  On the 8th day, exactly 256 people will feast on this juicy tidbit of slanderous information.

4.      If the rumor process continues until May 20, how many new people will hear the rumor on that day? 1,048,576!  Geez Louise!  That’s a lot of folks!

5.      If the rumor process continues for n days, how many new people will be told the rumor on the nth day?   

6.      What will be the total number of people who know the rumor on that day?  The number added is a direct power of 2.  So the number added on that day will be

 

 

Author & Contact
Michelle Houston
Email me

 

 

 

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