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 Write-up

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

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
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