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 onehalf 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 2^{n} + 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
onehalf of the 8000 have heard the rumor?
On the
11^{th} day, close to 4100 people will know!
3.
On what day will 256 new people be told the rumor? On the 8^{th} 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
