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Spreading Rumors

1. problem
Your best friend tells you a secret and makes you promise not to tell anyone. Although you are just dying to tell someone, you don't really want your friend's secret to be widely known. Is there much harm in telling just two people?

2. intuition
When spreading rumors, our sense is that there is little harm in telling just one or two people. We make them promise not to tell anyone else (in the same way that we promised), and figure it will stop there. Or, if they do tell someone, then the rumor won't spread very far. As we will see below, this is an example of where our intuition is quite poor: rumors spread very fast.

3. math
You tell your best friend's secret to two people. Now, 4 people know the secret: you, your best friend, and the two people you just told.
Let's assume that the two people to whom you told the secret each tell two people, for a total of four new people. Now, 8 people know the secret. If each of the last four people to receive the secret tell two people, eight new people know the secret. After four passes of the secret, a total of 16 people know the secret.

This process continues: the group of people who just learned the secret, each tell two more people. After just ten passes, 1,024 people will know the secret, and after 20 passes, 1,048,576 people will know the secret. After 28 passes, 268,425,456 people know, which is roughly the population of the United States. And after 33 passes, more than 8.5 billion people will know, which is more than the population of the entire planet.
The number of people who know a secret doubles after each pass (2, 4, 8, 16, 32, ...). That is, after N passes, 2^N people know the secret. This is known as exponential growth.
If we assume that each person will act in a similar manner to yourself, we see that the rumor will spread to a large number of people very quickly.

4. summary
When holding a secret we believe that little harm is done by telling just a few people, since the number of people that know the secret has only increased by a small number. The problem, of course, is that everyone thinks like this, and everyone tells a few people. The number of people who learn of the secret, therefore, grows very quickly, at a exponential rate. Here are a few more surprising examples of the impact of exponential growth.

Take a simple piece of notebook paper. Fold it in half. Fold it in half again, and again. How thick will the paper be after a total of 50 folds? A single sheet of paper is 0.1 mm thick. Folded once, the paper is 0.2 mm thick; folded twice it is 0.4 mm thick; folded three times it is 0.8 mm thick. The thickness is doubling each time (exponential growth). After ten folds, the paper is 102.4 mm thick (10.24 cm = 4.03 in). After twenty folds, the paper is 104,857.6 mm thick (approximately 105 m or 344 ft -- approximately the height of a 30-story building!). After 50 folds, the paper is 112,589,990,684,262.4 mm thick (approximately 112 billion km or 70 billion miles, almost the distance to the sun.

Once upon a time there was a king who loved to play chess. One day the king proposed that if anyone in his kingdom could beat him at chess, the king would grant them any reasonable wish. A poor farmer stepped forward to meet the king's challenge. Much to the king's surprise, the farmer beat the king quickly and with seeming ease. True to his word, the king agreed to grant the farmer's wish. Wanting to wish for something that seemed reasonable, the farmer suggested the following: "I propose that you place on the first square of the chess board one penny, and on the second square, two pennies, on the third, four pennies, and so forth, until the last square is reached." After a moments thought, the king granted the request, as it seemed that the farmer had, in fact, asked for very little money. As the king began to place the money on the chess board, however, he soon realized his terrible mistake. Let's see why.
The chess board consists of 8 x 8 = 64 squares. On the first row of the board, the king placed:
0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28

for a total of $2.55. Notice that the amount of money is doubling between squares (exponential growth). On the second row, the king placed:
2.56, 5.12, 10.24, 20.48, 40.96, 81.92, 163.85, 327.68

Combined with the first row, there was a total of $655.35 on the board. On the third row, the king placed:
655.36, 1310.72, 2621.44, 5242.88, 10485.76, 20971.52, 41943.04, 83886.08

Combined with the first two rows, there was a total of $167,772.16 -- and there were still five rows to complete! Continuing along like this, the completed board would have contained $368,934,881,474,191,032.32. The king, of course, ran out of money well before this and was unable to keep his promise.

Home Math Kids Science Kids

Spreading Rumors
1. problem	Your best friend tells you a secret and makes you promise not to tell anyone. Although you are just dying to tell someone, you don't really want your friend's secret to be widely known. Is there much harm in telling just two people?
2. intuition	When spreading rumors, our sense is that there is little harm in telling just one or two people. We make them promise not to tell anyone else (in the same way that we promised), and figure it will stop there. Or, if they do tell someone, then the rumor won't spread very far. As we will see below, this is an example of where our intuition is quite poor: rumors spread very fast.
3. math	You tell your best friend's secret to two people. Now, 4 people know the secret: you, your best friend, and the two people you just told. Let's assume that the two people to whom you told the secret each tell two people, for a total of four new people. Now, 8 people know the secret. If each of the last four people to receive the secret tell two people, eight new people know the secret. After four passes of the secret, a total of 16 people know the secret. This process continues: the group of people who just learned the secret, each tell two more people. After just ten passes, 1,024 people will know the secret, and after 20 passes, 1,048,576 people will know the secret. After 28 passes, 268,425,456 people know, which is roughly the population of the United States. And after 33 passes, more than 8.5 billion people will know, which is more than the population of the entire planet. The number of people who know a secret doubles after each pass (2, 4, 8, 16, 32, ...). That is, after N passes, 2^N people know the secret. This is known as exponential growth. If we assume that each person will act in a similar manner to yourself, we see that the rumor will spread to a large number of people very quickly.
4. summary	When holding a secret we believe that little harm is done by telling just a few people, since the number of people that know the secret has only increased by a small number. The problem, of course, is that everyone thinks like this, and everyone tells a few people. The number of people who learn of the secret, therefore, grows very quickly, at a exponential rate. Here are a few more surprising examples of the impact of exponential growth. Take a simple piece of notebook paper. Fold it in half. Fold it in half again, and again. How thick will the paper be after a total of 50 folds? A single sheet of paper is 0.1 mm thick. Folded once, the paper is 0.2 mm thick; folded twice it is 0.4 mm thick; folded three times it is 0.8 mm thick. The thickness is doubling each time (exponential growth). After ten folds, the paper is 102.4 mm thick (10.24 cm = 4.03 in). After twenty folds, the paper is 104,857.6 mm thick (approximately 105 m or 344 ft -- approximately the height of a 30-story building!). After 50 folds, the paper is 112,589,990,684,262.4 mm thick (approximately 112 billion km or 70 billion miles, almost the distance to the sun. Once upon a time there was a king who loved to play chess. One day the king proposed that if anyone in his kingdom could beat him at chess, the king would grant them any reasonable wish. A poor farmer stepped forward to meet the king's challenge. Much to the king's surprise, the farmer beat the king quickly and with seeming ease. True to his word, the king agreed to grant the farmer's wish. Wanting to wish for something that seemed reasonable, the farmer suggested the following: "I propose that you place on the first square of the chess board one penny, and on the second square, two pennies, on the third, four pennies, and so forth, until the last square is reached." After a moments thought, the king granted the request, as it seemed that the farmer had, in fact, asked for very little money. As the king began to place the money on the chess board, however, he soon realized his terrible mistake. Let's see why. The chess board consists of 8 x 8 = 64 squares. On the first row of the board, the king placed: 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28 for a total of $2.55. Notice that the amount of money is doubling between squares (exponential growth). On the second row, the king placed: 2.56, 5.12, 10.24, 20.48, 40.96, 81.92, 163.85, 327.68 Combined with the first row, there was a total of $655.35 on the board. On the third row, the king placed: 655.36, 1310.72, 2621.44, 5242.88, 10485.76, 20971.52, 41943.04, 83886.08 Combined with the first two rows, there was a total of $167,772.16 -- and there were still five rows to complete! Continuing along like this, the completed board would have contained $368,934,881,474,191,032.32. The king, of course, ran out of money well before this and was unable to keep his promise.