# A Mathematical Study of How Fast Rumours Spread

Much of the best mathematical inspiration comes from experience. It is hardly possible to believe in the existence of an absolute immutable concept of mathematical rigor dissociated form all human experience.
John von Neumann

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The speed at which rumours can spread is truly remarkable. It’s surprising to see how an event witnessed by just a handful of people can become the topic of conversation for an entire town in under two hours. This remarkable velocity of rumour propagation is often perplexing, and it’s difficult to comprehend how it happens so quickly.

However, upon closer inspection and analysis of the mathematics involved, the process of rumour spreading becomes much easier to comprehend. By breaking down the steps involved in how news is shared and passed on, the seemingly incredible speed of rumour dissemination becomes a clear and understandable phenomenon.

A man living in the capital comes to a town with about 60000 inhabitants at 9 a. m. and brings an interesting bit of news. He imparts this information to only three individuals at the house where he is staying, which takes approximately 15 minutes.

Thus by 9:15 a.m., the news has been conveyed to a total of four individuals – the man who brought the news and the three local residents with whom he shared it.

Subsequently, each of the three locals hastens to share the news with three other people, which also takes about 15 minutes. Thus, after half an hour, a total of 13 individuals are now aware of the news: the original four (the man from the capital and the three locals) plus the nine to whom they relayed the information (3 people each):

4 + (3 × 3)

= 13 persons.

Afterwards, each of the nine individuals who heard the news last promptly passes it on to three more acquaintances, once again taking around 15 minutes. By 9:45 a.m., the news has now reached a total of 40 residents: the original 13 who knew about it, as well as the 27 to whom the nine individuals passed it on (3 people each):

13 + (3 × 9)

= 40 residents.

If the rumour continues to spread in the same way, with each person sharing it with three others every 15 minutes, then the number of people who hear about it will be as follows:

By 10 a.m. the news will be known to

40 + (3 × 27)

= 121 persons.

By 10.15 a.m. the news will be known to

121 + (3 × 81)

= 364 persons.

By 10.30 a.m. the news will be known to

364 + (3 × 243)

= 1093 persons.

Indeed, within just 90 minutes, the news has already spread to almost 1100 people. However, considering the total population of the town (60000), it may not seem like a significant number. However, as the spreading of the rumour continues, the number of people who know about it will grow rapidly. Let’s see how quickly the rumour will continue to spread.

By 10.45 a.m. the news will be known to

1093 + (3 × 729)

= 3280 persons.

By 11 a.m. the news will be known to

3280 + (3 × 2187)

= 9841 persons.

In the next 15 minutes it will be the property of more than half of the town’s population.

9841 + (3 × 561)

= 29524 persons.

And this means that the news that only one man knew at 9 a. m. will be known to the entire town before it is 11.30 a. m.

As you can see, the number of people who know the rumour increases rapidly, and it doesn’t take long before it spreads to a significant portion of the town’s population. By 11:00 am, almost 10,000 people (or roughly 17% of the town’s population) would have heard the rumour. Let us see now how that is calculated. The whole thing boils down to the addition of the following numbers:

1 + 3 + (3 × 3) + (3 × 3 × 3) + (3 × 3 × 3 × 3) + (3 × 3 × 3 × 3 × 3) + ···.

Perhaps there is an easier way of computing this number, if we take into account the following peculiarity of the numbers we are adding:

• 1 = 1,
• 3 = 1 × 2 + 1,
• 9 = (1 + 3) × 2 + 1,
• 27 = (1 + 3 + 9) × 2 + 1,
• 81 = (1 + 3 + 9 + 27) × 2 + 1,
• 243 = (1 + 3 + 9 + 27 + 81) × 2 + 1, etc.

In other words, each number is equal to double the total of the preceding numbers plus 1.

Hence, to find the sum of all our numbers, from 1 to any number, it is enough to add to this last number, half of itself (minus 1). For instance, the sum total of

1 + 3 + 9 + 27 + 81 + 243 + 729

= 729 + half of 728

= 729 + 364

= 1093.

If the residents of the town were more inclined to share the news with others, the spread of the rumour would accelerate significantly. For instance, if each resident passed the news to five or even ten other people instead of three, the rumour would circulate much faster and reach a wider audience in a shorter amount of time. This is because the increased number of people who hear the rumour would then have the potential to share it with even more individuals, creating a snowball effect of information dissemination. In essence, the more talkative the residents are, the more rapid and extensive the spread of the rumour would be. In the case of five, the picture would be as follows:

• At 9 a.m. the news is known to ………………………………………… 1 person,
• By 9.15 a. m …………………. 1 + 5 …………………………………………. 6 persons,
• By 9.30 a. m ………………… 6 + (5 × 5) …………………………………. 31 persons,
• By 9.45 a. m ………………. 31 + (25 × 5)……………………………… 156 persons,
• By 10 a. m ……………….. 156 + (125 × 5)……………………………. 781 persons,
• By 10.15 a. m ……………. 781 + (625 × 5) …………………….. 3906 persons,
• By 10.30 a. m ………….. 3906 + (3125 × 5) ………………… 19531 persons,
• By 10.45 a. m ………… 19531 + (15625 × 5) ……………….. 97656 persons,

In short, it would be known to every one of the 60000 residents before 10.45 a. m.

It would spread a lot faster if each man shared the news with ten others. Here we would get these very fast-growing numbers:

• At 9 a.m. the news would be known to ………………… 1 person,
• By 9.15 a. m …………………………. 1 + 10 ………………….. 11 persons,
• By 9.30 a. m ……………………….. 11 + 100……………….. 111 persons.
• By 9.45 a. m ……………………… 111 + 1000 ……………. 1111 persons.
• By 10 a. m ………………………. 1111 + 10000…………. 11111 persons.
• By 10.15 a. m ………………… 11111 + 100000 ……… 111111 persons.

The last number is 111111, and that shows that the whole town would have heard the news shortly after 10 a.m. The news, in this case, would have taken a little over an hour to spread throughout the town.

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Reference. MATHEMATICS CAN BE FUN – YAKOV PERELMAN