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The population of Humorville is 9,800 people. In this town, jokes travel fast. In one hour, each person who hears a joke tells three other people who have not heard it and tells no one else. Last Friday, a visitor from out of town told the mayor a new joke at 10:00 AM.

How long did it take everyone in Humorville to hear the joke?

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closed as off-topic by A. P., Gareth McCaughan Jan 4 at 21:46

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It takes:

8 hours

because

at 10 o'clock, 1 person has heard the joke - the Mayor. He tells 3 people so that by 11:00, 4 people have heard the joke. By 12:00, these 3 have told 9 more in total, for grand total of 1+3+9=13 people. After $k$ hours $1+3+9+\dots+3^k$ people have heard it, and this sums to $\dfrac{3^{k+1}-1}2$. As $3^9=19683$, after 8 hours, 9841.5 people would have heard the joke, which is larger than the population of Humorville.

Alternatively

Assume the joke takes 5 minutes to tell, everyone would have heard the joke in 5 minutes!

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  • $\begingroup$ Nicely done and explained +1. I forgot to account for summation haha, oh wells. $\endgroup$ – Dorrulf Jan 4 at 21:53
  • $\begingroup$ Also, I don't get how your alternate works out... $\endgroup$ – Dorrulf Jan 4 at 21:54
  • $\begingroup$ @Dorrulf; everyone in Humorville would have taken 5 minutes to listen to the joke - not at the same time obviously! $\endgroup$ – JonMark Perry Jan 4 at 21:59
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What's so interesting about this problem is how basically only in the last hour do the majority of people hear the joke. This is counter-intuitive because of our brain's limited understanding of exponentials.

    9800               -   1      - 3^1 - 3^2 - 3^3 - 3^4 - 3^5 - 3^6 - 3^7 -  3^8
    (total population)    (mayor)  (1hr)  (2hr) (3hr) (4hr) (5hr) (6hr) (7hr) (8hr)

#REMAINING 9800 9799 9796 9787 9760 9679 9436 8707 6520 -41

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  • $\begingroup$ i'm trying to mark as a spoiler, but its not working. Can someone help me edit it? $\endgroup$ – user353877 Jan 4 at 21:53

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