Brain Teasers
E-mails
Mad Ade sends an e-mail to two friends, asking each of them to copy the e-mail and send it to two of their friends, those in turn send it to two of their friends, and so on.
How many e-mails would have been sent by the time 30 sets of e-mails had been forwarded?
How many e-mails would have been sent by the time 30 sets of e-mails had been forwarded?
Answer
2147483646Mad Ade sent the mail to 2 persons. Those 2 sent the mail to 2 persons each, total 4 persons. Now, those 4 person sent mail to a total of 8 persons, then 8 to 16 persons, 16 to 32 persons and so on.... Hence, it a series of 2, 4, 8, 16, 32 up to 30 numbers.
It is a geometric series with a common ratio 2 and first number is also 2. Summation of such series is given by A * (Rn - 1) / (R - 1) where
A = first term
R = common ratio
n = total numbers
So, total number of times mail sent by the time 30 sets of e-mails had been forwarded
= 2 * (2^30 - 1) / (2 - 1)
= 2 * (1073741824 - 1)
= 2 * 1073741823
= 2147483646
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Comments
Very nice riddle. A couple of minor points. First there is a typo: in the last few lines the 230 should be 2^30.
Secondly, not everyone is aware of the formula for evaluting the sum of a geometric series. It may therefore be better to simply derive directly the sum of 2^1 + 2^2 + 2^3 +....+2^30, instead of using the formula. This can be done as so: Let S equal the totla number of E-mails sent. Thus, S = 2+ 2^2 + 2^3 +....2^30. Thus, S = 2(1 + 2^2 + 2^3 +...+2^29) This implies that S= 2(1+ S - 2^30). Thus S = 2 +2S - 2^30. Solving for S, we have S = 2^30 - 2 = 2147483646
Secondly, not everyone is aware of the formula for evaluting the sum of a geometric series. It may therefore be better to simply derive directly the sum of 2^1 + 2^2 + 2^3 +....+2^30, instead of using the formula. This can be done as so: Let S equal the totla number of E-mails sent. Thus, S = 2+ 2^2 + 2^3 +....2^30. Thus, S = 2(1 + 2^2 + 2^3 +...+2^29) This implies that S= 2(1+ S - 2^30). Thus S = 2 +2S - 2^30. Solving for S, we have S = 2^30 - 2 = 2147483646
Beyond the fact that there aren't two billion people in the world that have email, so there would have to be repeats, the teaser isn't clear about what the first set is. I assumed the first set was Mad Ade's email, and thus got 2^30-1.
But Java, Mad-Ade sent 2 emails the first time. I agree with the given answer although it is trivial for anyone who has studied geometric sequences.
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