How many rounds for a message to reach everyone?

Question

One person sees a message and forwards it to $2$ new people. In the next round each of those forwards it to $2$ more, and so on, doubling the number of new recipients every round. Solve in your head.

How many rounds until a round reaches roughly the world population ( $\approx 8$ billion)? Aim: under 30 seconds.

Accepted Answer

In round n the number of new recipients is 2^{n}. We want 2^{n} \approx 8 	imes 10^{9}. Count doublings with 2^{10} \approx 10^{3}: - 2^{30} = (2^{10})^{3} \approx 10^{9} (actually \approx 1.07 	imes 10^{9}), short of 8 billion. - 2^{33} = 2^{30} \cdot 2^{3} \approx 8 	imes 1.07 	imes 10^{9} \approx 8.6 	imes 10^{9}, just past 8 billion. About 33 rounds. In only 33 steps a single message saturates a planet of 8 billion people, because each round multiplies the reach by 2 and 30 doublings already clears a billion. Sanity note: the cumulative number reached is 2^{n+1} - 1, so the totals bl…

How many rounds for a message to reach everyone?

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