Assume we have $n \ge 4$ people which everyone of them got a news. In every two steps these people call each other and transfer their all news they know. Prove that these people can know all the news in $2n-4$ calls.
Spreading news: Prove $2n-4$ by induction.
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$\begingroup$
induction
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4This is known as the [Gossip Problem](http://mathworld.wolfram.com/Gossiping.html). The proof is surprisingly lengthy...references can be found in the link (or online). – 2017-02-24
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0Related: [Gossip problem proof by induction](http://math.stackexchange.com/questions/977455/gossip-problem-proof-by-induction) – 2017-02-24
1 Answers
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For the $(n+1)$-st person $N$ it's enough to make one call to person $A$ to communicate the $N$'s news to $A$, then let the whole $n$-people group communicate all news to each other in $G(n)$ calls and finally to make another $A$ to $N$ talk to communicate all news to $N$.
I'm sure you can find the number of talks $G(n+1)$ now, related to $G(n)$.
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0This only shows that $G(n+1)≤G(n)+2$. It is also easy to prove that $G(n+1)≥G(n)+1$ so either $G(n+1)=G(n)+1$ or $G(n)+2$. Alas, $G(4)=G(3)+1$ so it really isn't obvious that the correct answer is (usually) $+2$. – 2017-02-24