Let $n>0$ be an integer. Each of the $n$ girls in a group is the only one who knows a certain piece of gossip. To share their information, they call each other two at a time, but each time, only one speaks and the other just listens to all the information her friend tells her.

Determine the minimum number of phone calls sufficient for each girl to know all the gossip.
Consider a series of phone calls that leads to each girl being aware of everything. We call $f$ the first girl to know everything.

Clearly, before $f$ has learned everything, there have been at least $n-1$ phone calls, one to retrieve each of the $n-1$ pieces of information she did not know (but not necessarily directly to $f$). On the other hand, as soon as $f$ finally knows everything and is the first girl in this case, each of the $n-1$ other girls must still wait for at least one more phone call to join the club of informed people (not necessarily given by $f$). Thus, it takes at least $2(n-1)$ phone calls for each girl to know all the gossip. Conversely, it is easy to see that this is indeed achievable in $2(n-1)$ phone calls: each of the $n-1$ girls calls a certain fixed girl $f$. Then, $f$ calls each of them back.