2. Each of  $n$ members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.

Hi, from my sketches I'm thinking the answer is $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks
1. **Claim**: The minimal number of phone calls required to convey all the information to each of the \( n \) members is \( 2n-2 \).

2. **Construction**: 
   - In the first \( n-1 \) moves, each member tells a designated person \( A \) their information. This means \( A \) will have all the information after \( n-1 \) calls.
   - In the next \( n-1 \) moves, \( A \) tells each of the other \( n-1 \) members all the information. Thus, after these \( n-1 \) calls, everyone will have all the information.

3. **Proof of Optimality**:
   - Define a member as "wise" if they have all the information.
   - Initially, no one is wise. To make at least one person wise, we need at least \( n-1 \) calls because each call can only transfer information from one person to another.
   - After one person becomes wise, we need to make the remaining \( n-1 \) members wise. Each call can only increase the number of wise people by 1. Therefore, we need at least \( n-1 \) additional calls to make all \( n \) members wise.
   - Hence, the total number of calls required is at least:
     \[
     (n-1) + (n-1) = 2n-2
     \]
   - Since we have a construction that achieves this in \( 2n-2 \) calls, this number is indeed the minimum.

\(\blacksquare\)

The final answer is \( \boxed{ 2n-2 } \).