Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.
1. **Base Case:**
   For \( n = 1 \), no phone calls are needed since the single person already knows their piece of information. Thus, the minimum number of calls is \( 0 \).

2. **Inductive Step:**
   Assume that for \( n \) people, the minimum number of phone calls required for everyone to know everything is \( 2n - 2 \). We need to show that for \( n + 1 \) people, the minimum number of phone calls required is \( 2(n + 1) - 2 = 2n \).

3. **Adding a New Person:**
   Consider adding a new person to the group of \( n \) people. Let this new person be \( P_{n+1} \). According to the induction hypothesis, the \( n \) people can share all information among themselves with \( 2n - 2 \) calls.

4. **First Call:**
   The new person \( P_{n+1} \) makes a call to any one of the \( n \) people (say \( P_1 \)). After this call, \( P_{n+1} \) knows everything that \( P_1 \) knows, which is all the information of the \( n \) people.

5. **Last Call:**
   Finally, one of the \( n \) people (say \( P_n \)) makes a call to \( P_{n+1} \). After this call, \( P_{n+1} \) shares all the information they have (which includes the information from the first call) with \( P_n \). Now, \( P_n \) knows everything that \( P_{n+1} \) knows, which is all the information of the \( n+1 \) people.

6. **Total Calls:**
   The total number of calls made is the \( 2n - 2 \) calls for the initial \( n \) people plus the 2 additional calls for the new person, which sums up to \( 2n \).

7. **Optimality:**
   To prove that \( 2n - 2 \) is the minimum number of calls, consider that each person must receive information from at least one other person. This requires at least \( n \) calls. Additionally, for \( n > 1 \), the first person must receive at least one additional call to ensure they know everything, and similarly for each subsequent person. Thus, the minimum number of additional calls required is \( n - 2 \). Therefore, the total minimum number of calls is \( n + (n - 2) = 2n - 2 \).

\(\blacksquare\)

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