a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$  are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts 

B: ` I can't tell what they are.' 
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.' 
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` Now I can tell what they are.'

What are the two numbers? 

b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?
Let's break down the problem step by step.

### Part (a)

1. **Define the Problem:**
   - Two positive integers \( u \) and \( v \) are chosen.
   - Logician \( A \) knows the sum \( S = u + v \).
   - Logician \( B \) knows the sum of squares \( Q = u^2 + v^2 \).

2. **Initial Information:**
   - \( B \) says, "I can't tell what they are."
   - This implies that there are multiple pairs \((u, v)\) that can produce the same \( Q \).

3. **First Exchange:**
   - \( A \) says, "I can't tell what they are."
   - This implies that there are multiple pairs \((u, v)\) that can produce the same \( S \).

4. **Second Exchange:**
   - \( B \) says, "I can't tell what they are."
   - This implies that even after knowing that \( A \) cannot determine the numbers, \( B \) still cannot determine the numbers.

5. **Third Exchange:**
   - \( A \) says, "I can't tell what they are."
   - This implies that even after knowing that \( B \) cannot determine the numbers, \( A \) still cannot determine the numbers.

6. **Fourth Exchange:**
   - \( B \) says, "Now I can tell what they are."
   - This implies that with the additional information from the previous exchanges, \( B \) can now uniquely determine the numbers.

### Analysis:

1. **Possible Pairs:**
   - We need to find pairs \((u, v)\) such that both \( S \) and \( Q \) have multiple representations initially but can be uniquely determined after several exchanges.

2. **Sum and Sum of Squares:**
   - Let's denote the sum \( S = u + v \) and the sum of squares \( Q = u^2 + v^2 \).

3. **Possible Values:**
   - We need to check pairs \((u, v)\) for small values to see which pairs fit the criteria.

### Calculation:

1. **Check Small Values:**
   - Let's start with small values and check the sums and sums of squares.

2. **Pairs and Their Sums:**
   - For \( u = 8 \) and \( v = 9 \):
     - \( S = 8 + 9 = 17 \)
     - \( Q = 8^2 + 9^2 = 64 + 81 = 145 \)

3. **Verification:**
   - Check if there are other pairs with the same \( S \) and \( Q \):
     - For \( S = 17 \), possible pairs are \((1, 16), (2, 15), (3, 14), (4, 13), (5, 12), (6, 11), (7, 10), (8, 9)\).
     - For \( Q = 145 \), possible pairs are \((8, 9)\) and \((9, 8)\).

4. **Uniqueness:**
   - After several exchanges, the only pair that fits both \( S = 17 \) and \( Q = 145 \) uniquely is \((8, 9)\).

### Conclusion:
The two numbers are \( 8 \) and \( 9 \).

### Part (b)

1. **Information Exchange:**
   - When \( B \) first says that he cannot tell what the two numbers are, \( A \) receives a large amount of information because it means that there are multiple pairs \((u, v)\) that can produce the same \( Q \).

2. **A's Knowledge:**
   - When \( A \) first says that he cannot tell what the two numbers are, \( B \) already knows that \( A \) cannot tell what the two numbers are because \( B \) knows that there are multiple pairs \((u, v)\) that can produce the same \( Q \).

3. **B's Advantage:**
   - The act of listening to \( A \) before answering the second time gives \( A \) more information on his second answer, which wouldn't be the case if he didn't listen.

The final answer is \( \boxed{ (8, 9) } \)