Let's analyze the problem step by step.

1. Given Information:

   • There are three logicians A, B, and C, each wearing a hat with a positive integer on it.
   • The number on one hat is the sum of the numbers on the other two hats.
   • A says, "My number is 10."
   • The logicians can see the numbers on the hats of the other two but not their own. They are error-free in their reasoning and know all the information so far.

2. Implications of A's Statement:

   • A says, "My number is 10," which implies that A has deduced that the number on their own hat is 10.
   • For A to be sure of this, A must have reasoned from seeing the numbers on B and C’s hats.
   • If A sees the numbers on B and C's hats, let's denote these numbers as $x$ (B's number) and $y$ (C's number). A can only be certain that their number is 10 if $x + y = 10$.
   • Therefore, B’s number $x$ and C’s number $y$ must add up to 10.

3. Solving for B and C's Numbers:

   • Since one of the numbers on the hats is the sum of the other two numbers, and A's number is 10, one of the following must be true:

     • $A = x + y$, which is 10, meaning $x + y = 10$.
     • $B = A + C$, which implies $B = 10 + y$.
     • $C = A + B$, which implies $C = 10 + x$.

   • However, given that A says their number is 10, and knowing that the sum of the other two numbers is 10, the only possible pair for $x$ and $y$ (i.e., B and C's numbers) is $x = 4$ and $y = 6$, or vice versa.

Thus, B and C have the numbers 4 and 6 (or 6 and 4).

Would you like more details, or do you have any further questions?

Here are 5 related questions to deepen your understanding:

1. How does A deduce that their number is 10?
2. Could A be wrong if they did not have perfect logic?
3. What would happen if A saw different numbers on B and C's hats?
4. How would the problem change if the sum of the two numbers did not equal one of the hats?
5. Can this problem be solved if the numbers were negative or non-integer?

Tip: In logic puzzles, always try to simulate the reasoning of each participant step by step.