Let Akbar and Birbal together have $n$ marbles, where $n > 0$.
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.” 
Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”
What is the minimum possible value of $n$ for which the above statements are true?

1. Let \( A \) and \( B \) represent the number of marbles Akbar and Birbal have, respectively. We are given that \( A + B = n \) and \( n > 0 \).

2. According to Akbar's statement, if he gives some marbles to Birbal, then Birbal will have twice as many marbles as Akbar. Let \( x \) be the number of marbles Akbar gives to Birbal. Then:
   \[
   A - x = k \quad \text{and} \quad B + x = 2k
   \]
   This implies:
   \[
   A - x + B + x = k + 2k \implies A + B = 3k \implies n = 3k
   \]
   Therefore, \( n \) must be divisible by 3.

3. According to Birbal's statement, if he gives some marbles to Akbar, then Akbar will have thrice as many marbles as Birbal. Let \( y \) be the number of marbles Birbal gives to Akbar. Then:
   \[
   A + y = 3m \quad \text{and} \quad B - y = m
   \]
   This implies:
   \[
   A + y + B - y = 3m + m \implies A + B = 4m \implies n = 4m
   \]
   Therefore, \( n \) must be divisible by 4.

4. To find the minimum possible value of \( n \) that satisfies both conditions, we need the least common multiple (LCM) of 3 and 4:
   \[
   \text{LCM}(3, 4) = 12
   \]
   Thus, the minimum possible value of \( n \) is 12.

5. Verify the solution with \( n = 12 \):
   - Let \( A = 6 \) and \( B = 6 \).
   - If Akbar gives 2 marbles to Birbal, then:
     \[
     A - 2 = 4 \quad \text{and} \quad B + 2 = 8
     \]
     Birbal has twice as many marbles as Akbar: \( 8 = 2 \times 4 \).

   - If Birbal gives 3 marbles to Akbar, then:
     \[
     A + 3 = 9 \quad \text{and} \quad B - 3 = 3
     \]
     Akbar has thrice as many marbles as Birbal: \( 9 = 3 \times 3 \).

Thus, \( n = 12 \) satisfies both conditions, and it is the minimum possible value.

The final answer is \( \boxed{12} \).