A group of n  friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the  12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
1. Let \( n \) be the total number of friends.
2. Each friend solved exactly 11 out of the 12 problems. Therefore, each friend missed exactly one problem.
3. Let \( a \) be the number of people who missed a short-answer problem \( S_i \) and \( b \) be the number of people who missed a full-solution problem \( F_j \).
4. Since each person missed exactly one problem, we have:
   \[
   a + b = n
   \]
5. We are given that the sum of the 32 entries in the table is 256. Each entry in the table represents the number of people who solved both a specific short-answer problem and a specific full-solution problem.
6. Each person who missed a short-answer problem \( S_i \) solved the remaining 7 short-answer problems and all 4 full-solution problems. Therefore, each such person contributes 7 entries to the table.
7. Each person who missed a full-solution problem \( F_j \) solved all 8 short-answer problems and the remaining 3 full-solution problems. Therefore, each such person contributes 8 entries to the table.
8. The total number of entries contributed by all people is:
   \[
   7a + 8b = 256
   \]
9. We now have the system of equations:
   \[
   \begin{cases}
   a + b = n \\
   7a + 8b = 256
   \end{cases}
   \]
10. Solve the first equation for \( a \):
    \[
    a = n - b
    \]
11. Substitute \( a = n - b \) into the second equation:
    \[
    7(n - b) + 8b = 256
    \]
12. Simplify and solve for \( n \):
    \[
    7n - 7b + 8b = 256 \\
    7n + b = 256 \\
    b = 256 - 7n
    \]
13. Substitute \( b = 256 - 7n \) into the first equation:
    \[
    a + (256 - 7n) = n \\
    a = n - (256 - 7n) \\
    a = 8n - 256
    \]
14. Since \( a \) and \( b \) must be non-negative integers, we need:
    \[
    8n - 256 \geq 0 \\
    n \geq 32
    \]
15. Substitute \( n = 32 \) into the equations to check if it satisfies both:
    \[
    a + b = 32 \\
    7a + 8b = 256
    \]
16. If \( n = 32 \):
    \[
    a = 8(32) - 256 = 0 \\
    b = 256 - 7(32) = 32
    \]
17. Therefore, \( a = 0 \) and \( b = 32 \) satisfy the equations.

The final answer is \( \boxed{32} \)