6. In a certain exam, there are 5 multiple-choice questions, each with 4 different answers to choose from, and each person selects exactly 1 answer per question. In 2000 answer sheets, it is found that there exists an $n$, such that in any $n$ answer sheets, there are 4 sheets where the answers of any two sheets differ in at least 2 questions. Find the minimum possible value of $n$.
Solve for the smallest possible value of $n$ as 25.
Let the 4 possible answers to each question be denoted as $1,2,3,4$. Each test paper's answers are recorded as $(g, h, i, j, k)$, where $g, h, i, j, k \in\{1,2,3,4\}$, and let
$$
\begin{array}{l}
\{(1, h, i, j, k),(2, h, i, j, k),(3, h, i, j, k),(4, h, i, j, k)\}, \\
h, i, j, k=1,2,3,4
\end{array}
$$

This yields 256 quadruples. Since $2000=256 \times 7+208$, by the pigeonhole principle, there are 8 test papers whose answers belong to the same quadruple. After removing these 8 test papers, there are still 8 test papers in the remaining 1992 that belong to the same quadruple; removing these 8 test papers, in the remaining 1984, there are again 8 test papers that belong to the same quadruple; removing these 8 test papers, together with the previously removed test papers, a total of 24 test papers are obtained. Among these 24 test papers, any 4 of them have at least two answers that belong to the same quadruple, which of course does not meet the requirements of the problem. Therefore, the smallest value of $n$ is greater than or equal to 25.
On the other hand, let
$$
\begin{aligned}
S= & \{(g, h, i, j, k) \mid g+h+i+j+k \equiv \\
& 0(\bmod 4), g, h, i, j, k \in\{1,2,3,4\}\}
\end{aligned}
$$

Then $|S| = 256$, and any two answers in $S$ have at most 3 questions the same. Removing 6 elements from $S$, when the remaining 250 answers are each chosen by exactly 8 people, a total of 2000 test papers are obtained, and among any 25 of these answers, there are always 4 that are different. Since they are all in $S$, they of course meet the requirements of the problem. This shows that $n=25$ can satisfy the requirements of the problem.
In summary, the smallest possible value of $n$ is 25.