Six, in a certain exam, there are 5 multiple-choice questions, each with 4 different answers to choose from. Each person selects exactly 1 answer for each 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 each pair of sheets has at most 3 questions with the same answers. Find the minimum possible value of $n$.
(Li Chengzhang, provided)
Solution: The minimum possible value of $n$ is 25.
Let the 4 possible answers to each question be denoted as $1,2,3,4$. The answers on each test paper are denoted as $(g, h, i, j, k)$, where $g, h, i, j, k \in \{1,2,3,4\}$. 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 with answers belonging to the same quadruple. After removing these 8 test papers, there are still 8 test papers with answers belonging to the same quadruple among the remaining 1992 test papers; after removing these 8 test papers, there are still 8 test papers with answers belonging to the same quadruple among the remaining 1984 test papers; after removing these 8 test papers, together with the previously removed test papers, there are 24 test papers. In these 24 test papers, any 4 of them have at least two with answers belonging to the same quadruple, which of course does not meet the requirements of the problem. Therefore, the minimum value of $n \geqslant 25$.
$$
\begin{array}{l}
\text { On the other hand, let } \\
S=\{(g, h, i, j, k) \mid g+h+i+j+k \equiv 0(\bmod 4), \\
g, h, i, j, k \in\{1,2,3,4\}\} .
\end{array}
$$
- Then $|S|=256$, and any two answers in $S$ have at most 3 questions the same. Removing 6 elements from $S$, when each of the remaining 250 answers is chosen by exactly 8 people, a total of 2000 test papers are obtained, and in any 25 of these answers, there are always 4 that are different, since they are all in $S$, of course, they meet the requirements of the problem. This shows that when $n=25$, the requirements of the problem can be met.
In summary, the minimum possible value of $n$ is 25.