6.59 A certain exam consists of 5 multiple-choice questions, each with 4 different answer options. Each person selects exactly 1 answer for each question. Among 2000 answer sheets, there exists an $n$ such that in any $n$ answer sheets, there are 4 sheets where each pair of sheets has at most 3 answers in common. Find the minimum possible value of $n$.


[Solution 1] By the Pigeonhole Principle, among 2000 exam papers, there must be 500 papers with the same answer to the 1st question. Among these 500 papers with the same answer to the 1st question, there must be 125 papers with the same answer to the 2nd question. Among these 125 papers with the same answers to the 1st and 2nd questions, there must be 32 papers with the same answer to the 3rd question. Among these 32 papers with the same answers to the 1st, 2nd, and 3rd questions, among the 4 different answers to the 4th question, there must be 1 answer chosen by at most 8 people. Removing this answer, at least 24 papers remain, where the answers to the first 3 questions are the same, and the answers to the 4th question have only 3 different answers.

For any 4 of these 24 papers, by the Pigeonhole Principle, there must be two papers with the same answer to the 4th question, thus these two papers have the same answers to the first 4 questions, which does not meet the requirements of the problem. This shows that the minimum possible value of $n$ is no less than 25.

On the other hand, there are $4^5 = 1024$ different possible answers. Select 1000 of these, each repeated once, to get 2000 papers. Clearly, among any 25 of these papers, there are always 13 that are different.

Use $0,1,2,3$ to represent the 4 different answers to each question, and record the answers on each paper as $(g, h, i, j, k), g, h, i, j, k \in \{0,1,2,3\}$. Divide all different answers into the following 4 groups:
$$
\begin{array}{c}
S_{m}=\{(g, h, i, j, k) \mid g+h+i+j+k \equiv m(\bmod 4)\}, \\
m=0,1,2,3 .
\end{array}
$$

It is easy to see that any two different answers in the same group have at most 3 questions with different answers. By the Pigeonhole Principle, among the above 13 different answers, at least 4 belong to the same group, which of course meets the requirements of the problem. This shows that $n=25$ can meet the requirements of the problem.
In summary, the minimum possible value of $n$ is 25.
[Solution 2] As in Solution 1, use $(g, h, i, j, k)$ to represent the answers on a paper, where $g, h, i, j, k \in \{0,1,2,3\}$. Divide the following 4 answers into one group:
$$
\{(g, h, i, j, 0),(g, h, i, j, 1),(g, h, i, j, 2),(g, h, i, j, 3)\},
$$

Thus, all 1024 different answers are divided into 256 such quartets. For any 2000 papers, they belong to these 256 quartets. Since $2000 = 256 \times 7 + 208$, by the Pigeonhole Principle, it is possible to sequentially select 3 groups of 8 papers each, such that each group of 8 papers belongs to the same quartet. Among these 3 groups of 24 papers, any 4 papers must have two that belong to the same quartet, thus they cannot meet the requirements of the problem, which shows that the minimum possible value of $n$ is no less than 25.
On the other hand, let
$$
S=\{(g, h, i, j, k) \mid g+h+i+j+k \equiv 0(\bmod 4)\},
$$

It is easy to see that $|S|=256$ and any two different answers in $S$ have at most 3 questions with the same answers. From $S$, select 250 different answers and make each answer chosen by exactly 8 people, to get 2000 papers. Among any 25 of these papers, there are always 4 that are different, which of course meets the requirements of the problem. This proves that $n=25$ can meet the requirements of the problem.
In summary, the minimum possible value of $n$ is 25.