6.53 16 students participate in a math competition. All the questions are multiple-choice, with 4 options for each question. After the exam, it is found that any two students have at most 1 question with the same answer. How many questions are there at most? Explain your reasoning.
[Solution 1] Use 16 points to represent 16 students, and connect a line between two points if the two students have the same answer to one question. Thus, there is at most one line between any two points, and the graph can have at most $C_{16}^{2}=120$ lines.

On the other hand, for the $i$-th question, let the number of people who choose answers $1,2,3,4$ be $m_{1}$, $m_{2}$, $m_{3}$, $m_{4}$, respectively, then $m_{1}+m_{2}+m_{3}+m_{4}=16$. The number of lines connected due to the same answer to the $i$-th question is
$$
\begin{aligned}
C_{m_{1}}^{2}+C_{m_{2}}^{2}+C_{m_{3}}^{2}+C_{m_{4}}^{2} & =\frac{1}{2}\left(m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}\right)-8 \\
& \geqslant \frac{1}{8}\left(m_{1}+m_{2}+m_{3}+m_{4}\right)^{2}-8=24 .
\end{aligned}
$$

Therefore, when there are $n$ questions, the number of lines is at least $24 n$, so we have
$$
24 n \leqslant 120 \text {. }
$$

From this, we know that there can be at most 5 questions.
When there are 5 questions, the answers in the table below meet the requirements of the problem. This set of answers is written in lexicographical order.
In summary, there can be at most 5 questions.
[Solution 2] From the table given in Solution 1, we know that there can be 5 questions.
On the other hand, since students who choose the same answer for a question must choose different answers for other questions, and there are only 4 options for each question, no more than 4 students can choose the same answer for each question. Since there are 16 students, each option for each question is chosen by exactly 4 people.

Therefore, for each student, there are 3 people who have the same answer as them for each question, and these groups of three have no common members. Thus, when there are $n$ questions, there are $3 n$ people who have the same answer as someone else for at least one question. Since $3 n \leqslant 15$, we have $n \leqslant 5$.
In summary, there can be at most 5 questions.