Example 2: 9 judges score 12 athletes participating in a bodybuilding competition. Each judge gives 1 point to the athlete they consider to be in 1st place, 2 points to the athlete in 2nd place, ..., and 12 points to the athlete in 12th place. The final scoring shows: the difference between the highest and lowest scores of each athlete's nine scores is no more than 3. Let the total scores of each athlete be $c_{1}, c_{2}, \cdots, c_{12}$, and $c_{1} \leqslant c_{2} \leqslant \cdots \leqslant c_{12}$. Find the maximum value of $c_{1}$.
(2004, Jiangsu Province Junior High School Mathematics Competition (Grade 9))
Explanation: It is impossible for 9 judges to give 1 point to five or more athletes, because among five or more athletes, at least one athlete must be rated no less than 5 by a judge. However, according to the problem, each of these five athletes is rated no more than 4 by each judge, which is a contradiction.

Therefore, 9 judges can give 1 point to at most four athletes.
We will discuss the following scenarios.
(1) If all judges give 1 point to a single athlete, then \( c_{1} = 9 \).
(2) If the nine 1 points given by the 9 judges are concentrated on two athletes, then one of these athletes must be rated 1 by at least five judges. According to the problem, the remaining judges give this athlete a score no greater than 4, thus,
\[
c_{1} \leqslant 5 \times 1 + 4 \times 4 = 21.
\]
(3) If the nine 1 points are concentrated on three athletes, then the total score of these three athletes is no more than
\[
9 \times 1 + 9 \times 3 + 9 \times 4 = 72.
\]
Thus, \( 3 c_{1} \leqslant c_{1} + c_{2} + c_{3} \leqslant 72 \).
Hence, \( c_{1} \leqslant 24 \).
(4) If the nine 1 points are distributed among four athletes, then the total score of these four athletes is
\[
9 \times 1 + 9 \times 2 + 9 \times 3 + 9 \times 4 = 90.
\]
Thus, \( 4 c_{1} \leqslant 90 \). Hence, \( c_{1} < 23 \).
In summary, \( c_{1} \leqslant 24 \).
The scenario where \( c_{1} = 24 \) is achievable. Let \( A_{i} (i = 1, 2, \cdots, 12) \) represent the athletes, and \( B_{j} (j = 1, 2, \cdots, 9) \) represent the judges. The values in Table 1 are the scores given by judge \( B_{j} \) to athlete \( A_{i} \).
Table 1
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline & \( A_{1} \) & \( A_{2} \) & \( A_{3} \) & \( A_{4} \) & \( A_{5} \) & \( A_{6} \) & \( A_{7} \) & \( A_{8} \) & \( A_{9} \) & \( A_{10} \) & \( A_{11} \) & \( A_{12} \) \\
\hline \( B_{1} \) & 1 & 4 & 3 & 2 & 5 & 6 & 7 & 9 & 10 & 8 & 11 & 12 \\
\hline \( B_{2} \) & 1 & 4 & 3 & 2 & 5 & 6 & 7 & 9 & 10 & 8 & 11 & 12 \\
\hline \( B_{3} \) & 1 & 4 & 3 & 2 & 5 & 6 & 7 & 9 & 10 & 8 & 11 & 12 \\
\hline \( B_{4} \) & 4 & 3 & 1 & 5 & 2 & 7 & 9 & 6 & 8 & 11 & 10 & 12 \\
\hline \( B_{5} \) & 4 & 3 & 1 & 5 & 2 & 7 & 9 & 6 & 8 & 11 & 10 & 12 \\
\hline \( B_{6} \) & 4 & 3 & 1 & 5 & 2 & 7 & 9 & 6 & 8 & 11 & 10 & 12 \\
\hline \( B_{7} \) & 3 & 1 & 4 & 2 & 5 & 9 & 6 & 7 & 11 & 10 & 8 & 12 \\
\hline \( B_{8} \) & 3 & 1 & 4 & 5 & 2 & 9 & 6 & 7 & 11 & 10 & 8 & 12 \\
\hline \( B_{9} \) & 3 & 1 & 4 & 2 & 5 & 9 & 6 & 7 & 11 & 10 & 8 & 12 \\
\hline Total & 24 & 24 & 24 & 30 & 33 & 66 & 66 & 66 & 87 & 87 & 87 & 108 \\
\hline
\end{tabular}