6.21 After 20 figure skaters perform, 9 judges rank the athletes from 1 to 20. It is known that the rankings given by different judges for each athlete do not differ by more than 3. Calculate the sum of the rankings each athlete receives, and arrange them in non-decreasing order: $C_{1} \leqslant C_{2} \leqslant C_{3} \leqslant \cdots \leqslant C_{20}$. Find the maximum value of $C_{1}$.
[Solution] Consider the distribution of 9 first places. If one athlete wins all 9 first places, then $C_{1}=9$. If two athletes share the 9 first places, then one of them must have won at least 5 first places, and his other ranks are at worst fourth, so $C_{1} \leqslant 1 \times 5+4 \times 4$ $=21$. If 3 athletes share the 9 first places, then due to their worst rank being fourth, the sum of ranks for the 3 athletes does not exceed 1 $\times 9+3 \times 9+4 \times 9=72$, so $C_{1}$ $\leqslant 24$. If 4 athletes share the 9 first places, then the total sum of ranks for the 4 athletes is 90, so $C_{1} \leqslant\left[\frac{90}{4}\right]=22$. It is evident that it is impossible for 5 or more athletes to share the 9 first places. Therefore, $C_{1} \leqslant 24$.

If the top 5 athletes $A, B, C, D, E$ receive ranks as shown in the right figure, and the other 15 athletes each receive 9 sixth places, seventh places, $\cdots$, twentieth places, then $C_{1}=24$. Thus, the maximum value of $C_{1}$ is 24.