Example 8 After 20 figure skaters have performed, 9 judges each assign them ranks from 1 to 20. It is known that for each skater, the difference between any two ranks does not exceed 3. If the sum of the ranks each skater receives is arranged in an increasing sequence: $C_{1} \leqslant C_{2} \leqslant \cdots \leqslant C_{20}$, find the maximum value of $C_{1}$.
(2nd All-Union Mathematical Olympiad)
If 9 judges all give a certain athlete the first place, then $C_{1}=9$. If two athletes are both judged as the first place, then one of them gets no less than 5 first places, and the other 4 ranks no higher than fourth. Therefore, $C_{1}=5 \times 1+$ $4 \times 4=21$.

If 3 athletes all get the first place, then their other ranks do not exceed fourth, and their total ranks do not exceed $1 \times 9+3 \times 9+4 \times 9=72$, so $C_{1} \leqslant 24$.

If 4 athletes all get the first place, then their total ranks do not exceed $1 \times 9+2 \times 9+3 \times 9+4 \times$ $9=90$, so the rank of one of them is no more than 24. The case where 5 or more athletes all get the first place is impossible. Therefore, $C_{1} \leqslant 24$.
Now, an example where $C_{1}=24$ is given:
The judges give the top 3 athletes the ranks
$$
1,1,1,3,3,3,4,4,4 \text {; }
$$

The ranks given to the last three athletes are
$$
2,2,2,5,5,5,6,6,6 \text {; }
$$

The ranks of the remaining athletes are given arbitrarily between the two. This way, $\left(C_{1}\right)_{\text {max }}=24$.