10. In a competition with 20 singers, 9 judges respectively assign them ranks from 1 to 20. It is known that for each singer, the difference between any two ranks does not exceed 3. If the sum of the ranks each singer receives is arranged in an increasing sequence: $C_{1} \leqslant C_{2} \leqslant \cdots \leqslant C_{20}$, then the maximum value of $C_{1}$ is . $\qquad$
10. 24.

If 9 judges all give a singer the first place, then $C_{1}=9$.

If two singers both get the first place, then one of them gets no less than 5 first places, while the other 4 ranks are no higher than fourth place, so $C_{1} \leqslant 5 \times 1+4 \times 4=21$.

If three singers all get the first place, then their other ranks are no higher than fourth place, and their total ranks do not exceed $1 \times 9+3 \times 9+4 \times 9=72$, so $C_{1} \leqslant 24$.

If four singers 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 $C_{1} \leqslant 24$.

The situation where five or more singers all get the first place is impossible, so $C_{1} \leqslant 24$.
Here is an example where $C_{1}=24$:
The judges give the three singers with rank sums $C_{1} 、 C_{2} 、 C_{3}$ the ranks $1,1,1,3,3,3,4,4,4$;

The judges give the three singers with rank sums $C_{4} 、 C_{5} 、 C_{6}$ the ranks $2,2,2,5,5,5,6,6,6$;

While the judges give the rest of the singers ranks between 7 and 20.
At this point, $C_{1}=24$.