10. 4. At a round table sit 30 people, each of whom is either a fool or a wise person. Ask all the people present the question: “Is the person to your right a wise person or a fool?” Wise people answer correctly, while fools may answer either correctly or incorrectly. It is known that the number of fools does not exceed $F$. Find the largest $F$ for which it is always possible to identify a wise person.
10. 4. $F=8$.

If $F=0$, then anyone sitting around the table can be pointed out. Now, let's assume $F \neq 0$. Group all the people sitting around the table: consecutive wise men form one group, and consecutive fools form another group, resulting in a total of $2k$ groups ($k$ groups of wise men, $k$ groups of fools). The number of wise men in the $i$-th group is denoted as $w_{i}$, and the number of fools in the $i$-th group is denoted as $f_{i} (1 \leqslant i \leqslant k)$. Therefore,
$$
w_{1}+w_{2}+\cdots+w_{k}+f_{1}+f_{2}+\cdots+f_{k}=30,
$$

and $f_{1}+f_{2}+\cdots+f_{k} \leqslant F$.
Consider the sequence formed by the people who consecutively answer “wise man” and the last person $x$ who says they are a wise man. A group of $w_{i}$ wise men gives a sequence of length not less than $w_{i}-1$, and $x$ is indeed a wise man. If $x$ is the $i$-th fool in a group of fools, then the length of the sequence does not exceed $f_{i}-1$. Therefore, if $\max _{i} w_{i}>\max _{i} f_{i}$, it can be concluded that the last person in the longest sequence of answers “wise man” is indeed a wise man. Because
$$
\begin{array}{c}
\max _{i} w_{i} \geqslant \frac{30-\left(f_{1}+\cdots+f_{k}\right)}{k} \geqslant \frac{30-F}{k} ; \\
\max _{i} f_{i} \leqslant\left(f_{1}+\cdots+f_{k}\right)-\dot{k}+1 \leqslant F-k+1,
\end{array}
$$

for $1 \leqslant k \leqslant F$, when $\frac{30-F}{k}>F-k+1$, a wise man can be pointed out among those sitting around the table. The above inequality is equivalent to
$$
\begin{array}{l}
k^{2}-(F+1) k+30-F>0 . \\
\text { When } F=9 \text {, it is not }
\end{array}
$$

certain that a wise man can be pointed out. In fact, consider the group of people sitting around the table (the figure shows the responses: $w$ for wise man, $f$ for fool; fools are marked with small black dots) at angles of $120^{\circ}, 180^{\circ}, 240^{\circ}, 200^{\circ}$. It is not difficult to prove that the sequence of responses remains the same after rotating these angles. Therefore, it is impossible to identify a wise man based on the known responses in this group.