10. Xiao Zhi scored above 80 points in an American middle school math test. He told Xiao Jiang his score, and Xiao Jiang could correctly deduce how many questions Xiao Zhi answered. If Xiao Ma scored a bit lower but still above 80 points, Xiao Jiang would not be able to deduce it. How many points did Xiao Ma score?
(This test has 30 multiple-choice questions, and the scoring formula is $\mathrm{S}=30+4 \mathrm{c}-\mathrm{w}$, where $\mathrm{S}$ is the score, $\mathrm{c}$ is the number of correct answers, and $\mathrm{w}$ is the number of wrong answers, with no points for unanswered questions)
Solve for the minimum such that $S=30+40-w>80$. The problem requires finding the smallest such $S$, for which the corresponding $c$ is unique. First, note that when $c$ increases by 1, $w$ decreases by 4, and the value of $S$ remains unchanged, but it must satisfy $(c+1)+(w+4) \leqslant 30$, i.e., $c+w \leqslant 25$. When $c+w \leqslant 25$, for each $S$, the value of $c$ cannot be uniquely determined. Therefore, only when $c+w \geqslant 26$ (1) can $c$ be uniquely determined. Secondly, it should be that $w \leqslant 3$ (2), otherwise, if $w>3$, $w$ can be reduced by 4 and $c$ by 1, and the value of $S$ remains unchanged (since from $S>80$ we get $c \geqslant 13$, which is also possible). Thus, to make $S$ as small as possible, we should, within the range allowed by inequalities (1) and (2), minimize $c$ and maximize $w$. This leads to $w=3, c=23$. Therefore, $S=30+4 \times 23-3=119$, meaning the horse scored 119 points.