11. (12 points) In the animal kingdom, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 tigers, 100 foxes, and 100 monkeys, divided into 100 groups, with each group consisting of 3 animals, exactly 2 of one kind and 1 of another. After the groups are formed,
the Kung Fu Panda asks each animal, “Is there a tiger in your group?”, and exactly 138 animals answer “yes”;
the Kung Fu Panda then asks each animal, “Is there a fox in your group?”, and exactly 188 animals answer “yes”.
How many monkeys told the truth both times? $\qquad$
【Solution】Solution: Let the number of monkeys with the tigers be $x$, the number of foxes with the tigers be $y$, the number of lying monkeys among the monkeys with the tigers be $m$ $(m \leqslant x)$, the number of lying monkeys among the monkeys with the foxes be $n$ $(n \leqslant 100-x)$, and the number of tigers with the monkeys be $z$, then $(x-m)+(100-y)+n=38$ (1) $m+(100$ $-x-n)+(100-z)=188$ (2),
Adding (1) and (2) and rearranging gives $z=74-y(3)$,

So $x$ monkeys are with $(74-y)$ tigers, $y$ foxes are with $(y+26)$ tigers, and $(100-x)$ monkeys are with $(100-y)$ foxes,

Since there are only 2 types of 3 animals in each group,
So $x \leqslant 2(74-y), y+26 \leqslant 2 y, (100-x) \leqslant 2(100-y)$,
So $100 \leqslant 348-4 y$, so $y \leqslant 62$,
So $100-y \geqslant 38$,
So $(x-m)+(100-y)+n \geqslant 38$ (equality holds if and only if $x=m, n=0$),
Combining (1)(2)(3) gives $y=62, z=12$,
Since $x \leqslant 2(74-y), (100-x) \leqslant 2(100-y)$,
So $x=24$,
So the number of monkeys telling the truth is $100-24=76$.
The grouping method is 24 monkeys and 12 tigers together, a total of 12 groups, 62 foxes and 88 tigers together, a total of 50 groups, 76 monkeys and 38 foxes together, a total of 38 groups,
Kung Fu Panda asks each animal, "Are there tigers in your group?", and exactly 138 animals answer "yes", indicating that 100 tigers and 38 foxes answered "yes"; 76 monkeys answered no;
Kung Fu Panda then asks each animal, "Are there foxes in your group?", and exactly 188 animals answer "yes". This indicates that 24 monkeys, 88 tigers, and 76 monkeys answered "yes", so the answer is 76.