11. A three-digit number in base seven, when expressed in base nine, has its digits in reverse order of the original number. Find this number.
Let the three digits of this number in base 7 be $x$, $y$, and $z$. According to the problem, we have
$$
49 x + 7 y + z = 81 z + 9 y + x,
$$

which simplifies to $24 x - y - 40 z = 0$,
$$
y = 8(3 x - 5 z).
$$

Since $y$ must be less than 7 and $3 x - 5 z$ is an integer, $y = 0$ and $3 x - 5 z = 0$ must hold for the equation to be true. Thus, $\frac{x}{5} = \frac{z}{3} = k$, which means $x = 5 k$ and $z = 3 k$. Since $x$ and $z$ must both be less than 7, $k = 1$, so we get $x = 5$, $y = 0$, and $z = 3$.