10. In a game, the "magician" asks a person to think of a three-digit number $(a b c)$ (where $a, b, c$ are the digits of the number in base 10), and asks this person to select 5 numbers $(a c b), (b a c), (b c a), (c a b)$, and $(c b a)$, and to find the sum $N$ of these 5 numbers, and tell the sum $N$ to the magician. Then the magician can reveal the number $(a b c)$ that the person thought of.

Now, let $N=3194$, please be the magician and find the number $(a b c)$.


10. Add (abc) to $N$, since $a$, $b$, and $c$ each appear exactly twice in every position, we get
$$
N+(a b c)=222(a+b+c) .
$$

Therefore, we should look for a multiple of 222, $222k$, which should be greater than $N$ (since $(abc) \neq 0$) and less than $N+1000$ (since $(abc)$ is a three-digit number), and make equation (1) hold, i.e., the sum of the digits of $222k - N$ equals $k$. If all these conditions are met, then $222k - N$ is the solution to the problem.
$$
\begin{array}{l}
\text { Since } 14 \cdot 222=3194 \\
+ \text { 1000, }
\end{array}
$$

So $k$ can only be chosen from 15, 16, 17, 18. Among these 4 numbers, only 16 satisfies the above conditions, i.e., $16 \cdot 222 - 3194 = 358$. And $3+5+8=16$, so $(abc) = 358$.