10. (12 points) In the $3 \times 3$ table of Figure 1, numbers have already been filled in. Choose a black cell as the starting point. If the number in this black cell and the number in an adjacent white cell undergo one of the four arithmetic operations (addition, subtraction, multiplication, or division, with the larger number always first), and the result is an integer multiple of the number in another adjacent black cell, you can move from this white cell to the next black cell. Each cell must be visited exactly once. (For example, in Figure 2, from 7 through 8 to 5, and the direction of the arrows in Figure 2 is a correct path). Find the correct path in Figure 1. If the first 3 cells in the correct path are filled with $A$, $B$, and $C$ respectively, then the three-digit number $\overline{\mathrm{ABC}}=$ $\qquad$
【Solution】Solution: Connect all possible paths with arrows, as shown in the figure. Among them, 1 has only one path to choose from, and the adjacent 8 has only two paths:
(1) If you take the $8 \rightarrow 3 \rightarrow 5$ direction, the next $\rightarrow 6 \rightarrow 7 \rightarrow 2$ is the only one, 9 and 4 cannot be connected; if you take the $5 \rightarrow 3 \rightarrow 8$ direction, 7 and 6 cannot be connected, so you cannot take a detour.
(2) If you go straight, then the first 3 grids are 834, and there are more than one paths afterward, for example: $8 \rightarrow 3 \rightarrow 4 \rightarrow 9 \rightarrow 5 \rightarrow 1$ $\rightarrow 6 \rightarrow 7$.

Therefore, the answer is 834.