15. Place the numbers 1, 2, $3, \cdots, 9$ into the 9 circles in Figure 4, such that the sum of the numbers in the three circles on each side of $\triangle ABC$ and $\triangle DEF$ is 18.
(1) Provide one valid arrangement;
(2) How many different arrangements are there? Prove your conclusion.
15. (1) Figure 9 gives one arrangement that meets the requirements.
(2) There are 6 different ways to fill in the numbers.

Let the sum of the three numbers in circles $A, B, C$ be $x$; the sum of the three numbers in circles $D, E, F$ be $y$; and the sum of the remaining three circles be $z$. Clearly, we have
$$
x+y+z=1+2+\cdots+9=45.
$$

In Figure 9, the sum of the numbers in the three circles on each of the six edges is 18, so we have
$$
z+3 y+2 x=6 \times 18=108.
$$
Subtracting (1) from (2) gives
$$
x+2 y=108-45=63.
$$

Adding the sums of the numbers in the three circles on each of the edges $AB, BC, CA$, we get
$$
2 x+y=3 \times 18=54.
$$

Solving equations (3) and (4) simultaneously, we get $x=15, y=24$, and thus $z=6$.
Among the numbers $1,2, \cdots, 9$, the only three numbers that sum to 24 are $7, 8, 9$, so the circles $D, E, F$ can only be filled with the numbers $7, 8, 9$, and there are 6 different ways to do this.

Clearly, once the numbers in these three circles are determined, the numbers in the remaining six circles are also determined according to the problem's requirements.
Therefore, the conclusion is that there are 6 different ways to fill in the numbers.