$a b c$
Example 3.1.1 Three-digit addition $\frac{+d e f}{g h i}$ where $a, b, c, d, e, f, g, h, i$ are exactly composed of 1, $2,3,4,5,6,7,8,9$, find the solutions for the above addition that make $g h i$ the largest and smallest.

Analysis: One-digit addition has two cases: with carry and without carry. Suppose in the three-digit addition, there are $k$ carries $(k=0,1,2)$. First, explore the necessary conditions that the addition satisfying the conditions should meet.
Solution: According to the condition $g+h+i=a+b+c+d+e+f-9k$,
we get $2(g+h+i)=a+b+c+d+e+f+g+h+i-9k=\sum_{i=1}^{9} i-9k=45-9k$.
$k$ must be an odd number, $k=1$, thus $g+h+i=18$.
657
The maximum possible value of $ghi$ is 981, and $\frac{+324}{981}$, so the maximum is 981.
Since $g \geqslant a+d$, $g \geqslant 3$,
$g=3$ gives $369,378,387,396$. In this case, $c+f$ must carry over, which is impossible.
286
$g=4$ has the smallest value of 459, and there is a solution $\frac{+173}{459}$, so the minimum is 459.