In the following multiplication, each letter represents a digit. The same letter always represents the same digit, and different letters represent different digits.

$$
\begin{gathered}
a b c d \cdot e f d g \\
\hline d e h g e \\
h f e f \\
a b c d \\
d f a a g \\
\hline d h h i f b b g
\end{gathered}
$$

What is the value of each letter?
I. $\mathbf{d}=\mathbf{1}$, because the third partial product is identical to the multiplicand.

II. $f \cdot a<10$, because the second partial product is a four-digit number. This implies that $a<5$ and $f<5$.

III. $h+h+$ remainder $=h$. But the remainder, based on (I) and (II), is either 1 or 2, and thus $h$ is either 9 or 8.

IV. If $h=8$, then $e=7$, and thus the first partial product dehge $=178 g 7$, which is only divisible by 7 if $g=5$. In this case, $a b c d=2551$, but this contradicts the fact that $b \neq c$.

V. Therefore, $\mathbf{h}=\mathbf{9}$, and thus $\mathbf{e}=\mathbf{8}$.

VI. The first partial product: $189 g 8$ is divisible by 8, so $\mathbf{g}=\mathbf{6}$.

VII. The multiplicand is therefore one-eighth of 18968, which means $a b c d=2371$, and thus

$$
\mathbf{a}=\mathbf{2}, \quad \mathbf{b}=\mathbf{3}, \quad \mathbf{c}=\mathbf{7}
$$

VIII. Since $f \neq 0$, and $f<5$, $f=4$. Therefore, the multiplication is

|  | 2 | 3 | 7 | 1 |  | 8 | 4 | 1 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 8 | 9 | 6 | 8 |  |  |  |  |  |
|  |  | 9 | 4 | 8 | 4 |  |  |  |  |
|  |  |  | 2 | 3 | 7 | 1 |  |  |  |
|  |  |  | 1 | 4 | 2 | 2 | 6 |  |  |
| 1 | 9 | 9 | 5 | 4 | 3 | 3 | 6 |  |  |

Accordingly, $\mathbf{i}=5$