## Task 1 - 160521

$$
\begin{array}{ccccc}
\mathrm{A} & \cdot & \mathrm{A} & = & \mathrm{B} \\
+ & & \cdot & & - \\
\mathrm{C} & \cdot & \mathrm{D} & = & \mathrm{E} \\
\hline \mathrm{F} & -\mathrm{G} & = & \mathrm{H}
\end{array}
$$

In the cryptogram above, digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are to be entered for the letters such that the same digits stand for the same letters and all the given arithmetic operations are correctly calculated.

Determine whether there is such an entry, whether it is the only one, and what it is in this case!
If there is such an entry, then according to the 1st line, $B$ is the square of $A(\neq B)$, so $B=4$ or $B=9$.

Furthermore, $B$ is the product of two single-digit numbers $C, D$, thus neither 0 nor 1. Therefore, none of the numbers $C, D$ is equal to 0 or 1.

Since $2 \cdot 3=6$, $E$ is at least 6. As $E$ is smaller than $B$ according to the 3rd column, $B=4$ is ruled out. It follows that $B=9$, so $A=3$.

Since $G(\neq 3)$ is thus three times $D(\neq 3)$, but greater than 0 and less than 10, the only possibility is $D=2, G=6$.

According to the third line, $F$ is greater than 6, so due to $F \neq B, B=9$, $F$ is either 7 or 8. According to the second line, $E$ is even, and according to the third column, $H$ is odd, so according to the third line, $F$ is odd. Therefore, $F=7$ and thus $H=1, E=8$, and $C=4$. Therefore, only the following entry can satisfy all conditions:

$$
\begin{array}{rrrr}
3 & \cdot & 3 & = \\
+ & \cdot & & - \\
4 & \cdot & 2 & = \\
\hline 7 & - & 6 & = \\
\hline
\end{array}
$$

It satisfies the conditions; because the digits entered for $A, B, C, D, E, F, G, H$ are 3, 9, 4, 2, 8, 7, 6, 1, all different, and the given arithmetic problems are correctly calculated.