Task 1 - 140811

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

Determine all solutions to the cryptogram, i.e., all possibilities for replacing the letters with digits such that all horizontal and vertical equations are satisfied! The same letters should represent the same digits, and different letters should represent different digits. Hint: The task is not to be solved by guessing alone, as is often the case in puzzle magazines; rather, considerations for the completeness and correctness of the solution should be provided.
(I) Assuming that for the digits assigned to the letters $A, \ldots, J$ the conditions of the problem are satisfied, it follows:

Since $A J$ and $B J$ are both two-digit numbers, and thus less than 100, their sum is less than 200. Since it is a three-digit number, its first digit is $A=1$. (1)

Since $A J < 100$, and $B J < 100$, it follows that $B J < 100 - A J < 100 - 10 = 90$, thus $B=9$. (2)

Since $A=1$, $A B C < 200$; and since $A B C : H = B J$, it follows that $H < 200 : 90 < 3$. Since $H$ is a leading digit, $H \neq 0$, and since $H \neq A=1$, it follows that $H=2$. (3)

Thus, $C H = H \cdot H A = 2 \cdot 21 = 42$, so $C=4$. (4)

Further, $B J = A B C : H = 194 : 2 = 97$, so $J=7$. (5)

Then, $D E = H A + A J = 21 + 17 = 38$, so $D=3$ (6) and $E=8$. (7)

Finally, $A F G = A B C - D E = 194 - 38 = 156$, so $F=5$ (8) and $G=6$ (9). Therefore, the solution to the cryptogram can only be (10):

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0666.jpg?height=186&width=714&top_left_y=1506&top_left_x=677)

(II) Since the assignments (1) to (9) assign different digits to the different letters A, ..., J, which, as can be seen from (10), correctly solve all horizontal and vertical problems of the cryptogram, the cryptogram has exactly the given solution.