## Task 1 - 160621

Ludwig says: "I can obtain the reader count 58125 of the mathematical student magazine 'alpha' as the result of the addition problem

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0214.jpg?height=177&width=422&top_left_y=488&top_left_x=820)

by substituting digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) for the letters, with the same digits for the same letters and different digits for different letters, and knowing that $I < R$ and the digits $E H P L$ read in this order form the number 1976.

Which digits should be substituted for the letters so that all these conditions are met?"

Check whether the determined substitution satisfies all the conditions and whether there are any other such entries!
When in an assignment all statements are true, it follows from the statements about the ten-thousands digit that $A=5$. From the statements about the units digit, it follows that $I+R=10$.

Of the possible representations of 10 as the sum of two different single-digit numbers

$$
10=1+9=2+8=3+7=4+6
$$

those are excluded in which the digits have already been used for other letters than $I$ and $R$, i.e., $E=1, H=9, P=7, L=6$. Therefore, only the representation $10=2+8$ remains.

Since $I<R$, it follows that $I=2, R=8$. Because a carry of exactly 1 occurs in the addition of the tens digits, it follows from the statements about the hundreds digits that $T=4$.

Thus, only the assignment $A L P H A$ (56795) HEITER (912418) can meet all the requirements.

It meets these requirements; because the digits assigned to different letters are all different, $E H P L=1976$ and $I<R$, and the addition

$$
\begin{array}{r}
567995 \\
+\quad 59 \\
+\quad 518 \\
\hline
\end{array}
$$

yields the sum 58125.