## Task 1 - 220911

Uwe says to Gert: "I have a two-digit number $z$, both of whose digits are different from 0. If I write these digits in reverse order and append the cross sum of $z$, I get the square of $z$."

Gert finds a number $z$ with these properties without using a multiplication table.

Show that the number $z$ can be uniquely determined from Uwe's statements without using a multiplication table, and give $z$!
Let the tens and units digit of a number $z$ with the required properties be $a$ and $b$, respectively. Then the sum of the digits $a+b$ has the same units digit as the number $z^{2}$ and therefore also as $b^{2}$. Thus, $a$ must be the units digit of $b^{2}-b$.

Apart from $b=0$, $b=1,5,6$ are also excluded, as they would lead to $a=0$. For the remaining possibilities, the following table checks whether $z^{2}$ is the number $z^{\prime}$ obtained by concatenating $b, a$, and $a+b$:

| $b$ | $b^{2}$ | $b^{2}-b$ | $a$ | $z$ | $z^{2}$ | $z^{\prime}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 2 | 4 | 2 | 2 | 22 | 484 | 224 |
| 3 | 9 | 6 | 6 | 63 | 3969 | 369 |
| 4 | 16 | 12 | 2 | 24 | 576 | 426 |
| 7 | 49 | 42 | 2 | 27 | 729 | 729 |
| 8 | 64 | 56 | 6 | 68 | 4624 | 8614 |
| 9 | 81 | 72 | 2 | 29 | 841 | 9211 |

Thus, $z=27$ is the only solution identified.