## Task 1 - 100821

The natural numbers from 1 to 9 are to be entered into the nine fields $A, B, C, D, E, F, G, H, K$ of the adjacent figure, each exactly in one of the fields, such that the sums $s_{1}, s_{2}$, and $s_{3}$ of the numbers in the fields $A, B, C, D$ or $D, E, F, G$ or $G, H, K, A$ are equal to each other.

a) What is the smallest and what is the largest value that these (equal) sums can take under the given conditions?

b) Give one way each in which this smallest and this largest value can be achieved!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0638.jpg?height=397&width=371&top_left_y=430&top_left_x=1365)
Let the numbers entered into the fields $A, B, C, D, E, F, G, H, K$ be denoted by $a, b, c, d, e, f, g, h, k$. Then, for

$$
s_{1}=a+b+c+d, \quad s_{2}=d+e+f+g, \quad s_{3}=g+h+k+a
$$

according to the problem statement, $s_{1}=s_{2}=s_{3} \quad$ (2) and

$$
a+b+c+d+e+f+g+h+k=45
$$

From (1), (2), and (3), it follows that $3 s_{1}=s_{1}+s_{2}+s_{3}=45+a+d+g$.

Therefore, $3 s_{1}$ and consequently $s_{1}$ are smallest (or largest) when the same applies to the sum $a+d+g$. The smallest (or largest) sum that can be formed from three different natural numbers 1, ..., 9 is $1+2+3=6$ (or $7+8+9=24$).

Thus, the smallest value of $s_{1}$ cannot be less than $(45+6): 3=17$ (or the largest cannot be greater than $(45+24): 3=23$).

If one can now find at least one of the entries described in the problem statement where $s_{1}=17$ (or $s_{1}=23$), then on the one hand, it is shown that these two values are indeed the smallest and largest values of $s_{1}$, and on the other hand, two such possibilities have been found as required in b).

Two such entries are, for example:

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0638.jpg?height=360&width=734&top_left_y=1733&top_left_x=664)