In each of the corner fields of the outer square, one of the numbers $2, 4, 6,$ and $8$ should be written, with different numbers in different fields. In the four fields of the inner square, the products of the numbers from the adjacent fields of the outer square should be written. In the circle, the sum of the numbers from the adjacent fields of the inner square should be written.

Which numbers can be written in the circle? Determine all possibilities.

(M. Dillingerová)

![](https://cdn.mathpix.com/cropped/2024_04_17_a44687afde4ea037c0fdg-2.jpg?height=400&width=400&top_left_y=594&top_left_x=834)

Hint. Can different fillings of the corner fields lead to the same sum in the circle?
The numbers 2, 4, 6, and 8 can be written in the corner fields in many ways. However, for different arrangements, we can end up with the same sum in the circle, as in the following cases:
![](https://cdn.mathpix.com/cropped/2024_04_17_a44687afde4ea037c0fdg-2.jpg?height=390&width=1308&top_left_y=1318&top_left_x=380)

The first and second cases differ by a $90^{\circ}$ rotation, the second and third are symmetric with respect to the horizontal axis of the square, etc. Different values in the circle will thus be obtained only when the numbers in the corner fields are not symmetric in any way.

We can simplify trying all possibilities by assuming one of the numbers, for example, 2, in a specific field, such as the top left. The other numbers are then filled in so that no two fillings are symmetric. The only symmetry that preserves the 2 in the top left field is the symmetry with respect to the diagonal passing through this field. With these requirements, we have the following three solutions:
![](https://cdn.mathpix.com/cropped/2024_04_17_a44687afde4ea037c0fdg-2.jpg?height=394&width=1310&top_left_y=2187&top_left_x=378)

The numbers 84, 96, and 100 can be written in the circle.

Note. For each filling of the corner fields, we can find 7 other fillings that are symmetric in some way. All fillings can thus be divided into eights, which certainly have the same sums in the circle. The numbers in the four corner fields can be filled in 24 ways in total. Therefore, the circle can contain at most $24: 8=3$ different numbers. By testing, we determine which numbers these are and that they are different from each other.