In all nine fields of the diagram, natural numbers are to be filled in such a way that the following conditions are met:

- each of the numbers $2, 4, 6$, and 8 is used at least once,
- four of the fields in the inner square contain the products of the numbers in the adjacent fields of the outer square,
- in the circle is the sum of the numbers in the adjacent fields of the inner square.

Determine which is the smallest and which is the largest number that can be written in the circle.

(M. Dillingerová)

![](https://cdn.mathpix.com/cropped/2024_04_17_5752a8a8cd9a6cffa1f9g-1.jpg?height=395&width=411&top_left_y=859&top_left_x=820)

Hint. Can the number in the circle be less than 20, or greater than 20000?
For a sufficiently large number in one of the corner fields of the outer square, the corresponding products in the inner square can be greater than any arbitrarily chosen number. Therefore, the sum in the circle can also be arbitrarily large.

We will determine the smallest sum that can be in the circle. It certainly cannot be any of the prescribed numbers: even the largest of them (8) is not greater than the sum of the remaining three $(2+4+6=12)$. The sum $\mathrm{v}$ in the circle, therefore, must be greater than or equal to the sum of all prescribed numbers, i.e., $2+4+6+8=20$.

If the sum $\mathrm{v}$ in the circle were 20, then the prescribed numbers would have to be in four adjacent fields of the small square. Since one of these numbers is 6, one of the adjacent fields of the outer square would have to be 6 or 3, and this would also have to be a divisor of the second adjacent field in the inner square. However, none of the numbers 2, 4, and 8 have such a divisor, so this possibility cannot occur.

The sum in the circle, therefore, must be greater than or equal to 21. The following filling demonstrates that the smallest number that can be written in the circle is 21.

![](https://cdn.mathpix.com/cropped/2024_04_17_5752a8a8cd9a6cffa1f9g-1.jpg?height=397&width=408&top_left_y=2183&top_left_x=824)

Note. Notice that the number in the circle is always the product of the sums of pairs of opposite numbers in the outer square. This insight can also be used in solving the problem.

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