6. (20 points) Does there exist a natural number whose square is equal to the sum of five pairwise distinct squares of integers, such that among them there is the number $7^{2}$?
Solution. Let's show a possible way to construct the desired number. We will use the formula $n^{2}+\left(\frac{n^{2}-1}{2}\right)^{2}=\left(\frac{n^{2}+1}{2}\right)^{2}$. That is,

$$
7^{2}+24^{2}=25^{2}, 25^{2}+312^{2}=313^{2}, 313^{2}+48984^{2}=48985^{2}
$$

Therefore, $48985^{2}=0^{2}+7^{2}+24^{2}+312^{2}+48984^{2}$.

Answer: Yes, for example, $48985^{2}=0^{2}+7^{2}+24^{2}+312^{2}+48984^{2}$.

| Content of the criterion | Points |
| :--- | :---: |
| The correct answer is obtained in a justified manner. | 20 |
| In the correct and justified solution process, there is a logical error or the solution is insufficiently justified. | 14 |
| The beginning of a possible solution to the problem is provided, some intermediate results are obtained, further solution is incorrect or absent. | 6 |
| The solution does not meet any of the criteria listed above. | 0 |
| Maximum score | 20 |