1108*. Two natural numbers are guessed. Mathematician A was told their sum, and Mathematician B was told the sum of their squares. If they combined their information, they would, of course, find both numbers. However, they discovered these numbers as a result of the following conversation:

B: - I don't know what these numbers are.

A: - Their sum is not less than 10.

B: - Then I know what these numbers are.

What are these numbers?
$\triangle$ Let the sought natural numbers be denoted by $x$ and $y$.

Since Mathematician V initially did not know what these numbers were, there exists another pair of natural numbers $a$ and $b$ such that

$$
x^{2}+y^{2}=a^{2}+b^{2}
$$

These pairs differ in that

$$
x+y \geqslant 10, \quad a+b<10
$$

Then $(x+y)^{2} \geqslant 100$. We obtain:

$$
\begin{aligned}
& \left(x^{2}+y^{2}\right)+\left(x^{2}+y^{2}\right) \geqslant x^{2}+y^{2}+2 x y=(x+y)^{2} \geqslant 100, \\
& 2\left(x^{2}+y^{2}\right) \geqslant 100, \quad x^{2}+y^{2} \geqslant 50 .
\end{aligned}
$$

On the other hand, $(a+b)^{2}<100$. We will have:

$$
50 \leqslant x^{2}+y^{2}=a^{2}+b^{2}<(a+b)^{2}<100 .
$$

From the obtained inequality

$$
50 \leqslant(a+b)^{2}<100
$$

it follows that

$$
5 \sqrt{2} \leqslant a+b<10
$$

Therefore, $a+b=8$ or $a+b=9$. Let's consider both possibilities.

1) Suppose $a+b=8$.

Then $a$ can take the values $1,2,3,4,5,6,7$, and $b$ can take the respective values $7,6,5,4,3,2,1$. Thus, the sum $a^{2}+b^{2}$ can only take the values 50, 40, 32, and 32. But since $x^{2}+y^{2} \geqslant 50$, then $a^{2}+b^{2} \geqslant 50$, so only $a^{2}+b^{2}=50$ fits. However, the number 50 can be represented as the sum of the squares of two natural numbers in only one way: $50=7^{2}+1^{2}$. (The case $50=5^{2}+5^{2}$ is impossible, since in this case $a=b=5$, but the inequality $a+b<10$ must hold.) Therefore, the possibility $a+b=8$ is ruled out.

2) Suppose $a+b=9$.

In the same way, we find that $a=8, b=1$ (or $a=1, b=8$). In this case,

$$
8^{2}+1^{2}=65=7^{2}+4^{2} .
$$

Thus, the numbers 7 and 4 are the sought ones.

Answer: 7 and 4.