Anna: I thought of two natural numbers, $x$ and $y$. You have to guess both with a single question. For this, you can perform operations with me involving $x$, $y$, and given integers, and I will tell you the final result.

Balázs: Can I use exponentiation?

Anna: No. Only addition, subtraction, and multiplication are allowed. For example, you can ask for the value of $x(y+2)-(3 x-1) y$. Is the task solvable? Can Balázs guess the two numbers with a single question?
I. solution. Balázs can guess Anna's numbers with a single question if he asks for the value of

$$
B(x, y)=(x+y)(x+y+1)-y=(x+y)^{2}+x
$$

Since $x$ and $y$ are natural numbers, therefore

$$
(x+y)^{2} \leq B(x, y) \text{ and } B(x, y) \geq 0 \left(\binom{1}{2}=\binom{0}{1}=0\right)$. In the solution, Balázs uses this result with the choice $a=x+y+1, b=x$, specifically, he "retrieves" the values of $a$ and $b$ from the calculated $m$ value for this number pair. In fact, he does not need the table, as for a given natural number $m$, he needs to calculate the largest number of the form $\binom{a}{2}$ that is not greater than $m$, similarly to how in the first solution he had to determine the largest square number not greater than $m$ from $m=a^{2}+b$ (where $a=x+y \geq x=b$).

5. The second solution can also be generalized, as it is true - and can be easily proven by complete induction - that for a given $n$, every natural number $m$ can be uniquely written as

$$
m=\binom{a_{n}}{n}+\binom{a_{n-1}}{n-1}+\ldots+\binom{a_{i}}{i}
$$

where $a_{n}>a_{n-1}>\ldots>a_{i} \geq i \geq 1$.

Then, with the choice

$$
\begin{aligned}
a_{n} & =x_{1}+x_{2}+\ldots+x_{n}+n \\
a_{n-1} & =x_{1}+x_{2}+\ldots+x_{n-1}+n-1 \\
\quad \vdots & \\
\quad a_{2} & =x_{1}+x_{2}+2 \\
a_{1} & =x_{1}+1
\end{aligned}
$$

the values of $a_{n}, a_{n-1}, \ldots, a_{2}, a_{1}$ can be determined from the value of $m$, and from there the values of $x_{1}, x_{2}, \ldots, x_{n}$ can also be determined.