Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him the value of $a_{1}x_{1}+\cdots+a_{n}x_{n}$. Then Bob chooses another list of positive integers $b_{1}$, $\cdots$, $b_{n}$ and asks Alice for $b_{1}x_{1}+\cdots+b_{n}x_{n}$. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?
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   One round does not suffice because if Bob only asks for one sum, there could be multiple sets of \( x_i \) that produce the same sum. For example, if Alice's numbers are such that the result can be represented in more than one way as an integral linear combination of the given \( a_k \), Bob would not be able to uniquely determine the numbers. For instance, if \( x_1 = 2 \) and \( x_2 = 1 \), and Bob asks for \( a_1 x_1 + a_2 x_2 \) with \( a_1 = 1 \) and \( a_2 = 2 \), the result is \( 2 \cdot 1 + 1 \cdot 2 = 4 \), which could also be obtained with different values of \( x_1 \) and \( x_2 \).

Therefore, Bob needs exactly two rounds to determine Alice's numbers.

The final answer is \(\boxed{2}\).