Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:

(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.

Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.

[i]Proposed by YaWNeeT[/i]
To solve this problem, we need to determine the smallest number of evaluations of the polynomial \( f(x) \) that Alice needs to determine the parity of \( f(0) \). 

1. **Understanding the Polynomial Properties**:
   - \( f(x) \) is a polynomial of degree less than 187.
   - For any integer \( n \), \( f(n) \) is an integer.

2. **Lagrange Interpolation**:
   - We can use Lagrange interpolation to express \( f(x) \) in terms of its values at specific points.
   - The Lagrange interpolation formula for \( f(x) \) given its values at \( x = 1, 2, \ldots, 187 \) is:
     \[
     f(x) = \sum_{k=1}^{187} f(k) \prod_{\substack{1 \le i \le 187 \\ i \ne k}} \frac{x - i}{k - i}
     \]
   - To find \( f(0) \), we set \( x = 0 \):
     \[
     f(0) = \sum_{k=1}^{187} f(k) \prod_{\substack{1 \le i \le 187 \\ i \ne k}} \frac{-i}{k - i}
     \]

3. **Parity Consideration**:
   - We need to determine the parity (even or odd) of \( f(0) \).
   - The parity of \( f(0) \) depends on the parity of the coefficients in the Lagrange interpolation formula.

4. **Simplifying the Product**:
   - The product term \( \prod_{\substack{1 \le i \le 187 \\ i \ne k}} \frac{-i}{k - i} \) can be simplified modulo 2.
   - Notice that \( \prod_{\substack{1 \le i \le 187 \\ i \ne k}} (-i) \) is a product of 186 terms, each of which is an integer from 1 to 187.

5. **Parity of Binomial Coefficients**:
   - The key observation is that the parity of \( f(0) \) is determined by the parity of the binomial coefficients in the Lagrange interpolation formula.
   - Specifically, we need to consider the binomial coefficients modulo 2.

6. **Using Lucas' Theorem**:
   - Lucas' theorem helps us determine the parity of binomial coefficients modulo 2.
   - According to Lucas' theorem, the binomial coefficient \( \binom{n}{k} \) modulo 2 is determined by the binary representations of \( n \) and \( k \).

7. **Evaluating the Parity**:
   - We need to evaluate the parity of \( f(k) \) for \( k = 1, 2, \ldots, 187 \).
   - By evaluating \( f(k) \) for all \( k \) in the set \( \{1, 2, \ldots, 187\} \), we can determine the parity of \( f(0) \).

8. **Conclusion**:
   - Since the degree of \( f(x) \) is less than 187, evaluating \( f(x) \) at 187 distinct points is sufficient to determine the polynomial completely.
   - Therefore, the smallest positive integer \( N \) such that Alice always knows for sure the parity of \( f(0) \) is \( \boxed{187} \).