To solve this problem, we need to determine the minimum number of guesses a contestant needs to guarantee correctly identifying the leader’s $n$-digit binary string, given the constraints on how the strings can differ.

### Explanation

1. Binary Strings and Hamming Distance:
The problem involves binary strings of length $n$ and the concept of Hamming distance, which measures the number of positions at which two strings differ. Specifically, the deputy leader lists all binary strings that differ in exactly $k$ positions from the leader’s string.

2. Determining the Guesses:
- If $n = 2k$, each string written by the deputy is unique in the sense that no other strings exist which differ by exactly $k$ positions. Therefore, the contestant must check two strings. The strings could potentially be symmetric—a situation arising from the properties of binary complements within a set of size $2k$. It results in no single binary string having a unique distance of $k$ from every other string without ambiguity.
- If $n \neq 2k$, there is less ambiguity. In such a scenario, examining one potential candidate string suffices because, among the possibilities, one and only one string will be the solution (the leader's string), ensuring the contestant can identify the leader's string definitively with a single guess.

3. Mathematical Justification:
- **Case $n = 2k$**: The contestant faces a symmetrical setup where the total number of different strings at a distance of $k$ equals the remaining possibilities of leader's string setup, necessitating 2 guesses.
- **Case $n \neq 2k$**: Asymmetry in possible distances allows the contestant to uniquely identify the leader's binary string with just 1 guess since a single guess resolves all possibilities.

Based on these assessments, the minimum number of guesses required is:
$$\boxed{2 \text{ if } n = 2k, \text{ and } 1 \text{ otherwise}}$$
This answer effectively accounts for all potential configurations of the problem, ensuring the uniqueness needed for a definitive guess regarding the leader’s string.