1. The leader of an IMO team chooses positive integers $n, k (n > k)$, and informs the deputy leader and the contestant. Then, the leader secretly tells the deputy leader a binary number string of $n$ digits. As a result, the deputy leader writes down all binary number strings of $n$ digits that differ from the leader's $n$-digit string in exactly $k$ positions (if $n=3, k=1$, and the leader chose 101 to tell the deputy leader, then the deputy leader should write $001, 111, 100$). The contestant is allowed to see the strings written by the deputy leader and must guess the leader's string. Find the minimum number of guesses (depending on $n, k$) to ensure a correct answer.
1. When $n=2 k$, the minimum number of guesses is 2; when $n \neq 2 k$, the minimum number of guesses is 1.

Let the binary number string chosen by the leader be $X$, and $X^{\prime}$ be a binary number string of length $n$ where each digit is different from $X$. Then the number string written by the assistant is the same whether the leader's chosen string is $X$ or $X^{\prime}$ (changing $k$ to $n-k$). Given this, assume $k \geqslant \frac{n}{2}$. Specifically, if $k=\frac{n}{2}$, the participant cannot distinguish between $X$ and $X^{\prime}$. Therefore, the participant must guess at least twice.

For $0<m<2 k$, consider any number string $Y$ such that $Y$ differs from $X$ in exactly $m$ digits. Without loss of generality, assume the first $m$ digits of $X$ and $Y$ are different.

Let $Z$ be a binary number string that differs from $X$ in exactly the first $k$ digits. Then $Z$ is written by the assistant. Since $Z$ and $Y$ differ in $|m-k|$ digits, and given $0<m<2 k$, we have $|m-k|<k$. Therefore, the participant knows that $Y$ is not the number string written by the leader.
Note that we have already assumed $k \geqslant \frac{n}{2}$.
When $n<2 k$, every number string $Y$ different from $X$ differs from $X$ in fewer than $2 k$ digits; when $n=$ $2 k$, except for $X$ and $X^{\prime}$, every number string differs from $X$ in fewer than $2 k$ digits.

Therefore, when $n=2 k$, the minimum number of guesses is 2; when $n \neq 2 k$, the minimum number of guesses is 1.