Question 106
The leader of an IMO team chooses positive integers $n$ and $k$ with $n \gt k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader's in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses 101, the deputy leader would write down 001, 111, and 100.) The contestant, who is allowed to look at the strings written by the deputy leader, tries to guess the leader's string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?