In a field, there are $n$ sheep and a wolf disguised as a sheep. Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. Initially, the wolf chooses certain sheep to become friends with. Each subsequent day, the wolf eats one of its friends. Each time the wolf eats a sheep $A$: (a) If a friend of $A$ was initially a friend, they are no longer friends with the wolf. (b) If a friend of $A$ is not a friend of the wolf, they become friends with the wolf. The wolf repeats this procedure until it has eaten as many sheep as possible.

Find the largest integer $m$ in terms of $n$ satisfying the following condition: there exists a choice of friendships among the sheep such that there are exactly $m$ ways to choose the sheep initially friendly with the wolf, so that the wolf can eat all the sheep.
We notice that if $A$ is the number of the wolf's friends, $M$ is the number of living sheep, and $P$ is the number of pairs of friends among the living sheep. Then $A+E+M$ modulo 2 does not change over time. Therefore, initially, $A$ must have the same parity as $E$ if the wolf wants to eat all the sheep. Thus, there are at most $2^{n-1}$ configurations that work.

If we consider the configuration where one sheep, named Gaston, knows everyone else. If initially, the wolf knows an odd number of sheep, it starts by eating while sparing Gaston as long as possible. By doing this, the number of the wolf's friends remains odd, because the sheep it eats have only one friend: Gaston. Thus, if it no longer has any friends other than Gaston, it must have Gaston as a friend. It eats Gaston, and then it can eat the other sheep one by one.

Thus $m=2^{n-1}$.