Question 25
There are $n$ sheep and a wolf in sheep's clothing. Some of the sheep are friends with each other (friendship is mutual). The wolf's goal is to eat all the sheep. First, it chooses some of the $n$ sheep and makes friends with them. Then, every day, it eats one sheep among its friends. Whenever it eats a sheep $A$:
(i) every friend of $A$ that was a friend of the wolf breaks off its friendship with the wolf;
(ii) every friend of $A$ that was not a friend of the wolf becomes a friend of the wolf.
This is repeated until the wolf has no friends left.
Find the largest positive integer $m$ (in terms of $n$) satisfying the following condition:
there exists a friendship relation among the $n$ sheep such that the wolf has in total $m$ different ways of choosing its initial friends that allow it to eat all the sheep.
(i) every friend of $A$ that was a friend of the wolf breaks off its friendship with the wolf;
(ii) every friend of $A$ that was not a friend of the wolf becomes a friend of the wolf.
This is repeated until the wolf has no friends left.
Find the largest positive integer $m$ (in terms of $n$) satisfying the following condition:
there exists a friendship relation among the $n$ sheep such that the wolf has in total $m$ different ways of choosing its initial friends that allow it to eat all the sheep.