Question 105
Fix an integer $n\gt 1$. Alberto and Barbara play the following game:
1. Alberto chooses a positive integer.
2. Barbara chooses an integer greater than $1$ that is a multiple or a divisor of Alberto's number, including that number itself.
3. Alberto returns Barbara's number to her, possibly adding or subtracting $1$.
Play continues by alternately repeating steps 2 and 3. Barbara wins if she manages to choose $n$ within $50$ moves. For which values of $n$ can Barbara win against any strategy of Alberto?
1. Alberto chooses a positive integer.
2. Barbara chooses an integer greater than $1$ that is a multiple or a divisor of Alberto's number, including that number itself.
3. Alberto returns Barbara's number to her, possibly adding or subtracting $1$.
Play continues by alternately repeating steps 2 and 3. Barbara wins if she manages to choose $n$ within $50$ moves. For which values of $n$ can Barbara win against any strategy of Alberto?