Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game:
(i) Alberto chooses a positive integer,
(ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself),
(iii) Alberto increases or decreases the Barbara’s number by $1$.
Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?
To determine for which values of \( n \) Barbara can win the game, we need to analyze the conditions under which she can reach \( n \) in at most 50 moves, regardless of Alberto's actions. Let's break down the problem step by step:

1. **Initial Choice by Alberto:**
   - Alberto chooses a positive integer \( a \).

2. **Barbara's Choice:**
   - Barbara chooses an integer \( b \) such that \( b > 1 \) and \( b \) is a multiple or submultiple of \( a \). This means \( b \) can be any divisor or multiple of \( a \).

3. **Alberto's Move:**
   - Alberto can increase or decrease \( b \) by 1, i.e., \( b \rightarrow b+1 \) or \( b \rightarrow b-1 \).

4. **Barbara's Objective:**
   - Barbara wins if she can reach the number \( n \) in at most 50 moves.

To determine the values of \( n \) for which Barbara can always win, we need to consider the structure of the game and the constraints on the moves.

### Analysis:
- Barbara needs to ensure that she can reach \( n \) regardless of Alberto's actions. This means she must be able to navigate through the integers in such a way that she can always counter Alberto's moves and still reach \( n \).

- Let's consider the properties of \( n \) that would allow Barbara to win:
  - If \( n \) is a multiple of 6, then Barbara can always choose numbers that are multiples of 6 or submultiples of 6, ensuring that she can reach \( n \) in a controlled manner.
  - If \( n \) is not a multiple of 6, then there might be scenarios where Alberto can force Barbara into a position where she cannot reach \( n \) within the allowed number of moves.

### Conclusion:
- Barbara can win if and only if \( n \) is a multiple of 6. This is because the structure of the game allows her to navigate through multiples and submultiples of 6 in a way that ensures she can always reach \( n \) within the given constraints.

The final answer is \( \boxed{ n } \) can be reached if and only if \( 6 \mid n \).