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? number theorygame strategywinning strategygame