number theory game of two, winning strategy in at most 50moves
Source: ITAMO 2000 p4
January 25, 2020
number theorygame strategywinning strategygame
Problem Statement
Let be a fixed integer. Alberto and Barbara play the following game:
(i) Alberto chooses a positive integer,
(ii) Barbara chooses an integer greater than which is a multiple or submultiple of the number Alberto chose (including itself),
(iii) Alberto increases or decreases the Barbara’s number by .
Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number in at most moves. For which values of can she win, no matter how Alberto plays?