Question 12
Let $N$ be a positive integer. Two players $A$ and $B$, taking turns, write numbers from the set $\{1, \ldots, N\}$ on a blackboard. $A$ begins the game by writing 1 on his first move. Then, if a player has written $n$ on
a certain move, his adversary is allowed to write $ n+1 $ or 2n (provided the number he writes does not exceed N). The player who writes N wins. We say that N is of type A or of type B according as A or B has a winning strategy.
(a) Determine whether N = 2004 is of type A or of type B.
(b) Find the least N > 2004 whose type is different from that of 2004.
a certain move, his adversary is allowed to write $ n+1 $ or 2n (provided the number he writes does not exceed N). The player who writes N wins. We say that N is of type A or of type B according as A or B has a winning strategy.
(a) Determine whether N = 2004 is of type A or of type B.
(b) Find the least N > 2004 whose type is different from that of 2004.