Question 107
A positive integer $N$ greater than $1$ is given. Two players, Player 1 and Player 2, play the following game:
(1) Player 1 says a positive integer $A$.
(2) Player 2 says a positive integer $B \gt 1$, where $B$ must be a divisor of $A$ or a multiple of $A$.
(3) Player 1 says one of the numbers $B-1$, $B$, $B+1$ as the new $A$.
From then on the game continues by alternately performing steps (2) and (3). If $N$ is among the first $50$ numbers said, Player 2 wins; otherwise Player 1 wins. Answer the following questions.
i. Prove that when $N=10$, Player 1 has a winning strategy.
ii. Prove that when $N=24$, Player 2 has a winning strategy.
iii. Find all positive integers $N$ greater than $1$ for which Player 2 has a winning strategy.
(1) Player 1 says a positive integer $A$.
(2) Player 2 says a positive integer $B \gt 1$, where $B$ must be a divisor of $A$ or a multiple of $A$.
(3) Player 1 says one of the numbers $B-1$, $B$, $B+1$ as the new $A$.
From then on the game continues by alternately performing steps (2) and (3). If $N$ is among the first $50$ numbers said, Player 2 wins; otherwise Player 1 wins. Answer the following questions.
i. Prove that when $N=10$, Player 1 has a winning strategy.
ii. Prove that when $N=24$, Player 2 has a winning strategy.
iii. Find all positive integers $N$ greater than $1$ for which Player 2 has a winning strategy.