I'm having difficulty understanding the problem. I hope someone can clarify for me. Thanks!
Is it asking if, for a given n, there is at least one way for me to win? Or is it asking if there's a way for me to win such that it excludes any ways of the other player winning? An illustration of the difference is n=5. This is winnable by me or the other player, depending on the quality of the moves he makes. But it's not winnable exclusively by me based on some "ideal" first move. Contrast this with n=3. There is an ideal first move, where I remove 3, which completely prevents the other player from winning.