Two players are playing a game

Question

Bob and alice are playing a game: There are 1999 balls. The players take turns to remove some of the balls (Bob first), the only restriction one has to take at least one ball and at most half of the balls. Who leaves only one ball behind, loses the game. Which player can win?

Answer 1

The first person always wins with the following strategy.

Take $464$ balls to leave $1535$. Your opponent must take at least one and at most $767$ to leave between $768$ and $1534$. Therefore, you can always get him down to $767$.

Likewise, after the next round you can get him down to $383$, then $191$, $95$,$47$, $23$, $11$, and then $5$.

Once your opponent is at $5$ he cannot win because he must take either one or two balls leaving you with $3$ or $4$. In either case, you leave him with two. He must take one and is left with one so he loses.