[Winning and Losing Positions]

The game starts with the number 1000. On a turn, a player is allowed to subtract from the current number any natural number that is a power of two and does not exceed it $\left(1=2^{0}\right)$. The player who reaches zero wins.
Analyzing from the end, we find the winning positions. These are numbers divisible by 3. The first player wins. The first move can be, for example, subtracting $1, 4, 16$.