Folklore

Two players alternately increase a natural number so that with each increase, the difference between the new and old values of the number is greater than zero but less than the old value. The initial value of the number is 2. The player who, as a result of their move, reaches 1987 wins. Who wins with correct play: the first player or their partner?
In the sequence $1987,993,496,248,124,62,31,15,7,3$, each subsequent number is the quotient of the division of the previous number by 2 with a remainder. We will prove that this sequence consists of winning numbers (i.e., a player who names one of these numbers has a winning strategy). The number 1987 is a winning number by the condition. Suppose the number $2k$ or $2k+1 (k>2)$ is a winning number. Then $k$ is also a winning number. Indeed, if one player names the number $k$, the other can only name a number from the interval $[k+1,2k-1]$, after which the first player can name both $2k$ and $2k+1$. The player starting the game must name the number 3 and, following the indicated strategy, will win.

## Answer

The first player.