11.5. We have the number 1. Petya (starting the game) and Vasya take turns performing the following operations on the current number: in one move, they multiply it by one of the numbers $2,3,4 \ldots 10$, and then add one of the numbers 1, 2, 3... 10 to it. The winner is the one who first gets a number not less than 1000. Who will win with correct play?
# 11.5. Answer: Petya

Let's use retrospective analysis (analysis from the end). The player who has a number from 99 to 999 on their turn wins, the player who has 49, ..., 98 loses, the player who has 4, ..., 48 wins, the player who has 2, 3 loses, and the player who has 1 wins. Thus, the player who starts the game, Petya, wins.

![](https://cdn.mathpix.com/cropped/2024_05_06_5f156093764be7ae3ebag-1.jpg?height=462&width=448&top_left_y=2096&top_left_x=1455)