3. There are 865 plus signs written on the board. Petya and Vasya are playing such a game. A move consists of either replacing one plus with a minus, or erasing one plus and two minuses, or replacing two pluses with two minuses. They take turns, with Petya going first, and the one who cannot make a move loses. Who will win with correct play?
Answer: Petya will win.

Solution. Petya's first move should be to replace one plus with a minus. As a result, the number of pluses on the board will become divisible by 3. After any of Vasya's moves, the number of pluses will change by 1 or 2, and Petya will be able to move in such a way that the number of pluses is again divisible by 3 - for this, he only needs to replace one or two pluses with minuses.

By acting in this manner, Petya will ensure that after some of his moves, the number of pluses will become 0, and Vasya will not be able to make a move.