Shapovalov A.V.

The numbers $1,2,3, \ldots, 1000$ are written on a board. Two players take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by 3, the player who made the first move wins; otherwise, his partner wins. Who will win with correct play?
Let's indicate how a beginner's partner can guarantee a win.

At the beginning of the game, the partner of the beginner should erase numbers that are multiples of 3 until none are left. Since the number of numbers not exceeding 1000 and divisible by 3 is 333, the partner of the beginner will need no more than 333 moves to ensure that none of the remaining numbers on the board are divisible by 3 (some of the multiples of 3 may also be erased by the beginner).

After this, the partner of the beginner makes their moves arbitrarily until there are three numbers left on the board. Each of these will give a remainder of 1 or 2 when divided by 3, so among the three remaining numbers on the board, there will definitely be two that give the same remainder. The partner of the beginner should leave these two (the sum will not be divisible by 3).

## Answer

The partner.