8.2. Once Alexei and Daniil were playing such a game. If a number \( x \) is written on the board, it can be erased and replaced with \( 2x \) or \( x - 1000 \). The player who gets a number not greater than 1000 or not less than 4000 loses. Both players aim to win. At some point, the boys stopped playing. Who lost if the first number was \( 2017 \)?
Answer: no one lost.

Solution: note that if a number is less than 2000 but greater than 1000, then by multiplying by 2, you can get a number that is less than 4000. If a number is less than 4000 but greater than 2000, then by subtracting 1000 (possibly twice), you can get a number between 1000 and 2000. Thus, the only number from which no move can be made is 2000.

We will prove that no one could have obtained 2000. Note that the initial number is not divisible by 5. If we multiply it by 2 or subtract 1000 from it, the new number will again not be divisible by 5.

Thus, Alexei and Danila could have continued their game indefinitely, and no one would have lost.

Criteria:

Noted only that from numbers less than 2000 and greater than 2000, a move can always be made - 3 points. Proved only that 2000 cannot be obtained - 3 points.