98434 – We Solve Problems

#### Dissections (other)

Two people are playing. The first player writes out numbers from left to right, randomly alternating between 0 and 1, until there are 1999 numbers in total. Each time after the first one writes out the next digit, the second changes two numbers from the already written row $($ when only one digit is written, the second misses its move $)$. Is the second player always able to ensure that, after his last move, the arrangement of the numbers is symmetrical relative to the middle number?

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