1000 white stones and 1 black stone are arranged in a row. A move consists in selecting one black stone and changing the color of the 2 neighbouring stones (or changing the color of 1 neighboring stone if the selected stone is at the end of the row).

Find all possible initial positions of the black stone, so that all stones can be made black by moves.


I suspect

the only solution is when there is a single black stone with 500 white stones on each side.

You win by

start in the middle. Then selecting black stone(s) farthest from the middle until the ends are black. Then start in the middle again and repeat until all the stones are black.

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