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But the guy he swapped with used to be diagonal from that center before the inner ring rotated so although he moved twice, he remains jealous.

That's true. But my assumption was that the author allows a second move for a given neighbor whereby he has to move up/down, side/side from the spot he acquired after the first move.

I know that doesn't fit the spirit of the puzzle statement. The puzzle isn't written too well. It leaves certain options open (like double occupancy, more than one move, etc.)
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