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If the assumption is that every apartment is occupied by exactly one person, then I think the answer is that it can't be done.

Color the squares alternately black and white, like on a chessboard, starting with 1 black, 2 white and so on. Note that:

1) There are 13 black squares and 12 white squares, and
2) Everybody who currently lives on a white square wants to move to a black square, and vice versa.

This means that there are only 12 apartments on white squares, but 13 people who want to move into them. Similarly there are 13 black square apartments but only 12 people to fill them. So if we are using the assumption above, there can be no solution.
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