IMO Shortlist 1999 problem A3

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2. travnja 2012.
A game is played by n girls (n \geq 2), everybody having a ball. Each of the \binom{n}{2} pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice nice if at the end nobody has her own ball and it is called tiresome if at the end everybody has her initial ball. Determine the values of n for which there exists a nice game and those for which there exists a tiresome game.
Izvor: Međunarodna matematička olimpijada, shortlist 1999