Couldn't This Wait Till Next Year?

A kidnapper kidnaps 2015 people and takes them to his lair. He makes each of them sit on a round table and declares that he's going to leave everybody unharmed, but not before they play a game.

Each of the players is given a certain number of cards from 1 to 2015. No two players get the same number of cards. A move consists of a player giving a card to one of the two players adjacent to him. The game is over when each of the players has the same number of cards.

Find the last three digits of the minimum number of moves it will take for the game to be over.

Details and assumptions

  • Since the players don't want to stay in the bad guy's lair, they play optimally, so that they canave as soon as possible. The question asks for the minimum number of moves for which there exists a specific strategy which always ends the game.

  • Two moves cannot occur simultaneously.

  • The players can discuss a strategy before beginning.

  • Each player knows the number of cards any other players has.

  • The players are sitting beside a round table, so every player has exactly two players beside him.

  • This problem is not entirely original.


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