There are 2n cards labeled 1, 2, ..., 2n respectively, and the cards are distributed randomly between two players so that each has n cards. Each player takes turns to place one card, and you win if you put down a card so that the current sum of all the played cards is divisible by 2n+1.
For example, if n=10, and the previously placed cards are 5, 8, 9, 19, then if player A now places 1, he wins since 5+8+9+19+1 = 42 is divisible by 2*10+1=21.
Assuming both players want to win, what strategy should one adopt in order to win? Following the strategy, is there a consistent winner of this game?
Player 2 will win. On each of his turns, he has (k+1) choices, only k of which can lead to winning moves for Player 1 on his next turn. So Player 1 never has a winning move as long as Player 2 makes such a move. On the other hand, if the game goes to the last move, Player 2's last move wins him the game.
Solution
