Consider cards labeled 1, ..., 2n. The cards are shuffled and dealt to two playe

Question

Consider cards labeled 1, ..., 2n. The cards are shuffled and dealt to two players A and B so that each gets n of the cards. Let x be the sum of the labels on the cards that have been played; initially, x = 0. Starting with A, play alternates between the two players. At each play, a player adds one of his or her remaining cards to x. The first player who makes x divisible by 2n + 1 wins. Prove that for every deal, player B has a strategy to win. (Hint: Prove that B can always make it impossible for A to win on the next move).

Explanation / Answer

Given a stack of cards with 2n = 12, I observed and noted the shift in positions as seen below. Position 0 1 2 3 4 5 6 7 8 9 10 11 Card A B C D E F G H I J K L After one perfect in-shuffle Position 0 1 2 3 4 5 6 7 8 9 10 11 Card G A H B I C J D K E L F Number of Positions Moved +6 -1 +5 -2 +4 -3 +3 -4 +2 -5 +1 -6 I noticed that after one in-shuffle that the movement of the cards created a mirror image of movement. After n cards the pattern of movement begins to reflect but with opposite signs. To create a function to describe the movement in position, I had to consider two separate cases based on where the card was originally in the deck. Case One: x

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