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