Question 88
Let $k$ be a positive integer. The little one and the magician on the skywalk play a game. Initially, the little one places $N = 2^k$ distinct balls in a row on the table, and covers each of them by a cup. On each turn, the little one chooses two cups, then the magician can either swap the balls in the two cups, or do a fake move so that the balls in the two cups stay the same. The little one cannot distinguish whether the magician fakes a move on not, nor can she observe the balls inside the cups.
After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one wins. Prove that the little one has a winning strategy.
After $M = k \times 2^{k-1}$ turns, the magician opens all cups so the little one can check the ball in each of the cups. If the little one can identify whether the magician fakes a move or not for each of the $M$ turns, then the little one wins. Prove that the little one has a winning strategy.