Question

一叠 $2n$ 张卡片从上到下连续编号为 $1$ 到 $2n$ 。取走上面的 $n$ 张卡片，保持顺序，形成牌堆 $A$ 。剩下的卡片形成牌堆 $B$ 。现在把卡片重新叠成一叠，方法是分别从牌堆 $B$ 和牌堆 $A$ 的顶部交替取牌。在这个过程中，编号为 $(n + 1)$ 的卡片是新牌堆的底牌，编号为 $1$ 的卡片放在它上面，如此继续，直到牌堆 $A$ 和 $B$ 都用完。如果重新叠牌后，每个牌堆中至少有一张卡片占据它在原牌堆中的同一位置，则称这叠牌为神奇的。例如，八张卡片形成一叠神奇牌，因为编号为 $3$ 和编号为 $6$ 的卡片保留了原来的位置。求在编号为 $131$ 的卡片保留原位置的神奇牌堆中，卡片的张数。

The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are now restacked into a single stack by taking cards alternately from the tops of pile $B$ and pile $A,$ respectively. In this process, card number $(n + 1)$ is the bottom card of the new stack, card number $1$ is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is called magical. For example, eight cards form a magical stack because cards number $3$ and number $6$ retain their original positions. Find the number of cards in the magical stack in which card number $131$ retains its original position.

答案

答案：392

知识点：排列奇偶性一次方程

难度评级：2560

解答：

从底部往上读，新牌堆为 $n+1,\ 1,\ n+2,\ 2,\ \ldots,\ 2n,\ n$ 。因此牌堆 $B$ 的卡片以相反顺序占据从顶部数的偶数位置，牌堆 $A$ 的卡片以相反顺序占据奇数位置：原位置 $i \le n$ （牌堆 $A$ ）的卡片移到位置 $2(n - i) + 1$ ，而原位置 $i \gt n$ （牌堆 $B$ ）的卡片移到位置 $2(2n - i) + 2$ 。

由于 $131$ 是奇数，编号为 $131$ 的卡片若要保持原位置，只能来自牌堆 $A$ ，因而 $131 = 2(n - 131) + 1$ ，解得 $n = 196$ 。的确， $131 \le 196$ ，且这叠牌是神奇的，因为编号为 $262$ 的来自牌堆 $B$ 的卡片也保持不动： $2(2n - 262) + 2 = 262$ 。这叠牌共有 $2n = 392$ 张。

The new stack, read from the bottom up, is $n+1,\ 1,\ n+2,\ 2,\ \ldots,\ 2n,\ n.$ So pile $B$ 's cards occupy the even positions from the top in reverse order, and pile $A$ 's cards occupy the odd positions in reverse order: a card at original position $i \le n$ (pile $A$ ) moves to position $2(n - i) + 1,$ while a card at position $i \gt n$ (pile $B$ ) moves to position $2(2n - i) + 2.$

Since $131$ is odd, card $131$ can keep its position only if it comes from pile $A,$ so $131 = 2(n - 131) + 1,$ which gives $n = 196.$ Indeed $131 \le 196,$ and the stack is magical because card $262$ from pile $B$ also stays fixed: $2(2n - 262) + 2 = 262.$ The stack has $2n = 392$ cards.

2005 AIME II 第 6 题

答案：392

难度评级：2560

解答：

其他年份的第 6 题