Problem:
The cards in a stack of cards are numbered consecutively from through from top to bottom. The top cards are removed, kept in order, and form pile . The remaining cards form pile . The cards are now restacked into a single stack by taking cards alternately from the tops of pile and pile , respectively. In this process, card number is the bottom card of the new stack, card number is on top of this card, and so on, until piles and 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 and number retain their original positions. Find the number of cards in the magical stack in which card number retains its original position.
Solution:
Note that, after the restacking, all the cards from pile occupy even-numbered positions and their order is reversed. Similarly, all the cards from pile will be placed in odd-numbered positions, and their order is also reversed. A card in position for will be moved to position in the restacking, and, for , the card will be moved to position . For a card to remain in the st position, it must be in pile . Then , and . Note that the stack is magical because cards number and retain their original positions.
The problems on this page are the property of the MAA's American Mathematics Competitions