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Karl Fulves - The Return Trip

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In Faro & Riffle Technique an unsolved faro shuffle problem was stated. Given a deck arranged in a known order, the cards were shuffled in an unknown sequence of in and out shuffles. Examining the new order of the cards, could one specify a faro sequence that would return the deck to its original order?

Solutions for decks containing 2exp"n" cards, odd-size decks and some even decks are known. The present manuscript deals with solutions for decks containing 4n + 2 cards. The discussion is mostly simple and calls for no math background. If you are looking for mathematical elegance, this manuscript is not the place.