The Mathematics of the Flip and Horseshoe Shuffles

Abstract

We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is “reversed,” and then the cards are interlaced. Flip shuffles are when the reversal comes from turning the half of the deck over so that we also need to account for face-up/face-down configurations, while horseshoe shuffles are when the order of the cards are reversed but all cards still face the same direction. We show that these shuffles are closely related to faro shuffling and determine the order of the associated shuffling groups. An application of this theory is given through a card trick based on applying different shuffles of eight cards.