The mathematics of the flip and horseshoe shuffles

TL;DR

This paper introduces new perfect shuffles called flip and horseshoe shuffles, analyzing their mathematical properties and their relation to faro shuffling, including the order of their associated groups.

Contribution

It defines and studies the mathematical structure of flip and horseshoe shuffles, extending the understanding of perfect shuffles and their group properties.

Findings

Flip shuffles involve face-up/face-down configurations.

Horseshoe shuffles reverse card order while maintaining orientation.

Both shuffles are closely related to faro shuffling and their group orders are determined.

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 flipping the half 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.