Many modern-day casinos use a mechanical shuffler to shuffle cards. These shufflers have a fixed number of shelves $m$ (typically $m=10$). These mechanical shufflers randomize a deck by assigning a shelf (uniformly at random) to each incoming card, which is placed at the top or the bottom of the assigned shelf with equal probability. At the end, the shelves are concatenated. This shuffling scheme was first analyzed by Diaconis, Fulman, and Holmes in 2013 and is commonly called the shelf-shuffling.

How well do these shufflers perform? There are several ways to measure how effective a shuffling scheme is. One such way is to ask how many cards one can guess correctly after shelf-shufflings. One can further study the number of correct guesses one can make if one is provided with some feedback after each guess. These naturally lead to several questions: what is the best strategy to maximize the expected number of correct guesses (with and without feedback)? What is the average number of correct guesses under an optimal strategy? How does it fluctuate? Furthermore, to make a meaningful comparison, it is also imperative to understand the same questions for a uniformly random deck of cards, which serves as a benchmark.

In this talk, we will begin with a gentle survey of the various questions of interest in this field and their connections with Markov chains on the symmetric group. In the latter part of the talk, we will discuss a central limit theorem for the number of correct guesses in a uniform random deck with complete feedback, and then we will specialize to the shelf shuffling with a single shelf ($m=1$), and we will answer some of the above questions in this special case.