The Card Guessing Game: A generating function approach

TL;DR

This paper analyzes a card guessing game after a single riffle shuffle, providing a precise expected correct guesses formula, a systematic method for higher moments, and insights into optimal strategies for multiple shuffles.

Contribution

It offers an improved expected guesses formula, a generating function framework for moments, and clarifies optimal strategies for multiple shuffles, extending prior work.

Findings

Derived an exact expression for expected correct guesses after one shuffle.

Developed a generating function approach for higher moments.

Showed that the optimal strategy for one shuffle does not extend to multiple shuffles.

Abstract

Consider a card guessing game with complete feedback in which a deck of $n$ cards ordered $1,\dots, n$ is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses cards from the top of the deck one at a time under the optimal strategy until no cards remain. We provide an expression for the expected number of correct guesses with arbitrary number of terms, an accuracy improvement over the results of Liu (2021). In addition, using generating functions, we give a unified framework for systematically calculating higher-order moments. Although the extension of the framework to $k\geq2$ shuffles is not immediately straightforward, we are able to settle a long-standing McGrath's conjectured optimal strategy described in Bayer and Diaconis (1992) by showing that the optimal guessing strategy for $k=1$ riffle shuffle does not necessarily apply to $k\geq2$ shuffles.