Shuffling cards for blackjack, bridge, and other card games

TL;DR

This paper investigates the number of riffle shuffles needed to adequately mix decks of cards for games like blackjack and bridge, analyzing the complexity of related polynomial computations and the limitations of approximation methods.

Contribution

It proves the #P-completeness of computing descent polynomials for decks with repeated cards and evaluates approximation methods, combining theoretical and experimental insights.

Findings

Computing descent polynomials is #P-complete.

Bell curve approximations are insufficient for accurate mixing analysis.

The paper combines theoretical proofs with experimental validation.

Abstract

This paper is about the following question: How many riffle shuffles mix a deck of card for games such as blackjack and bridge? An object that comes up in answering this question is the descent polynomial associated with pairs of decks, where the decks are allowed to have repeated cards. We prove that the problem of computing the descent polynomial given a pair of decks is $#P$-complete. We also prove that the coefficients of these polynomials can be approximated using the bell curve. However, as must be expected in view of the $#P$-completeness result, approximations using the bell curve are not good enough to answer our question. Some of our answers to the main question are supported by theorems, and others are based on experiments supported by heuristic arguments. In the introduction, we carefully discuss the validity of our answers.