A rule of thumb for riffle shuffling

TL;DR

This paper investigates the number of riffle shuffles needed to sufficiently mix a deck of n cards when only specific features like suits or colors are considered, providing simplified formulas and a practical rule of thumb.

Contribution

It introduces new formulas and an asymptotic rule of thumb for estimating shuffles needed based on the features of interest, simplifying previous complex calculations.

Findings

Number of shuffles reduces from 3/2 log_2(n) to log_2(n) for certain features.

Derived closed-form expressions for the shuffling process.

Proposed an accurate asymptotic rule of thumb for practical use.

Abstract

We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For these features, the number of shuffles drops from 3/2 log_2(n) to log_2(n). We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.