How many shuffles to randomize a deck of cards?

@article{Trefethen2000HowMS,
  title={How many shuffles to randomize a deck of cards?},
  author={Lloyd N. Trefethen and Lloyd M. Trefethen},
  journal={Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences},
  year={2000},
  volume={456},
  pages={2561 - 2568}
}
  • L. TrefethenL. Trefethen
  • Published 8 October 2000
  • Computer Science
  • Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
A celebrated theorem of Aldous, Bayer and Diaconis asserts that it takes ∼3/2 log2 n riffle shuffles to randomize a deck of n cards, asymptotically as n → ∞, and that the randomization occurs abruptly according to a 'cut–off phenomenon'. These results depend upon measuring randomness by a quantity known as the total variation distance. If randomness is measured by uncertainty or entropy in the sense of information theory, the behaviour is different. It takes only ∼log2 n shuffles to reduce the… 

Figures from this paper

Information Loss in Top to Random Shuffling

  • Dudley Stark
  • Computer Science
    Combinatorics, Probability and Computing
  • 2002
It is found that the relative entropy of a deck of n cards after m successive top to random shuffles converges to an explicitly given expression when m = [n log n+cn] for a constant c.

Riffle shuffles of decks with repeated cards

By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either

Progressive Randomization of a Deck of Playing Cards: Experimental Tests and Statistical Analysis of the Riffle Shuffle

The question of how many shuffles are required to randomize an initially ordered deck of cards is a problem that has fascinated mathematicians, scientists, and the general public. The two principal

A Better Way to Deal the Cards

This work considers the case of decks with repeated cards and decks which are dealt into hands, as in bridge and poker, and derives asymptotic formulas for the randomness of the resulting games.

MATHEMATICAL DEVELOPMENTS FROM THE ANALYSIS OP RIFFLE SHUFFLING

1. Introduction The most common method of mixing cards is the ordinary riffle shuffle, in which a deck of n cards (often n = 52) is cut into two parts and the parts are riffled together. A sharp

Card-Shuffling Analysis with Markov Chains

In this essay we shall discuss mathematical models of card-shuffling. The basic question to answer is ”how many times do you need to shuffle a deck of cards for it to become sufficiently

Information Loss in Riffle Shuffling

The asymptotic behaviour of the relative entropy (to stationarity) for a commonly used model for riffle shuffling a deck of n cards m times shows that the deck becomes random in this information-theoretic sense after m = 3/2 log2n shuffles.

The Point Of Point Crossover: Shuffling To Randomness

It is shown that there is a cut-off phenomenon in the rate at which the sample get randomized, and if a statistical criterion such as Kendall's W coefficient or the average Kendall's T coefficient is used to measure randomness (rather than total variation distance), the sample can be said to be random (upto statistical significance) in O(ln N) steps.

Shuffle analogues for complex reflection groups G(m, 1, n)

Analogues of 1-shuffle elements for complex reflection groups of type G(m, 1, n) are introduced. A geometric interpretation for G(m, 1, n) in terms of "rotational" permutations of polygonal cards is

Enumeration of the distinct shuffles of permutations

A shuffle of two words is a word obtained by concatenating the two original words in either order and then sliding any letters from the second word back past letters of the first word, in such a way

References

SHOWING 1-10 OF 47 REFERENCES

Trailing the Dovetail Shuffle to its Lair

We analyze the most commonly used method for shuffling cards. The main result is a simple expression for the chance of any arrangement after any number of shuffles. This is used to give sharp bounds

An affine walk on the hypercube

Mixing of random walks and other diffusions on a graph

We survey results on two diffusion processes on graphs: random walks and chip-firing (closely related to the “abelian sandpile” or “avalanche” model of self-organized criticality in statistical

Riffle shuffles, cycles, and descents

The theorems are derived from a bijection discovered by Gessel that allows new formulae and approximations for the number of permutations in a permutation with given cycle type and number of descents.

Fluctuation Theory for the Ehrenfest urn

The Ehrenfest urn model with d balls, or alternatively random walk on the unit cube in d dimensions, is considered in discrete and continuous time, together with related models. Attention is focused

Random walks on finite groups and rapidly mixing markov chains

© Springer-Verlag, Berlin Heidelberg New York, 1983, tous droits reserves. L’acces aux archives du seminaire de probabilites (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml),

The cutoff phenomenon in finite Markov chains.

  • P. Diaconis
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1996
This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn), and shows that chains with polynomial growth (drunkard's walk) do not show cutoffs.

Markov Chains and Stochastic Stability

This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

Elements of Information Theory

The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.

An Introduction To Probability Theory And Its Applications

A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.