# 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} }

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…

## 40 Citations

### Information Loss in Top to Random Shuffling

- Computer ScienceCombinatorics, 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

- Mathematics
- 2006

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

- MathematicsOpen Journal of Statistics
- 2019

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

- Mathematics, EngineeringAm. Math. Mon.
- 2010

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

- Mathematics
- 2003

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

- Mathematics
- 2005

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

- Computer ScienceCombinatorics, Probability and Computing
- 2002

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

- MathematicsGECCO
- 2002

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)

- Mathematics
- 2017

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

- Mathematics
- 2009

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…

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