Shuffling is a procedure used to

Illustrated guide to several shuffling methods

Magician's tool with much shuffling simulation

Mathematics of shuffling:

Real World Shuffling In Practice

*Ivars Peterson's MathTrek

Real world (historical) application:

How We Learned to Cheat at Online Poker: A Study in Software Security

{{Non trick-taking card games Card game terminology Card shuffling

randomizeRandomization is the process of making something random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and do ...

a deck of playing card
A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a Pap ...

s to provide an element of chance in card game
A card game is any game
A game is a structured form of play
Play most commonly refers to:
* Play (activity), an activity done for enjoyment
* Play (theatre), a work of drama
Play may refer also to:
Computers and technology
* Googl ...

s. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome.
__TOC__
Techniques

Overhand

One of the easiest shuffles to accomplish after a little practice is the overhand shuffle. Johan Jonasson wrote, "The overhand shuffle... is the shuffling technique where you gradually transfer the deck from, say, your right hand to your left hand by sliding off small packets from the top of the deck with your thumb." In detail as normally performed, with the pack initially held in the left hand (say), most of the cards are grasped as a group from the bottom of the pack between the thumb and fingers of the right hand and lifted clear of the small group that remains in the left hand. Small packets are then released from the right hand a packet at a time so that they drop on the top of the pack accumulating in the left hand. The process is repeated several times. The randomness of the whole shuffle is increased by the number of small packets in each shuffle and the number of repeat shuffles performed. The overhand shuffle offers sufficient opportunity for sleight of hand techniques to be used to affect the ordering of cards, creating a stacked deck. The most common way that players cheat with the overhand shuffle is by having a card at the top or bottom of the pack that they require, and then slipping it to the bottom at the start of a shuffle (if it was on top to start), or leaving it as the last card in a shuffle and just dropping it on top (if it was originally on the bottom of the deck).Riffle

A common shuffling technique is called the ''riffle,'' or ''dovetail'' shuffle or ''leafing the cards'', in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Many also lift the cards up after a riffle, forming what is called a bridge which puts the cards back into place; it can also be done by placing the halves flat on the table with their rear corners touching, then lifting the back edges with the thumbs while pushing the halves together. While this method is more difficult, it is often used incasino
A casino is a facility for certain types of gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shopping, cruise ships, and other tourist attractions. Some casinos are also known for hosting live entertai ...

s because it minimizes the risk of exposing cards during the shuffle. There are two types of perfect riffle shuffles: if the top card moves to be second from the top then it is an in shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling
Shuffling is a procedure used to randomization, randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often ...

, otherwise it is known as an out shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling
Shuffling is a procedure used to randomization, randomize a deck of playing cards to provide an element of chance in card games. Shuffling is ofte ...

(which preserves both the top and bottom cards).
The Gilbert–Shannon–Reeds modelIn the mathematics of shuffling playing cards, the Gilbert–Shannon–Reeds model is a probability distribution on riffle shuffle permutations that has been reported to be a good match for experimentally observed outcomes of human shuffling, and tha ...

provides a mathematical model of the random outcomes of riffling that has been shown experimentally to be a good fit to human shuffling and that forms the basis for a recommendation that card decks be riffled seven times in order to randomize them thoroughly. Later, mathematicians Lloyd M. Trefethen and Lloyd N. Trefethen authored a paper using a tweaked version of the Gilbert–Shannon–Reeds model showing that the minimum number of riffles for total randomization could also be six, if the method of defining randomness is changed.
Hindu

Also known as the "Indian", "Kattar", "Kenchi" (Hindi
Hindi (Devanagari: , हिंदी, ISO 15919, ISO: ), or more precisely Modern Standard Hindi (Devanagari: , ISO 15919, ISO: ), is an Indo-Aryan language spoken chiefly in Hindi Belt, North India. Hindi has been described as a Standard la ...

for scissor) or "Kutti Shuffle". The deck is held face down, with the middle finger on one long edge and the thumb on the other on the bottom half of the deck. The other hand draws off a packet from the top of the deck. This packet is allowed to drop into the palm. The maneuver is repeated over and over, with newly drawn packets dropping onto previous ones, until the deck is all in the second hand. Indian shuffle differs from stripping in that all the action is in the hand ''taking'' the cards, whereas in stripping, the action is performed by the hand with the original deck, ''giving'' the cards to the resulting pile. This is the most common shuffling technique in Asia and other parts of the world, while the overhand shuffle is primarily used in Western countries.
Pile

Cards are simply dealt out into a number of piles, then the piles are stacked on top of each other. Though this is deterministic and does not randomize the cards at all, it ensures that cards that were next to each other are now separated. Some variations on the pile shuffle attempt to make it slightly random by dealing to the piles in a random order each circuit.Corgi

Also known as the Chemmy, Irish, wash, scramble, beginner shuffle, smooshing, schwirsheling, or washing the cards, this involves simply spreading the cards out face down, and sliding them around and over each other with one's hands. Then the cards are moved into one pile so that they begin to intertwine and are then arranged back into a stack. This method is useful for beginners, but the shuffle requires a large surface for spreading out the cards. Statistically random shuffling is achieved after approximately one minute of smoothing. Smooshing has been largely popularized by Simon Hofman.Mongean

The Mongean shuffle, or Monge's shuffle, is performed as follows (by a right-handed person): Start with the unshuffled deck in the left hand and transfer the top card to the right. Then repeatedly take the top card from the left hand and transfer it to the right, putting the second card at the top of the new deck, the third at the bottom, the fourth at the top, the fifth at the bottom, etc. The result, if one started with cards numbered consecutively $\backslash scriptstyle\; 1,\; 2,\; 3,\; 4,\; 5,\; 6,\; \backslash dots,\; 2n$, would be a deck with the cards in the following order: $\backslash scriptstyle\; 2n,\; 2n-2,\; 2n-4,\; \backslash dots,\; 4,\; 2,\; 1,\; 3,\; \backslash dots,\; 2n-3,\; 2n-1$. For a deck of given size, the number of Mongean shuffles that it takes to return a deck to starting position, is known . Twelve perfect Mongean shuffles restore a 52-card deck.Faro

Weaving is the procedure of pushing the ends of two halves of a deck against each other in such a way that they naturally intertwine. Sometimes the deck is split into equal halves of 26 cards which are then pushed together in a certain way so as to make them perfectly interweave. This is known as a ''Faro Shuffle''. Thefaro shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling
Shuffling is a procedure used to randomization, randomize a deck of playing cards to provide an element of chance in card games. Shuffling is oft ...

is performed by cutting the deck into two, preferably equal, packs in both hands as follows (right-handed):
The cards are held from above in the right and from below in the left hand. Separation of the deck is done simply lifting up half the cards with the right hand thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and slammed into each other as to align them. They are then pushed together by the short sides and bent (either up or down). The cards then alternately fall into each other, much like a zipper
A zipper, zip, fly, or zip fastener, formerly known as a clasp locker, is a commonly used device for binding the edges of an opening of textile, fabric or other flexible material, such as on a garment or a bag. It is used in clothing (e.g., ja ...

. A flourish can be added by springing the packets together by applying pressure and bending them from above, as called the bridge finish. The faro is a controlled shuffle which does not randomize a deck when performed properly.
A perfect faro shuffle, where the cards are perfectly alternated, is considered one of the most difficult sleights by card magicians, simply because it requires the shuffler to be able to cut the deck into two equal packets and apply just the right amount of pressure when pushing the cards into each other. Performing eight perfect faro shuffles in a row restores the order of the deck to the original order only if there are 52 cards in the deck and if the original top and bottom cards remain in their positions (1st and 52nd) during the eight shuffles. If the top and bottom cards are weaved in during each shuffle, it takes 52 shuffles to return the deck back into original order (or 26 shuffles to reverse the order).
Mexican spiral

The Mexican spiral shuffle is performed by cyclic actions of moving the top card onto the table, then the new top card under the deck, the next onto the table, next under the deck, and so on until the last card is dealt onto the table. It takes quite a long time, compared with riffle or overhand shuffles, but allows other players to fully control cards which are on the table. The Mexican spiral shuffle was popular at the end of the 19th century in some areas ofMexico
Mexico, officially the United Mexican States, is a country
A country is a distinct territorial body or political entity
A polity is an identifiable political entity—any group of people who have a collective identity, who are organi ...

as a protection from gamblers and con men arriving from the United States.
Faking

Magicians
Magician or The Magician may refer to:
Performers
* A practitioner of Magic (supernatural)
* A practitioner of Magic (illusion)
* Magician (fantasy), a character in a fictional fantasy context
Entertainment
Books
* ''The Magician'', an 18th-cent ...

, sleight-of-hand artists, and card cheats employ various methods of shuffling whereby the deck appears to have been shuffled fairly, when in reality one or more cards (up to and including the entire deck) stays in the same position. It is also possible, though generally considered very difficult, to "stack the deck" (place cards into a desirable order) by means of one or more riffle shuffles; this is called "riffle stacking".
Both performance magicians and card sharps regard the Zarrow shuffle Herb Zarrow (November 4, 1925 – May 11, 2008) was an American Magic (illusion), magician influential in the profession for his inventions of unique sleight of hand and card tricks. His skills were held in the highest regard in professional magici ...

and the Push-Through-False-Shuffle as particularly effective examples of the false shuffle. In these shuffles, the entire deck remains in its original order, although spectators think they see an honest riffle shuffle.
Machines

Casino
A casino is a facility for certain types of gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shopping, cruise ships, and other tourist attractions. Some casinos are also known for hosting live entertai ...

s often equip their tables with shuffling machine
A shuffling machine is a machine for randomly shuffling packs of playing cards.
Because standard shuffling techniques are seen as weak, and in order to avoid "inside jobs" where employees collaborate with gamblers by performing inadequate shuffles ...

s instead of having croupier
A croupier or dealer is someone appointed at a gambling
File:A photo of a gambling stand in Paris.jpg, A gambling stand in Paris
Gambling (also known as betting) is the wagering of money or something of Value (economics), value (referred to a ...

s shuffle the cards, as it gives the casino a few advantages, including an increased complexity to the shuffle and therefore an increased difficulty for players to make predictions, even if they are collaborating with croupiers. The shuffling machines are carefully designed to avoid biasing the shuffle and are typically computer-controlled. Shuffling machines also save time that would otherwise be wasted on manual shuffling, thereby increasing the profitability of the table. These machines are also used to lessen repetitive-motion-stress injuries to a dealer.
Players with superstitions often regard with suspicion any electronic equipment, so casinos sometimes still have the croupiers perform the shuffling at tables that typically attract those crowds (e.g., baccarat
Baccarat (; german: Burgambach) is a commune
A commune is an intentional community of people sharing living spaces, interests, values, beliefs, and often property
Property (''latin: Res Privata'') in the Abstract and concrete, abstrac ...

tables).
Randomization

There are exactly 52factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

(expressed in shorthand as 52 !) possible orderings of the cards in a 52-card deck. In other words, there are 52 × 51 × 50 × 49 × ··· × 4 × 3 × 2 × 1 possible combinations of card sequence. This is approximately (80,658 vigintillion) possible orderings, or specifically 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. The magnitude of this number means that it is exceedingly improbable that two randomly selected, truly randomized decks will be the same. However, while the exact sequence of all cards in a randomized deck is unpredictable, it may be possible to make some probabilistic predictions about a deck that is not sufficiently randomized.
Sufficiency

The number of shuffles that are sufficient for a "good" level of randomness depends on the type of shuffle and the measure of "good enough randomness", which in turn depends on the game in question. For most games, four to seven riffle shuffles are sufficient: for unsuited games such asblackjack
Blackjack, formerly also Black Jack and Vingt-Un, is the American member of a global family of banking games known as Twenty-One (card game), Twenty-One, whose relatives include the British game of Pontoon (card game), Pontoon and the European ...

, four riffle shuffles are sufficient, while for suited games, seven riffle shuffles are necessary. There are some games, however, for which even seven riffle shuffles are insufficient.
In practice the number of shuffles required depends both on the quality of the shuffle and how significant non-randomness is, particularly how good the people playing are at noticing and using non-randomness. Two to four shuffles is good enough for casual play. But in club play, good bridge
A bridge is a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules t ...

players take advantage of non-randomness after four shuffles, and top blackjack players supposedly track aces through the deck; this is known as "ace tracking", or more generally, as "shuffle track
Shuffle tracking is an advantage gambling technique where a player tracks certain cards or sequences of cards through a series of shuffling, shuffles. Shuffle tracking is typically done in blackjack games, although it can be done in other card games ...

ing".
Research

Following early research atBell Labs
Nokia Bell Labs (formerly named Bell Labs Innovations (1996–2007), AT&T Bell Laboratories (1984–1996) and Bell Telephone Laboratories (1925–1984)) is an American industrial research and scientific development company
A company, ab ...

, which was abandoned in 1955, the question of how many shuffles was required remained open until 1990, when it was convincingly solved as ''seven shuffles,'' as elaborated below. Some results preceded this, and refinements have continued since.
A leading figure in the mathematics of shuffling is mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

and magician Persi Diaconis
Persi Warren Diaconis (; born January 31, 1945) is an American mathematician of Greece, Greek descent and former professional Magic (illusion), magician. He is a Professor of Statistics and Mathematics at Stanford University.
He is particularly k ...

, who began studying the question around 1970, and has authored many papers in the 1980s, 1990s, and 2000s on the subject with numerous co-authors. Most famous is , co-authored with mathematician Dave Bayer, which analyzed the Gilbert–Shannon–Reeds modelIn the mathematics of shuffling playing cards, the Gilbert–Shannon–Reeds model is a probability distribution on riffle shuffle permutations that has been reported to be a good match for experimentally observed outcomes of human shuffling, and tha ...

of random riffle shuffling and concluded that the deck did not start to become random until five good riffle shuffles, and was truly random after seven, in the precise sense of variation distance described in Markov chain mixing time; of course, you would need more shuffles if your shuffling technique is poor. Recently, the work of Trefethen et al. has questioned some of Diaconis' results, concluding that six shuffles are enough. The difference hinges on how each measured the randomness of the deck. Diaconis used a very sensitive test of randomness, and therefore needed to shuffle more. Even more sensitive measures exist, and the question of what measure is best for specific card games is still open. Diaconis released a response indicating that you only need four shuffles for un-suited games such as blackjack
Blackjack, formerly also Black Jack and Vingt-Un, is the American member of a global family of banking games known as Twenty-One (card game), Twenty-One, whose relatives include the British game of Pontoon (card game), Pontoon and the European ...

.
On the other hand, variation distance may be too forgiving a measure and seven riffle shuffles may be many too few. For example, seven shuffles of a new deck leaves an 81% probability of winning New Age Solitaire where the probability is 50% with a uniform random deck. One sensitive test for randomness uses a standard deck without the joker
Joker, The Joker or The Jokers may refer to:
* Joker (playing card)
* Jester, a person employed to tell jokes and provide entertainment
Fictional characters Print
* Joker (character), a DC comics character
** The Joker (comic book), ''The Joker'' ...

s divided into suits with two suits in ascending order from ace to king, and the other two suits in reverse. (Many decks already come ordered this way when new.) After shuffling, the measure of randomness is the number of rising sequences that are left in each suit.
Algorithms

If a computer has access to purely random numbers, it is capable of generating a "perfect shuffle", arandom permutation A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simu ...

of the cards; beware that this terminology (an algorithm that perfectly randomizes the deck) differs from "a perfectly executed single shuffle", notably a perfectly interleaving faro shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling
Shuffling is a procedure used to randomization, randomize a deck of playing cards to provide an element of chance in card games. Shuffling is oft ...

. The Fisher–Yates shuffle
The Fisher–Yates shuffle is an algorithm for generating a random permutation of a finite sequence—in plain terms, the algorithm shuffling, shuffles the sequence. The algorithm effectively puts all the elements into a hat; it continually deter ...

, popularized by Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist
A computer scientist is a person who has acquired the knowledge of computer science
Computer science deals with the theoretical foundations of information, ...

, is simple (a few lines of code) and efficient ( O(''n'') on an ''n''-card deck, assuming constant time for fundamental steps) algorithm for doing this. Shuffling can be seen as the opposite of sorting
Sorting is any process of arranging items systematically, and has two common, yet distinct meanings:
# Collating order, ordering: arranging items in a sequence ordered by some criterion;
# categorization, categorizing: grouping items with simil ...

.
There are other, less-desirable algorithms in common use. For example, one can assign a random number to each card, and then sort the cards in order of their random numbers. This will generate a random permutation, unless any of the random numbers generated are the same as any others (i.e. pairs, triplets etc.). This can be eliminated either by adjusting one of the pair's values randomly up or down by a small amount, or reduced to an arbitrarily low probability by choosing a sufficiently wide range of random number choices. If using efficient sorting such as mergesort
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algor ...

or heapsort
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of com ...

this is an O(''n'' log ''n'') average and worst-case algorithm.
Online gambling

These issues are of considerable commercial importance inonline gambling
Online gambling (or Internet gambling) is any kind of gambling
Gambling (also known as betting) is the wagering something of ("the stakes") on an with an uncertain outcome with the intent of winning something else of value. Gambling thus ...

, where the randomness of the shuffling of packs of simulated cards for online card games is crucial. For this reason, many online gambling sites provide descriptions of their shuffling algorithms and the sources of randomness used to drive these algorithms, with some gambling sites also providing auditors' reports of the performance of their systems.
See also

*Card manipulation
Card manipulation is the branch of Magic (illusion), magical illusion that deals with creating effects using sleight of hand techniques involving playing cards. Card manipulation is often used in magical performances, especially in Close-up magic ...

*Mental poker
Mental poker is the common name for a set of cryptographic problems that concerns playing a fair game over distance without the need for a trusted third party. The term is also applied to the theories surrounding these problems and their possible so ...

* Solitaire (cipher)
References

* * * * * * * *Footnotes

External links

Physical card shuffling:Illustrated guide to several shuffling methods

Magician's tool with much shuffling simulation

Mathematics of shuffling:

Real World Shuffling In Practice

*Ivars Peterson's MathTrek

Real world (historical) application:

How We Learned to Cheat at Online Poker: A Study in Software Security

{{Non trick-taking card games Card game terminology Card shuffling