Shuffling

"Shuffle" redirects here. For other uses, see Shuffle (disambiguation).

Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome.

A riffle shuffle

Shuffling techniques

Several techniques are used to shuffle a deck of cards. Some techniques are easy to learn while others achieve better randomization or are better suited to special decks.

Riffle

After a riffle shuffle, the cards cascade

A common shuffling technique is called the riffle or dovetail shuffle, 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. This 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 in casinos 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 2nd from the top then it is an in shuffle, otherwise it is known as an out shuffle (which preserves both the top and bottom cards).

Riffle shuffling does, however, carry a risk of damaging cards from excessive bending. Casinos often replace their playing cards to prevent cheating from players that detect deformations in the cards. However, collectible card game cards are considerably less replaceable than playing cards, and CCG cards can be damaged from riffle shuffling, even when protected with card sleeves.

Stripping or overhand

Another procedure is called stripping, overhand, or slide shuffle, where small groups of cards are removed from the top of a deck and placed in the opposite hand (or just assembled on the table) in reverse order.

Hindu shuffle

Also known as "Kattar" or "Kenchi" (Hindi for scissor). 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. Hindu 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 shuffle

Cards are simply dealt out into a number of piles, then the piles are stacked on top of each other. This ensures that cards that were next to each other are now separated. The pile shuffle does not provide a good randomization of the cards (but this can be enormously improved by dealing to the piles in a different order each circuit). It is sometimes used in collectible card games where other forms of shuffling might damage rare cards.

Corgi, Chemmy, Irish or Wash shuffle

Also known as the scramble, beginner shuffle, 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 and small children or if one is inept at shuffling cards. However, the beginner shuffle requires a large surface for spreading out the cards and takes longer than the other methods. The now-often-used name Corgi originated in Yorkshire, England. It has spread and is used in many parts of the UK, and often heard in international poker rooms or tournaments such as the WSOP or EPT. It is quite common in casino poker rooms for dealers to use this method upon introducing a brand new deck, which are packaged in ranked order by suits, before shuffling it by some other means (i.e., a riffle shuffle or shuffling machine).

Mongean shuffle

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 $\scriptstyle 1, 2, 3, 4, 5, 6, \dots, 2n$, would be a deck with the cards in the following order: $\scriptstyle 2n, 2n-2, 2n-4, \dots, 4, 2, 1, 3, \dots, 2n-3, 2n-1$.

For a deck of given size, the number of Mongean shuffles that it takes to return a deck to starting position, is known (sequence A019567 in OEIS). Twelve perfect Mongean shuffles restore a 52-card deck.

Weave and Faro shuffles

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.

The faro shuffle 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 flourish can be added by springing the packets together by applying pressure and bending them from above. 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).

Shuffling machines

Main article: Shuffling machine

Because standard shuffling techniques are seen as weak, and in order to avoid "inside jobs" where employees collaborate with gamblers by performing inadequate shuffles, many casinos employ automatic shuffling machines. They also save time that would otherwise be spent shuffling, allowing several more hands per hour to be played and increasing the profitability of the table. These machines are also used to lessen repetitive motion stress injuries to a dealer. Note that the shuffling machines have to be carefully designed, as they can generate biased shuffles otherwise: the most recent shuffling machines are computer-controlled, though they have not yet fully been integrated into gaming.

Randomization

There are exactly 52 factorial (expressed in shorthand as 52!) possible orderings of the cards in a 52-card deck. This is approximately 8×10⁶⁷ possible orderings. The magnitude of this number means that it is exceedingly improbable that two randomly selected, truly randomized decks, will ever, in the history of cards, 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.

A famous paper by mathematician and magician Persi Diaconis and mathematician Dave Bayer on the number of shuffles needed to randomize a deck 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.^{[citation needed]} Diaconis released a response indicating that you only need four shuffles for un-suited games such as blackjack.^[1][2] 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 (Mann, especially section 10).

One sensitive test for randomness uses a standard deck without the jokers 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.^[3]

In practice the number of shuffles that you need depends both on how good you are at shuffling, and 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 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 tracking".^{[citation needed]}

Shuffling algorithms

Main article: Fisher–Yates shuffle

In a computer, shuffling is equivalent to generating a random permutation of the cards. There are two basic algorithms for doing this, both popularized by Donald Knuth.

The first is simply to assign a random number to each card, and then to 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 assigning new random numbers to these cases, or reduced to an arbitrarily low probability by choosing a sufficiently wide range of random number choices. If using efficient sorting such as mergesort or heapsort, this is an O(n log n) algorithm.

The second, generally known as the Knuth shuffle or Fisher–Yates shuffle, is a linear-time algorithm which involves moving through the pack from top to bottom, swapping each card in turn with another card from a random position in the part of the pack that has not yet been passed through (including itself). Providing that the random numbers are unbiased, this will always generate a random permutation.

In online gaming

These issues are of considerable commercial importance in online gambling, 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
• Mental poker
• Overhand shuffle
• Solitaire (cipher)

References

• Aldous, David; Diaconis, Persi (1986), "Shuffling Cards and Stopping Times", American Mathematical Monthly (The American Mathematical Monthly, Vol. 93, No. 5) 93 (5): 333–348, doi:10.2307/2323590, JSTOR 2323590 
• Bayer, Dave; Diaconis, Persi (1992), "Trailing the Dovetail Shuffle to its Lair", Annals of Applied Probability 2 (2): 295–313, doi:10.1214/aoap/1177005705, http://projecteuclid.org/euclid.aoap/1177005705 
• Diaconis, Persi (1988), Group Representations in Probability and Statistics (Lecture Notes Vol 11), Institute of Mathematical Statistics, pp. 77–84, ISBN 978-0-940600-14-0 
• Diaconis, Persi (2002), Mathematical Developments from the Analysis of Riffle Shuffling, Technical Report 2002-16, Stanford University Department of Statistics, http://www-stat.stanford.edu/reports/papers2002.html ^{[dead link]}
• Diaconis, Persi; Graham, R. L.; Kantor, William M. (1983), "The Mathematics of Perfect Shuffles", Advances in Applied Mathematics 4 (2): 175–196, doi:10.1016/0196-8858(83)90009-X, http://www-stat.stanford.edu/~cgates/PERSI/papers/83_05_shuffles.pdf 
• Mann, Brad (1993?), How many times should you shuffle a deck of cards?, Dartmouth College Chance Project http://www.dartmouth.edu/~chance/index.html, http://www.dartmouth.edu/~chance/teaching_aids/books_articles/Mann.pdf 
• Trefethen, L. N.; Trefethen, L. M. (2000), "How many shuffles to randomize a deck of cards?", Proceedings of the Royal Society, Series A 456 (2002): 2561–2568, doi:10.1098/rspa.2000.0625 
• van Zuylen, Anke; Schalekamp, Frans (2004), "The Achilles' heel of the GSR shuffle: A note on New Age Solitaire", Probability in the Engineering and Informational Sciences (Cambridge University Press) 18 (3): 315–328, ISSN 0269-9648, http://itcs.tsinghua.edu.cn/~anke/NewAgeS.pdf 

Footnotes

1. ^ "Shuffling the cards: Math does the trick". Science News. Friday, November 7, 2008. http://www.sciencenews.org/view/generic/id/38434/title/Shuffling_the_cards_Math_does_the_trick. Retrieved 14 November 2008. "Diaconis and his colleagues are issuing an update. When dealing many gambling games, like blackjack, about four shuffles are enough." 
2. ^ Assaf, Sami; Persi Diaconis, and K. Soundararajan. "A Rule of Thumb for Riffle Shuffling" (PDF). t.b.a.. http://www-stat.stanford.edu/~cgates/PERSI/papers/redblack.pdf. Retrieved 14 November 2008. 
3. ^ Anke Van Zuylen and Frans Schalekamp (July 2004), "The Achilles Heel of the GSR-Shuffle: A note on New Age Solitaire", Probability in the Engineering and Informational Sciences 18 (3): 315–328, http://itcs.tsinghua.edu.cn/~anke/NewAgeS.pdf, retrieved 14 November 2009. 

External links

Physical card shuffling:

• Illustrated guide to several shuffling methods
• Magicians tool with lots of shuffling simulation

Mathematics of shuffling:

• Real World Shuffling In Practice
• Shuffle - MathWorld - Wolfram Research
• Ivars Peterson's MathTrek: Card Shuffling Shenanigans

Real World (Historical) Application:

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