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Encyclopedia > Shuffle
The riffle

## Shuffling techniques

Several techniques are used to shuffle a deck of cards. While some techniques achieve a better randomization, other techniques are easier to learn and easier to handle or better suited for special situations. Randomization is the process of making something random. ...

### Riffle

The most common shuffling technique is called a riffle, 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 intertwined. 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 a bit more difficult, it is often used in casinos because it minimizes the risk of exposing cards during the shuffle. The Trump Taj Mahal in Atlantic City, New Jersey. ...

### Stripping or Overhand

Another procedure is called stripping or overhand or slide shuffle, where small groups of cards are removed from the top or bottom of a deck and replaced on the opposite side (or just assembled on the table in reverse order). Although this is a much less effective randomizing procedure, it remains a very common shuffle, especially amongst occasional players.

### Pushing

Pushing 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 and is quite difficult to master. Eight perfect Faro shuffles in succession will bring the deck back to its original order.

### Hindu 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, and the maneuver is repeated over and over until the deck is all in the second hand.

### Pile shuffle

Cards are arranged in piles by putting the top card from the deck in turn on one of several piles. Then the piles are stacked on top of each other. This ensures that cards that were next to each other are now separated. This is the only method of "human" shuffling approved in bridge when four piles are used (each pile is then assigned to the four players in this game). "Machine Shuffling" is also allowed when required (see below). Contract bridge, usually known simply as bridge, is a trick-taking card game of skill, and partly of chance, for four players, who form two partnerships (sides). ...

### Beginner shuffle

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.

This is also used periodically in casinos, where it is called a "wash" or "scramble". A typical sequence between hands of poker, for example, is a wash, two riffles, a strip, a third riffle, and a cut, which an experienced dealer can accomplish in as little as five seconds. [citation needed] Poker Room at the Trump Taj Mahal, Atlantic City, New Jersey Poker is a card game, the most popular of a class of games called vying games, in which players with fully or partially concealed cards make wagers into a central pot, which is awarded to the player or players...

### 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 1,2,3,4,5,6,...,2n, would be a deck with the cards in the following order: 2n,2n − 2,2n − 4,...,4,2,1,3,...,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). The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

### 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 will then alternately fall into each other, much like a zipper. A flourish can by added by springing the packets together by applying pressure and bending them from above. Closeup of the zipper on a pair of jeans This article is about the fastening device called zipper zip, zipper, or the acronym ZIP, see zip (disambiguation). ...

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 about the right pressure when pushing the cards into each other. If one does perform 8 perfect Faro Shuffles in a row, the order of the deck will return to the original order, if there are 52 cards in the deck.

## False shuffles

Magicians, sleight-of-hand artists, and card cheats employ various methods of shuffling whereby the deck appears to have been shuffled fairly, when in reality the order of the cards stays exactly the same. 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". Magician redirects here. ... Sleight of hand, also known as prestidigitation (quick fingers) or legerdemain (pronounced in French, from lightness of hand or deformation of le jeu de main i. ... A card sharp (also commonly known as card shark) is a person who purposely cheats at cards with the aim of making money. ...

## Shuffling machines

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 which perform continuous shuffles on a pack of cards, and can produce any number of cards on demand. 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. The Trump Taj Mahal in Atlantic City, New Jersey. ...

## Randomization

There are approximately 8 x 1067 (52 factorial) possible ways to order the cards in a 52-card deck. 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. In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ... This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities. ...

The mathematician and magician Persi Diaconis is an expert on the theory and practice of card shuffling, and an author of a famous paper on the number of shuffles needed to randomize a deck, concluding that it did not start to become random until five good riffle shuffles, and was truly random after seven. (You would need more shuffles if your shuffling technique is poor, of course.) Recently, the work of Trefethen et al. has questioned some of Diaconis' results, concluding that six shuffles is 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. Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... Magician redirects here. ... Persi W. Diaconis (born January 31, 1945) is an American mathematician and former professional magician. ...

Here is an extremely sensitive test to experiment with. Take a standard deck without the jokers. Divide it 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.) Shuffle to your satisfaction. Then go through the deck trying to pull out each suit in the order ace, two, three ... When you reach the top of the deck, start over. How many passes did it take to pull out each suit? The Fool in Tarot of Marseille The Joker is a special card found in most modern decks of playing cards. ...

What you are seeing is how many rising sequences are left in each suit. It probably takes more shuffles than you think to both get rid of rising sequences in the suits which were assembled that way, and add them to the ones that were not!

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. 2–4 shuffles is good enough for casual play. But in club play, good bridge players take advantage of non-randomness after 4 shuffles, and top blackjack players literally track aces through the deck; this is known as "ace tracking", or more generally, as "shuffle tracking". Contract bridge, more usually known as Bridge, is a trick_taking card game for four players who form two partnerships, or sides. The partners on each side sit opposite one another. ... Blackjack! The face cards (Jack, Queen, and King) count as 10 points, and the Ace counts as 1 or 11. ...

## Shuffling algorithms

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 two of the random numbers generated are the same. This can be eliminated either by retrying these cases, or reduced to an arbitrarily low probability by choosing a sufficiently wide range of random number choices. A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. ... Flowcharts are often used to represent algorithms. ... Donald Knuth at a reception for the Open Content Alliance. ...

The second, generally known as the Knuth shuffle or Fisher-Yates shuffle[1], is a linear-time algorithm (as opposed to the previous O(n log n) algorithm if using efficient sorting such as mergesort or heapsort) 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 computational complexity, an algorithm is said to take linear time, or O(n) time, if the time it requires is proportional to the size of the input, which is usually denoted n. ... It has been suggested that Landau notation be merged into this article or section. ... In computer science, merge sort or mergesort is a sort algorithm for rearranging lists (or any other data structure that can only be accessed sequentially, e. ... Heapsort is one of the best general-purpose sorting algorithms, a comparison sort and part of the selection sort family. ...

Notice that great care needs to be taken in implementing the Knuth shuffle; even slight deviations from the correct algorithm will produce biased shuffles. For example, working your way through the pack swapping each card in turn with a random card from any part of the pack is an algorithm with nn different possible execution paths, yet there are only n! permutations. A counting argument based on the pigeonhole principle will clearly show that this algorithm cannot produce an unbiased shuffle, unlike the true Knuth shuffle, which has n! execution paths which match up one-to-one with the possible permutations. Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... The inspiration for the name of the principle: pigeons in holes. ... A bijective function. ...

Whichever algorithm is chosen, it is important that a source of truly random numbers is used as the input to the shuffling algorithm. If a biased or pseudo-random source of random numbers is used, the output shuffles may be non-random in a way that is hard to detect, but easy to exploit by someone who knows the characteristics of the supposedly "random" number source.

Performing a flourish with a deck of playing cards. ... The Solitaire cryptographic algorithm was designed by Bruce Schneier for use in Neal Stephensons Cryptonomicon (where it was initially called Pontifex). ...

## References

• D. Aldous and P. Diaconis, "Shuffling cards and stopping times", American Mathematical Monthly 93 (1986), 333–348
• Trefethen, L. N. and Trefethen, L. M. "How many shuffles to randomize a deck of cards?" Proceedings of the Royal Society London A 456, 2561–2568 (2000)

Results from FactBites:

 Shuffle - Wikipedia, the free encyclopedia (1763 words) Shuffling is often followed by a cut, to ensure that the shuffler has not manipulated the outcome. The most common shuffling technique is called a riffle, 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 intertwined. The mathematician and magician Persi Diaconis is an expert on the theory and practice of card shuffling, and an author of a famous paper on the number of shuffles needed to randomize a deck, concluding that it did not start to become random until five good riffle shuffles, and was truly random after seven.
More results at FactBites »

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