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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.
For a deck of given size, the number of Mongean shuffles that it takes to return a deck to starting position, is known sequence A in the OEIS.
Twelve perfect Mongean shuffles restore a card deck. 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, 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.
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 of Mexico as a protection from gamblers and con men arriving from the United States.
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 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 as a particularly effective example of the false shuffle.
In this shuffle, the entire deck remains in its original order, although spectators think they see an honest riffle shuffle. Casinos often equip their tables with shuffling machines instead of having croupiers 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 Baccarat tables.
There are exactly 52 factorial expressed in shorthand as 52! 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.
The number of shuffles which 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: 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 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 ".
Following early research at Bell Labs , which was abandoned in , the question of how many shuffles was required remained open until , when it was convincingly solved as seven shuffles, as elaborated below.
A leading figure in the mathematics of shuffling is mathematician and magician Persi Diaconis , who began studying the question around ,  and has authored many papers in the s, s, and s on the subject with numerous co-authors.
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.
On the other hand, variation distance may be too forgiving a measure and seven riffle shuffles may be many too few.
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.
If a computer has access to purely random numbers, it is capable of generating a "perfect shuffle", a random permutation 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 Fisher—Yates shuffle , popularized by Donald Knuth , 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. 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.
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 or heapsort this is an O n log n average and worst-case algorithm.
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.
The number one key to avoiding the exposure of cards is to keep the cards low to the table. The cards should never be more than an inch or so off the table.
The intuitive method by which someone squares up, say, a pile of papers is to lift it off the table and tap its edges against the table.
Keep yourself from falling into this habit when dealing with cards. It shows off the bottom of the pack to anyone who cares to look, and many of the other cards sticking out at odd angles will expose their indices.
Develop the habit of squaring up the pack with your fingers without picking it up off the table. The riffle shuffle begins by splitting the pack into two.
Hold the bottom half of the pack in landscape orientation long edge parallel to the edge of the table closest to you , keeping it flat against the table and secure with one hand.
Then slide the top half of the pack off with your other hand, keeping it close to the other half of the pack, and pulling it in the direction of the long edges of the pack, until you have two half-decks sitting side by side next to each other.
Next, orient the two half-decks in an inverted V the point of the V pointing away from you. Move the decks toward one another, keeping them square with your index fingers on the short edges of the deck opposite you, your thumbs on the long edges of the deck inside the V, and your other fingers on the long edges of the deck on the outside of the V.
Then, perform the actual riffle by arching the corners of the cards closest to one another, bending them between your index fingers, which are moved to rest on top of the deck in the corners of the cards, and your thumbs, which remain in the same position.
Gradually release the pressure from your thumbs, which will cause the cards to begin falling off the bottom of the deck, pressed past your thumbs by your index fingers.
If the two packs are close enough, their corners should interleave. With practice, the cards will naturally alternate between the two packs, thoroughly intermixing the two packs.
Now, complete the shuffle by rotating the two interleaved packs so that they are parallel to one another but still intermixed. Push the two packs together until you can square them up into one shuffled pack.
The strip shuffle is, on its own, not a very powerful shuffling technique. In combination with the riffle shuffle, however, it helps to further randomize the deck by rearranging blocks of the deck, helping to break up runs of cards that remained together through the three riffles.
The strip shuffle is, essentially, the beginning of a riffle shuffle. Hold the pack in landscape orientation, then pull the top fifth or so of the deck off the top, keeping it close to the remainder of the deck, and set it down next to the pack.
Then do the same with the next fifth of the deck, placing it on top of what was the top fifth, and so on, until the entire deck has been gone through in this way.
This means that they mix the cards up, so the order is random and the chances of any hand appearing is fair to all players. There are lots of different shuffle styles which a dealer might employ.
Some shuffles are designed to facilitate cheating, by stripping cards from the deck or by stacking it — placing the cards in a particular order to influence the draw.
Others are designed to showcase the skill of the dealer, and for the entertainment of the players at the table. A standard card deck is usually split into the four suits when first opened, arranged in numerical order.
In order to play games with the cards, they need to be in a random order. Therefore, the cards are shuffled before a game begins. The dealer mixes up the cards using one of a number of techniques or a mixture of several before the cards can be dealt.
Shuffling techniques include the Overhand shuffle, Riffle, Hindu shuffle, Corgi shuffle and the Pile shuffle. Cards may then be shuffled after each round of play, to keep things random, or the deck may be used until all cards run out before a fresh deck is called for.
Shuffling can be used to conceal cheating, for example by stripping cards from the deck or by marking or counting the location of certain cards.
To prevent cheating by the dealer or by the players, some casinos will use a shuffle machine instead of manual shuffling. The mechanism mixes up the cards prior to play, removing the chance that the deck could be tampered with.
A card shuffle requires a certain amount of technique, and some dealers will employ special tricks and styles to entertain the punters.
They split the deck at some point near the middle, and the bottom half of the pack is added to the top of the pile.
Most of these machines were manually run by turning a crank which would activate the inner gears and rollers.
Randomness could be improved by increasing the number of shuffling turns performed by the operators or by increasing the number of boxes, combs or partitioning chambers in the machines.
Some devices were simple boxes with combs that would simulate a manual shuffling like riffle shuffling. In , Charles and William Gunzelmann filed a patent for a simple rhombus -shaped apparatus where the cards were inserted in an upper chamber.
The operator would then turn the box upside down and repeat the operation. A glass windows permitted seeing that all cards had fallen into the compartment.
After , inventors focused on the design of machines that could directly deal the cards, an idea that was already present in Ranney's machine back in In , a dealing table was patented by Laurens Hammond.
His patent description provides interesting insights regarding the problems related to previous machines: He also criticized the randomness of previous shuffling methods and pointed out the risk of predicting the final sequence.
The patent also contains mathematical explanations regarding the inner state of his machine. A motor drove a rotating frame that would distribute 13 cards to each player.
The machine was going through 53 cycles to distribute the 52 cards. During each cycle, a selector plate with 52 notches rotated by one step.
There were four possible depths for the notches and a lug touching the notches would determine which player would receive the card.
Each card was taken from the top of deck and sent to the corresponding player's receptacle using a conveyor track.
The first cycle was used to rotate the plate and ensured that the distribution would start with a new sequence. One property of the machine is that the same player could be served during two or three consecutive cycles.
To increase randomness, the author proposes to use a set of different selector plates or to use another deck being shuffled while people are playing.
The machine was fast enough to shuffle a whole deck in a few seconds. If only one plate was used, the same dealing sequence would appear after 52 deals there were 52 possible starting points on the plate; the starting point was not randomly chosen as the plate always rotated by one step in the same direction during each cycle.
The problem of ensuring randomness using mechanical means was hard to resolve. In the early s, Robert McKay proposed an ingenious machine containing a chamber with 52 balls of different diameters for each player, there were 13 balls with the same size.
This wheel would then rotate, slot by slot, and a rod in contact with the ball would "detect" its diameter. A distribution mechanism could then use the diameter information and take the appropriate action to deal the card to the correct player.
Together with the lottery machines, the shuffling devices continued to evolve. In , Ralph Potter invented an electromechanical machine that would read perforated cards and generates random sequences.
These lights symbolized cards and roulette values. Players pressed on buttons to indicate their choices to the machine. To some extent, his device was one of the first attempts to make a computerized pseudo-random generator and game console.
During the rest of the s, many inventions tried to address the dealing problem, mainly by using rotating frames that would distribute cards to each player around the table.
Rotating parts were common in the shuffling machines; designers often used gears and plates with notches or holes whose purposes were similar to the sequence-generator plate of Hammond's machine.
These shufflers shared some similarities with the machines used in cryptography such as Enigma. This German encryption device used during World War II contained rotors that stepped each time a key was typed and produced an encrypted version of the letter.
Both domains must fulfill mathematical requirements regarding randomness to avoid known patterns, repeated sequences and other kind of statistical weaknesses or biases.
After World War II, engineers tried to generate random sequences using electrical devices. Signals from electrical noise sources like a hot cathode gas discharge tube or a resistor would typically be sent through filters and amplifiers to output one or several random streams.
Such device is described in a patent by Newby et al. According to the patents filled during the s and s, designers created simple devices where a basic shuffling operation was repeated several times by feeding the output deck back into the machine instead of having one complex pass implying many tricky mechanical operations ending up with a poor shuffling and lower reliability.
Some of them tried to reproduce what was manually done during riffle shuffling with cards interleaving each others. Cards picking using rollers in contact with the top or bottom of the deck were still heavily used at that time.
In , Thomas Segers patented his "electronic card dealer" which was not working with real cards but simulating random selections. According to the patent, the design contains multivibrators , logical-and gates and a tube oscillator.
The inventor also indicates that transistors could have been used in the circuit. In , David Erickson and Richard Kronmal proposed a shuffler based upon a logic circuit with binary gates.
The flap forwarded the card into the proper container and was moved by a coil controlled by the pseudo-random generator. Synchronization was important and several methods were used to ensure that the card would follow the correct path.
Until the s, there were not many innovations. In , Edward Sammsel proposed a machine that extracted the cards from the bottom of two deck holders and put them in a second compartment.
Another extractor would eject the card that was taken by the dealer. Photosensors detected how many cards were present in each compartment and if the card was taken by the dealer.
In this case, another card would be processed from the initial holders. From Wikipedia, the free encyclopedia.