Shuffle
From Free net encyclopedia
- For other uses, see Shuffle (disambiguation).
Image:Card shuffling.jpg A deck of playing cards is randomized by a procedure called shuffling to provide an element of chance in card games. Shuffling is often followed by a cut, to ensure that the shuffler has not manipulated the outcome.
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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.
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.
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 manuever 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 the card game of "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).
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.
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 form 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 <math>1, 2, 3, 4, 5, 6, ..., 2n</math>, would be a deck with the cards in the following order: <math>2n, 2n-2, 2n-4, ..., 4, 2, 1, 3, ..., 2n-3, 2n-1</math>.
For a deck of given size, the number of Mongean shuffles that it takes to return a deck to starting position, is known Template:OEIS.
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".
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.
Randomization
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.
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?
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".
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.
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.
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.
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.
See also
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)
External links
- Real World Shuffling In Practice
- http://www.math.washington.edu/~chartier/Shuffle/
- http://www.nature.com/nsu/001005/001005-8.html
- Shuffle - MathWorld - Wolfram Research
- Generating random permutations
- Ivars Peterson's MathTrek: Card Shuffling Shenanigans
- Wiki:LinearShuffle and Wiki:HandShuffle discuss some of the mathematics of hand shuffling and computer shuffling algorithms.de:Mischen (Spielkarten)