Riffle shuffle playing cards11/20/2022 However, under ordinary circumstancesįigure 1. For example, a pack of 52 cards recycles after only 8 perfect “out-shuffles” (i.e. #Riffle shuffle playing cards seriesInstead, the sequences of cards resulting from a series of perfect riffle shuffles cycle through a fixed number of permutations leading back to the original card order. Indeed, in a perfect riffle shuffle of an even-numbered deck, whereby the deck is cut exactly in half and 1 card is dropped alternately from each pile, there would be no randomization at all. Clearly, a single riffle shuffle cannot randomize an ordered deck because the order of cards from each pile is maintained. The process can be performed either by hand or mechanically by an auto shuffler, like the device shown in Figure 1 used to acquire some of the data reported in this paper. To execute a riffle shuffle, one separates (“cuts”) the deck into two piles, then interleaves the cards by dropping them alternately from each pile to reform a single deck. The standard way to mix a deck of cards randomly is to shuffle it, for which purpose the riffle shuffle is perhaps the most widely studied form. (Mathematically, there is on average 1 chance in n of guessing correctly the value of any unrevealed card in a deck of n randomly distributed cards). From a practical standpoint, a deck is considered random if players are unable to predict any sequence of cards following a revealed card. Most card games are conducted under the assumption that the deck in play has been initially randomized. Depending on what one considers a distinct game, experts in the subject estimate the number of card games to be between 1000 and 10,000. Among the most ancient forms of gambling are card games, which developed initially in Asia but became popular in Europe after the invention of printing. Probability as a coherent mathematical theory is said to have been “born in the gaming rooms of the seventeenth century” in attempts to solve one or another betting problem. This paper reports what the author believes to be the most thorough experimental examination to date of the randomization of shuffled cards, using statistical tests previously employed in nuclear physics to search for violations of physical laws by testing different radioactive decay processes for non-randomness. Not only mathematicians and scientists, but the general public as well have shown much interest in the card randomization problem, as reported in popular science periodicals and major news media. random walk diffusion theory theory of phase transitions), quantum physics, computer science, and other fields in which randomly generated data sequences are investigated. The methods employed transcend pure mathematics, and have implications for statistical physics (e.g. Proposed solutions to the problem of determining the number of shuffles required to randomize a deck of cards have drawn upon concepts from probability theory, statistics, combinatorial analysis, group theory, and communication theory. Introduction: The Card Randomization Problem Whereas mechanical shuffling resulted in significantly fewer rising sequencesġ. #Riffle shuffle playing cards manualNumber of manual shuffles matched very closely the theoretical predictionsīased on the Gilbert-Shannon-Reed (GSR) model of riffle shuffles, Number of shuffles and 4) the mean number of rising sequences as a function of Sensitive to different patterns indicative of residual order 2) as aĬonsequence, the threshold number of randomizing shuffles could vary widelyĪmong tests 3) in general, manual shuffling randomized a deck better than mechanical shuffling for a given Manually and by an auto-shuffling device were recorded sequentially andĪnalyzed in respect to 1) the theory of runs, 2) rank ordering, 3) serialĬorrelation, 4) theory of rising sequences, and 5) entropy and information theory.Īmong the outcomes, it was found that: 1) different statistical tests were Permutations of 52-cardĭecks, each subjected to sets of 19 successive riffle shuffles executed Randomizing shuffles, and 3) whether manual or mechanical shuffling randomizesĪ deck more effectively for a given number of shuffles. Whether different statistical tests yield different threshold numbers of Which of the two theoretical approaches made the more accurate prediction, 2) This paper reports a comprehensive experimentalĪnalysis of the card randomization problem for the purposes of determining 1) Which differed in how each defined randomness, has led to statistically different The two principal theoretical approaches to the problem, Question of how many shuffles are required to randomize an initially orderedĭeck of cards is a problem that has fascinated mathematicians, scientists, and
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