## Fewer Shuffles Suffice 101

Posted
by
kdawson

from the off-to-buffalo dept.

from the off-to-buffalo dept.

An anonymous reader writes

*"You may have heard that it takes about seven shuffles to mix up a deck of cards to near randomness. Turns out, though, that most of the time, perfect randomness is more than you need. In blackjack, for example, you don't care about suits. The same mathematician who developed the original result now says that for many games, four shuffles is enough. And the result isn't only important for card sharks. It helps reveal the math underlying Markov Chain Monte Carlo simulations, telling applied mathematicians when they can stop their simulations."*
## Help! (Score:5, Funny)

My reply was shuffled 4 times and it's now at a completely random position!## 4 shuffles... (Score:3, Funny)

4 shuffles should be enough for everyone.

## Re: (Score:3, Insightful)

3, if the dealer is clumsy or awkward.

Unlike blackjack, many of the poker variants pay better the more hands you can 'play'. Slow shufflers ruin my hand/hour numbers, and subsequently my $/hour.

## Re: (Score:1)

in poker the suit matters so, the new limit doesn't apply.

## Re: (Score:2)

in poker the suit matters so, the new limit doesn't apply.

The suit is an additional data point effected by the randomization of shuffling. The number of shuffles still applies according to the maths in TFA.

## Re: (Score:3, Informative)

Any decent brick and mortar casino will have an auto shuffler. Even the local dog tracks here in FL that run card games use them.

## Re: (Score:1)

I guess that would be easily got around by running the cards through the machine a second time.

## Re: (Score:3)

Any decent brick and mortar casino will have an auto shuffler. Even the local dog tracks here in FL that run card games use them.

Having been a frequent visitor to casinos in the recent past, I can say that the use of auto-shufflers varies widely depending on the casino, or even different tables in the same casino. Some players, especially blackjack players, have a superstition that the auto-shuffler is somehow "rigging" the game to give the house a bigger advantage. I have seen casinos with auto-shufflers at every table hand shuffling cards because the reaction from players had been so negative.

I have to say that this discovery appli

## Re: (Score:3, Informative)

This paranoia seems kind of funny considering a well practiced dealer can become a "mechanic" and deal exactly what he wants to the participants, with only the keenest eyes catching it.

It's funny because some of the same sentiment is heard in online poker circles from players that distrust the shuffling algorithms used. In the past this distrust was actually a valid sentiment, with predictable patterns being found in specific online card rooms. But these days it's still a very very small risk given that the

## Re: (Score:3, Insightful)

I just find it kind of ironic that people are willing to gamble with a random deck of cards, yet unwilling to overlook the smallest risk of a rigged deck.

Since you discuss the online vs brick/mortar differences. Your final comment is even more profound, imho.

I just find it kind of ironic that people are willing to gamble with a random deck of cards, yet unwilling to overlook the smallest risk of a rigged deck.

I think its far more likely for an online deck in poker or multi-deck blackjack to be more (dare I say truly?) randomized than a manual shuffle based on super fast CPU crunched RNG's.

At the end of the day, the number of hands are finite, but it's a really big number for a single 52 card deck.

I found that in Hold'em, there are 1326 possible starting hands and in Stud, 2,598,960 possible.

Source1 [top15poker.com].

Source2 [wikipedia.org].

## Re:TGIF (Score:4, Interesting)

Theology.)## In Soviet Russia... (Score:2, Funny)

## Re: (Score:2)

When you count how many cards you have in quantum poker, you no longer know which cards you hold. Conversely, if you check what cards you are holding, you no longer know how many they are.

## Think of the Children! (Score:2)

It helps reveal the math underlying Markov Chain Monte Carlo simulations, telling applied mathematicians when they can stop their simulations.

Please! Stop your simulations already! Think of the Children!

## oh no (Score:2, Funny)

Quick, somebody report a bug to Microsoft. Free Cell and Hearts need a patch!

## Re: (Score:2)

Well, they added the ribbon toolbar to Paint and Wordpad, maybe they'll spend a few minutes on these too!

## Good one... (Score:3, Funny)

"We're all enthusiastic," Diaconis says, "because you can describe it to

your mom, the math is hard, and the results are interesting."RTFA just for it to turn out to be a Your-Mom joke. Thanks guys. You really got me.

## Re: (Score:1, Funny)

And we all know what "the math" is a euphemism for...

## Re: (Score:1, Funny)

Arithmetic?

## Re: (Score:1, Troll)

I RTFPDF and here's the decrypted prison code

http://i36.tinypic.com/35na4hd.png [tinypic.com]

## xkcd (Score:4, Funny)

## Could have told you that decades ago. (Score:2)

## Re: (Score:2, Informative)

never ascribe to a heck of a shuffling that which can be adequately explained by stacking the deck and bottom dealing.

## Re: (Score:1)

guaranteeingthat that full house won't go back to the same person. It's equally important to ensure that it might go back to them as it is to ensure that it won't. Randomness is about randomness, not about getting different hands.## Re: (Score:2)

I would say the chances would be the same as any other random cards end up being dealt to one person. We always talk about how "rare" it is for a specific hand to be dealt out of a deck, but in reality its just as likely as getting any other 5 cards dealt to you. Its just the ridiculous high number of combinations available for a deck make it seem like they're rare. You have just as much chance of being dealt a royal flush as you do any other specific 5 cards. Its just that there are more combinations of 5

## How is this random? (Score:3, Interesting)

I've seen this assertion, and never quite understood it. I mean, if you're doing a perfect interleave shuffle, dividing the cards into two piles A and B and then weaving them together ABABABAB and so on, in what sense is that random? No matter how many times you iterate, it's still a purely deterministic process and you can easily predict the order of cards in the deck post-shuffle. So how do you get a random non-predictable card order out of this?

I can understand that in real life, you're not going to shuffle perfectly, there'll be a few more cards in one pile than the other, your interleave will occasionally do something like ABBBAABA instead of being perfect, and so forth, but in that case I don't see how you can say "Oh, it'll be random after 7 shuffles," because it'll depend on the amount of imperfection. And even then, this still doesn't strike me as actual random behavior; it's still deterministic, it just doesn't matter because a human observer isn't capable of observing the information he'd need to predict card order. But that information's still *there*, and a theoretical perfect observer will still be able to know what the card order is. With a truly random sequence, there is *no* way to determine the order, even given a perfect observer.

## Re: (Score:3, Insightful)

## Re: (Score:3, Informative)

I think you needed to glance at the article:

We're not taking about perfect shuffling here.

## Re: (Score:1)

but they're saying "a few" its hardly quantitative either, how can you do maths on "a few"?

## Re: (Score:2)

You can't - it's all bullshit.

When these retards talk about "random" they really mean "statistically non-uniform".

The more effort you put into making things "random" the less random they actually are. The goal of card simulations, shuffling, etc. is to make each hand statistically like every other hand. This is done to ensure the money flows in as expected.

If you were dealing with an actual random deck, you would be able to assume only the accepted probabilities of a statistically non-uniform deck. Once

## Re: (Score:2)

Really, you should shuffle the deck a random number of times. Determine that random number by covering your (exposed) genitals with delicious honey and counting the number of insects that are stuck in the honey 1 hour later.

But that could easily be influenced by variables outside your control(time of day, place of exposure, temperature, relative humidity, season, alignment of the planets?), and thus, totally not random. It boggles the mind!

## Re: (Score:2)

No, it's more random.

If you seek to control it, you're interested in statistical non-uniformity.

## a random number of times (Score:2)

Dood - it doesn't get any more random after a while...

There are other problems with your idea as well, but I can't be bothered...

## Re: (Score:2)

Define "random".

Something is either random, or not.

(Hint: NOTHING is random)

(HINT: No, not even that. We just don't understand quantum physics yet.)

## Re: (Score:2)

(HINT: No, not even that. We just don't understand quantum physics yet.)Hmm, lets see...

quantum physics: better part of a century of experimental confirmation in thousands of independent tests with accuracy exceeding any other scientific theory.

your assertion: just your say so.

Yeah, gotta go with quantum physics here.

Executive summary: The physical world really is inherently random. Deal.

## Re: (Score:2)

Accuracy and precision in our observations.

Almost no understanding of the causes of what we're observing.

## Re: (Score:1)

## Re: (Score:2)

Computers don't normally shuffle with the "divide into two stacks and interleave" algorithm. They tend to do more of a "remove a random item from A and place in stack B, repeat until A is empty" algorithm.

And as for deterministic, you forgot the variable of where the deck is split (size of A relative to B) which is another source of imperfection.

Layne

## Re: (Score:2)

My algorithm makes a random deck in a single pass (assuming cards, 52 iterations - 51 with a "take the last item" optimization). Yours would require additional passes to ensure that it was random.....I haven't run the numbers, but I would guess somewhere in the neighborhood of 4x52 iterations (formerly 7x52).... :D

Layne

## Re: (Score:3, Insightful)

## Re: (Score:3, Interesting)

I thought about spending several paragraphs and a couple of examples explaining this, but from past experience I have learned that probability is sometimes counter-intuitive and some people just never get it.

No joke, you ever try to explain the Monte Hall logic of changing doors? I've had people fight to the bitter end on that one. I've drawn pictures. I even wrote a little .Net app with three doors and a picture of a goat just to help people comprehend...

-Rick

## Re: (Score:1)

## Re: (Score:3, Insightful)

There's a subtle point that isn't obvious -- the revealed door is never the winning door.

You have a 2/3 chance of guessing wrong. But in that case, the other wrong door will be revealed, so swapping means you win.

Meanwhile, you have a 1/3 chance of guessing correctly, and therefore not swapping only gives you a 1/3 chance of winning.

## Re: (Score:2)

That's a pretty good explanation. I would have modded you insightful, but my mouse slipped.

## Re: (Score:1)

No joke, you ever try to explain the Monte Hall logic of changing doors? I've had people fight to the bitter end on that one. I've drawn pictures. I even wrote a little .Net app with three doors and a picture of a goat just to help people comprehend...

-Rick

You're doing it wrong. What I've found works is to extrapolate to ten, one hundred, ten thousand, or a million doors, and describe it as Monty Hall going down the row of doors, looking behind them, then opening them, except one. I've never had anybody argue the point with the million doors.

## Re: (Score:2)

No joke, you ever try to explain the Monte Hall logic of changing doors? I've had people fight to the bitter end on that one. I've drawn pictures.

What I've found works is to extrapolate to ten, one hundred, ten thousand, or a million doors, and describe it as Monty Hall going down the row of doors, looking behind them, then opening them, except one. I've never had anybody argue the point with the million doors.

I don't get how changing the number of doors makes the principle any clearer...

I was one of those "fight it to the bitter end" types (there was a pint of Ben & Jerry's on the line) - the bit that finally helped me understand was the realization that the choice of which door gets opened is constrained by your first-choice door - then I just ran the possible combinations (1 in 3 chance that my initial door choice was the right door, in which case switching results in failure, 2 in 3 chance that my initial

## Re: (Score:2)

## Re: (Score:1)

## Re: (Score:3, Insightful)

## Re: (Score:1)

I don't need to read it. It's clear just looking at the summary that it is plagiarized.

How many shuffles necessary to get randomness? Four. [xkcd.com]

QED.

- RG>

## Re:How is this random? (Score:5, Informative)

Yes, they model the imperfect interleave, and they assume that more cards will fall from the larger of the two stacks. The randomness comes, of course, from the fact that the number of cards which fall from each stack into each "leaf" is effectively non-deterministic and unobservable.

Your perfect observer would also argue that rolling a die is a deterministic process: he needs only to observe the force (acceleration) I apply to my hand/arm, and can in principle reconstruct the path of the die. However we assume that the observer can't do this, as long as I'm putting some effort into the shaking. [As a small semantic point, note that I could put up a screen and block your observer's view of my hand; by your definition, I have now made the die a "truly random" number generator even though I haven't really changed anything. We need to be careful saying things like "perfect observer" because there isn't really any such thing, just like there is no such thing as an unstoppable force, or impenetrable barrier.]

Regarding the seven shuffles thing, the result is rather robust to variation in the number of cards which drop. This is because the eigenstates of the deck-system, corresponding to unmixed-states, decay geometrically with respect to applying the shuffle operator. Intuitively, every time you apply a shuffle, it becomes less likely to see a given pre-existing pattern in the cards. Let's say it becomes x times as likely. If the shuffles are independent, then after seven it'll be approximately x^7 times as likely. You may object that you could shuffle back to the pre-existing pattern; the fact is, that probability is very small, and the theory does account for it.

Now, x^7 is going to be a pretty small number, whether x=0.5 or x=0.2 (but not if x=0.9). Establishing the upper bound on the relevant x is of course, part of the paper...

## Re: 7 vs 4 Shuffles (Score:2)

I used to play complicated variants of Solitaire. I needed pretty much every one of those shuffles and then usually one more to make up for the terrible shuffle that was done really horribly.

In these variants, one small blockage of 3 cards stuck together from last game due to an incorrect shuffle can lose you the next round.

I file this under Texas SharpShooter.

http://en.wikipedia.org/wiki/Texas_sharpshooter_fallacy [wikipedia.org]

"Let's discard games from the set of all games until they qualify under four shuffles!"

## Re: (Score:2)

But maybe the "blockage" occurred purely by chance, even though you did shuffle well. ;-)

That's the problem with randomness... you can never be sure: http://www.random.org/analysis/dilbert.jpg [random.org]

## Re: (Score:2)

The randomness comes, of course, from the fact that the number of cards which fall from each stack into each "leaf" is effectively non-deterministic and unobservable.Sure it's observable. Record it and play back the film. They don't let you do that in a casino, so it's "random enough" for their purposes, but you can't turn a truly random process into a predictable one by observing it on a finer timescale.

[As a small semantic point, note that I could put up a screen and block your observer's view of my han## Re: (Score:1)

there are no such observers for radioactive decay

I'll dare make the claim that it's just a matter of human technology not being advanced enough to create such observers. If the abscence of a sufficiently accurate observer defines true randomness, then shuffling a deck of card 4/7/whatever times creates true randomness if no cam is recording the shuffling. Otherwise, true randomness simply don't exist.

## Re: (Score:2)

I'll dare make the claim that it's just a matter of human technology not being advanced enough to create such observers.

Nope. Observed phenomena in quantum mechanics exist that could not exist if the weirder claims of quantum mechanics were not true, including the inherent perfect unpredictability (i.e. randomness ) of certain phenomena. In other words, if atomic decay could be even theoretically predicted, certain experiments that have been done would have had different results.

## Re: (Score:2)

Nobody lets you shuffle, do they?

While you may be technically right that a deck shuffle may not be truly random in a technical sense, the randomness they're talking about is certainly good enough for any practical situation.

Note that your playing back the video of either the die or the card shuffle is cheating. I can "predict" radioactive decay too if I'm allowed to film the particles flying out of the nuclei and play them back later. The trick is to predict the process

beforeit happens, not after you've## Re: (Score:2)

You need to make a distinction between outcomes of a process, and the process itself. Even if you can fully understand the shuffle by reviewing several gigabytes of video after-the-fact, doesn't mean that it was

implementeddeterministically. (On a similar note, a perfect deterministic understanding of that shuffle won't help you with the next shuffle... why is that?)Similarly, suppose that we can perfectly observe radioactive decay in a recording. That's fine, but the atoms still decayed randomly...

A full

## Re: (Score:2)

"Randomness is objective, not dependent on the observer, and there are theoretical observers who could predict the outcome of a die roll; there are no such observers for radioactive decay, or the emission of Hawking radiation, or of shot noise."

This is a completely philosophical claim and non-scientific. There are two primary camps for probability interpretations. (1) The "frequentist" camp, where probability is an objective statement about the ratio of successes over many trials (i.e., no statement can be

## Re: (Score:2)

Bayesianism != subjective.

That section is awful. It opens up by

definingBayesians as "subjectivists" and spews on about that a while, before finally (in the last paragraph!) saying that "the use of Bayesian probability involves specifying a prior probability...",which is the actual definition of Bayesianism!At least they mention de Finetti, but for an antidote look at e.g. http://en.wikipedia.org/wiki/Empirical_Bayes [wikipedia.org], which is unfortunately at a much higher technical level than the bullshit section in Pro

## Re: (Score:1)

fix it! Imagine that!## Re: (Score:2)

Yeah, and after I put one or two (precious!) hours into correcting it, it'll likely be reverted by some pompous jackoff (some people are really religious about the Bayes=subjective thing) at which point I'll have the option of ditching my time, or getting into a protracted edit war. I'd rather spend the time writing a paper or running a simulation.

You know, some people don't have time to waste on amateur drama about scholarly topics. Imagine

that!## Re: (Score:3, Informative)

Sure it's observable. Record it and play back the film. They don't let you do that in a casino, so it's "random enough" for their purposes, but you can't turn a truly random process into a predictable one by observing it on a finer timescale.

You seem to be under the impression that observability creates non-randomness. It doesn't. It only creates non-randomness *if* observations done *before* the randomizing process can predict the results.

Suppose I had a device which used radioactive decay to produce perfect random whole numbers between 1 and any arbitrary number up to 52. I set an entire deck of cards in front of me, laid out side by side. I ask the device to pick a number from 1 to 52, take that card, and turn it over to one side. I then do

## Re: (Score:2)

As a small semantic point, note that I could put up a screen and block your observer's view of my hand; by your definition, I have now made the die a "truly random" number generator even though I haven't really changed anything.

I think that's an essential point about randomness. Or at least thermodynamical entropy, with which I'm more familiar. Entropy is really a measure of how little you know about a system, and the better you define a system, the smaller its entropy.

Entropy can be calculated as k*log(W), where W is the number of microstates corresponding to your idea of the system. For example, there are more ways of arranging molecules into 1 liter of liquid water, than into 1 liter of frozen water. Thus we say that liquid

## Re: (Score:2)

In fact, if you do a perfect ABABAB shuffle enough times, the deck will return to its original position.

It's called the Faro Shuffle:

http://en.wikipedia.org/wiki/Perfect_shuffle [wikipedia.org]

But you're right... a perfect shuffle is uniform. A very sloppy shuffle will leave big chunks of cards in the same order. So there is some indeterminate middle ground, I guess.

## Re:How is this random? (Score:5, Informative)

Since I've RTFA a little (I know, I know, this is

slashdot), and IAAM (mathematician) allow me to try to answer.Whether a deck of cards is "random" or not is a subtle (and somewhat meaningless question). Afterall, the deck might follow a pattern but if I don't know the pattern then it appears random to me. In fact, this is exactly how decks of cards work: you've assigned each card with a number from 1 to 52, and you deal the cards by picking the card that's assigned number 1 first, number 2 second, etc. If I don't know how the numbers are assigned, then I can't tell what's coming next so it looks random. On the other hand, if I know what order you've put the cards in, then nothing is a surprise.

Instead of considering whether a deck is "random" or not, we're more interested in how well one can predict what the order of the cards are: either without seeing any of the cards, or after seeing the first few deals. For instance, if I know that you only order your decks in increasing order of rank with the suits randomly ordered, then seeing that the first deal is an Ace of Hearts tells me what the next 12 cards must be. All because I know how you like to order your cards.

This example, of course, never happens. But if instead of being certain about how you order things, what if I knew that you were more likely to order in a certain way? What if you ordered them as above 50% of the time, and the other 50% you riffle shuffled them? Then seeing an Ace of Hearts on the first deal doesn't make me certain about what's coming next, but I have a pretty good idea. Seeing a Two of Hearts re-affirms my hunch, but I still can't be completely certain.

This still isn't quite a real-life example, but it's getting close. If we

knowthat the person shuffling favors certain orders over other, then we can predict what's coming next with better than chance accuracy. So the idea of "randomizing" a deck of cards is to re-order them without having any bias in the new order that we choose.The way to minimize the bias is to select a permutation of 52 cards, with each permutation equally likely to be chosen. So each permutation has a probability of 1/52! chance of being picked (that's 1 / (52*51*50*49*...*1) ). This "uniform distribution" is the best way to keep someone from being able to predict what card is next, even if they've already seen the previous cards. That's because we don't have any bias in how we are ordering, so there's no extra information for them to take advantage of.

When we do a riffle shuffle we are choosing a new ordering of the cards. Obviously we are choosing our re-ordering in a biased way: we're more likely to have cards from the top and bottom interleaved than we are to reverse the order of the deck for instance. So we have a certain distribution of probabilities on the possible permutations, and this distribution is not uniform.

But what if we riffle shuffle again? Given our original deck order, we now have certain probabilities of choosing the various permutations as our new order. And as it turns out, we're a little less likely to be biased in favor of certain permutations. If we keep riffle shuffling over and over again we're smoothing out our bias and heading towards a uniform distribution.

The question of "how many times do we need to shuffle?" is really "how many times do we need to shuffle to be pretty close to the uniform distribution?" There are technical definitions for what it means to be "close to the uniform distribution", but that's the idea.

So a deck of cards has been "randomized" if I tell you the order it started in, I tell how what procedure I'm going to use to pick a new re-ordering, and you

stillcan't tell what order the deck is likely to be in because my procedure is going to choose any of the possible re-orderings with equal probability. Note that you don't get to peak at the deck after each shuffle, you only get to see it at the start.As for the imperfection in the shuffling, TFA tells you the model they use: The c

## Re: (Score:2)

## I for one.... (Score:2, Funny)

## First post (Score:4, Funny)

## Re: (Score:1)

facepalm!

entropy does not decrease

never

## Re: (Score:2)

and can you predict when it will decrease? decreases are

randomfluctuations, you can see when they have happend but you cant see when they are going to happen.If i have random inputs there is no 'blind' (probably the wrong word, but what i mean is without being able to see the cards) algorithm that can predictably make the data less random, without

## Re: (Score:2)

facepalm!

entropy does not decrease

never

Every time I shuffle my deck of one cards it always produces the same order.

## Re: (Score:2)

Incorrect sir!

You are simply defining the lower entropy instances (entropy after n shuffles, after n+1, etc) as indications of HIGHER entropy in the series.

Taken as a single element, which is what matters when playing cards, a shuffled deck can easily have lower entropy than it had prior to being shuffled.

In fact, good sir, I propose that for any statistically non-uniform (this is the goal, NOT randomness) deck n, deck n+1 (obtained by shuffling deck n exactly once) has exactly a 49.9999% chance of being hi

## Completely untrue (Score:2)

Shuffle tracking and sequencing techniques are well developed, and (very) skilled players regularly do these techniques to get an edge against the house.

The house doesn't shuffle thoroughly to ensure randomness, it does so to thwart advantage players.

## Simple substitution cipher program (Score:2)

If you'd like to play with solving simple substitution ciphers using both dictionary attacks and hill-climbing methods (similar to the method described in the paper), try Decrypto. It's open source, too.

http://www.blisstonia.com/software/Decrypto [blisstonia.com]

## There is another variable (i.e. I call BS, YMMV) (Score:4, Informative)

I've played far more than my share of cards, from CCG [wikipedia.org]s and other proprietary games to standard 4-suit 52-card playing cards (learning to shuffle 200-card decks in Magic:TG before we discovered that a 60 card deck was optimal sure made me good at shuffling!), and let me say this:

some people shuffle better than others.Quality of shuffling varies widely; If I concentrate, I can get a clean broken-in deck to shuffle perfectly alternating cards from each half (though this is undesirable as it is not random). On the other end of the spectrum, many people shuffle very large chunks alternating, which is only as random as the cards are clean (which is to say, usually not very random).

Methods of shuffling also vary. There is the standard "Riffle" shuffle that was probably used in this study, there is overhand shuffling (taking small piles of cards from one or both sides of the deck and assembling them in a different order elsewhere), and there are several other methods. Because my riffle can sometimes be too precise, I will actually alternate riffle and overhand shuffles, performing three of each when I shuffle a deck.

In Magic: The Gathering, it is common to table-shuffle, which is essentially dealing out the cards into a set number of piles (usually 4-6 as they each divide a 60 card deck evenly, thus letting you ensure the cards are all there). This assures absolutely no clumping of dirty cards. Since it isn't very random, it should be followed by proper shuffling. (M:TG tournament rules now require three riffle shuffles since some people insist upon table-shuffling to preserve their expensive cards.) I use this method when dealing with dirty standard cards, too.

The WikiPedia page on Shuffling [wikipedia.org] is actually amazingly informative, covering different shuffling methods, fake shuffle tricks (for magic tricks or cheating), shuffle-tracking (for gamblers), and far more math than the article linked in this sciencenews.org article. Give it a gander.

## Re: (Score:2)

Shuffle standards vary depending on the region. As an American, I'm used to riffle shuffle as the standard method. If you try to use another method, even in a casual game, you will likely get complaints from other players that you haven't really shuffled the cards. But in Australia, I found that most people used overhead shuffling. More than once my use of the riffle shuffle was commented on as being usual. However, I did note that the professional poker dealers at the casino in Melbourne did use a ri

## Re: (Score:3, Informative)

Riffle shuffling is "professional-grade" in that it is the most thorough, and it is standard in the States. Throughout Asia and other parts, the "Hindu shuffle," which is very similar to the overhand method, is the most prevalent (as explained at WikiPedia:Shuffling#Hindu shuffle [wikipedia.org]).

Most of the Asians and Australians I've played with actually use Hindu rather than overhand, so I'd guess that's what you saw. The difference is in the delivery of the cards from one pile to the other; in overhand, you're droppi

## Bridge. (Score:2)

I play bridge. All 52 cards matter. 7 shuffles? Hell no!

On an average evening, 24 people shuffle a deck each. On average, the distributions of the cards is far from what you'd expect from a good (say by a computer) shuffle....

## Monte Carlo? (Score:2)

If your random number generator is truly random [fourmilab.ch], a single pass will scatter your dataset. The problem Monte Carlo simulations can run into is not the number of passes through the randomizer but relying on a crappy

## Single deck, 7 perfect shuffles (Score:2)

Get you back where you started.

Try it sometime.

## How many? (Score:2)