Seven Perfect Shuffles Randomize a Deck of Cards. But How Many Sloppy Ones?

https://www.quantamagazine.org/seven-perfect-shuffles-randomize-a-deck-of-cards-but-how-many-sloppy-ones-20260617/

by layer8 • 1 month ago

102 59 1 month ago

quantamagazine.org

layer82• 1 month ago

The paper: https://arxiv.org/abs/2510.22783

capitol_• 1 month ago

Shouldn't a perfect shuffle just reorder the cards without adding entropy?

You would need sloppy ones to introduce randomness.

soared2• 1 month ago

I don’t know on perfect shuffles but for the sloppy shuffles, the deck is cut at a random location between each shuffle.

aureate• 1 month ago

See the paragraph beginning "Yet terms and conditions also apply."

HPsquared3• 1 month ago

It's modelled with randomness, each card is taken from left or right with a probability, it's not a deterministic model.

myrmidon• 1 month ago

You misunderstood because the title is ambiguous.

This talks about seven consecutive riffle shuffles ("cut the deck and interleave the piles"): Those are not a "perfect shuffle" (i.e. same probability for every permutation) by themselves, only after doing them several times consecutively (which is kinda suprising by itself).

jtbayly• 1 month ago

A "perfect shuffle" according to the article:

The riffle shuffle has to follow a realistic but strict model where cards are randomly interleaved from the left or right pile one by one. (Each card gets dropped from either the left or the right pile with a probability that’s proportional to the number of cards remaining in that pile. This means that the cards don’t simply alternate between left and right, which would result in a predictable structure; instead, the order might go “left, right, right, left, right, left, left.”)

empath753• 1 month ago

Yeah, a "perfect" shuffle is known as a faro shuffle and it's the basis of a lot of magic tricks, but it's a weird looking shuffle and it sort of ruins the tricks once you can recognize it.

AndyNemmity6• 1 month ago

not a lot of magic tricks. quite few actually. and it's interesting that it ruins the trick for you, because it's still a quite hard maneuver.

as a magician, I'm always still impressed when I see perfect faros.

have_faith• 1 month ago

And 8 perfect shuffles resets it back to starting order (perfect being cards interlaced 1 by 1)

brookst2• 1 month ago

So just do -1 shuffles and save yourself a lot of effort?

HPsquared3• 1 month ago

Quite the assumption here: "cards are randomly interleaved from the left or right pile one by one. (Each card gets dropped from either the left or the right pile with a probability that’s proportional to the number of cards remaining in that pile."

... Why would it be proportional to the number of cards in each pile? (Edit: I suppose the person doing the shuffling might adjust the rate of cards coming from each hand ... But not perfectly and continuously)

tobr• 1 month ago

But not perfectly and continuously

Isn’t that where the randomness comes in?

HPsquared3• 1 month ago

The randomness comes from sampling the probabilities. The strange assumption is that the probabilities are exactly proportional to number of cards currently in each stack.

fwlr• 1 month ago

If there is one card in this pile and no cards in the other, the probability of dropping the card from this pile is one. If instead there are some cards still in the other, a) the probability is less than one, and b) we move one step closer to the first state. So by construction it must be proportional - perhaps a poorly behaved proportionality, but that is still enough for the math to work.

soared2• 1 month ago

Upper limit of 14. I’m curious then - when playing cards with friends we start with a semi -random, but definitely clumped, deck. It gets shuffled a couple times.

How random is that deck? How many “cold spots” does it have? Just how not random of decks are people playing with, and ultimately does that even matter if players lack the knowledge or skill to change their play because of that knowledge?

fisian2• 1 month ago

There's this numberphile video where Persi Diaconis tells a story about bridge and how the players got used to not totally random shuffled decks: https://youtu.be/RIGJH12vVCY?si=cMSx6YEr8MMFgYg6 (Starts at around 3:30).

RNanoware• 1 month ago

Anecdotally, I find that certain card games are more enjoyable with the imperfections of human shuffling: when clumps naturally arise after playing, packing, and unpacking the game several times. An element of organic personality arises when you see a sequence of cards from a previous game. That human element is lost when a computer perfectly shuffles a deck into a never-before-seen orientation.

brookst2• 1 month ago

Games that sort the cards are the worst / most interesting for this. Gin rummy, etc, where the end result of a game is sorted groups of same-numbers and runs. You can really tell when then shuffling has just transposed a few cards.

recursivecaveat• 1 month ago

Magic the gathering has this problem. You have 2 types of cards, and drawing an imbalanced mixture is pretty detrimental. During play you tend to sort them into 2 piles though. Consequently it's a not uncommon sight to see people manually interweaving their cards after a match, then shuffling. Logically, this is either pointless or cheating depending on the quality of that shuffle, but people do it anyways haha.

PaulHoule4• 1 month ago

I am in a tarot group where we have a lot of decks that people share. Many tarot users believe that a deck develops a personality specific to the user and because of that I got my own deck which I take to the group.

There's the general belief that all magical tools develop significance for the user over time, something that my wife who is a "secular green witch" who doesn't believe in psi at all would tell you all about.

Scientifically though, if somebody isn't a good shuffler their deck is not going to be well shuffled and they'll get readings that deviate from what you'd get from a well shuffled deck. It's harder to shuffle a tarot deck well because it has more cards and these are frequently larger. (Personally my riffle shuffle is awful and probably not much better than an overhand)

A new deck usually has the major arcana together and in order and other cards might be sorted by suit and then number. We do a 5 card spread and if your have a new and poorly shuffled deck of course you are going to have more spreads where you get both the Emperor and the Empress or the 4 of Swords and the 7 of Swords.

bombcar• 1 month ago

There's even a form of bridge related to this IIRC, if a hand is passed out (nobody has opening to start the bidding), you stack the hands on top of each other and don't shuffle - because players organize their cards by suits, it always results in fun.

pmyteh• 1 month ago

Yes - it's called goulashing. The cards are stacked and dealt out in groups of three and four to produce wild distributions.

The most interesting feature of it for me is that the carefully-developed bridge bidding systems don't work properly as they're implicitly based on a normal random distribution of cards. With a goulash it's likely that both sides have hands with 7+ cards in one suit, and the game is to pre-emptively bid high in "your" suit before they can do so in theirs. So you would ideally like different signaling arrangements that indicate suit length rather than quality.

mrbluecoat• 1 month ago

TL;DR "roughly 14"

ecolonsmak• 1 month ago

"...unique tracking label for every card in the deck"

I'd like more details on how this was accomplished on a practical level. Got me thinking about how to embed trackers thin enough to go into a playing card that would operate like a mesh network then the deck could self report once it's properly randomized making a green light go off indicating play may begin.

layer82• 1 month ago

They didn’t do this practically, the “tracking label” is just an analogy to convey what they did mathematically. The word “barcode” is also only used because it might be more accessible to the layperson than “bit sequence”.

empath753• 1 month ago

There are actually "marked" decks you can buy that come with an iphone app that tell you exactly where every card on the deck is by looking at the side.

WorldMaker2• 1 month ago

Marked decks are an ancient tradition for both cheaters and magicians. There are also ways to mark a deck that aren't obvious to most people with a casual inspection and that don't need an app to read from the edge.

hdndjsbbs• 1 month ago

This is just the authors explanation to explain how to encode where a card ends up. The cards don't actually have barcodes, they have a binary-encoded number where a 0 indicates the left pile and 1 indicates the right pile during a specific round of the shuffle. The number encodes the journey that card makes during the shuffle. It's not an actual barcode.

chadgpt3• 1 month ago

AI written? Em dashes, it's not X it's Y

zeroonetwothree• 1 month ago

Username checks out

vessenes• 1 month ago

Ironically seven perfectly interleaved riffle shuffles will return a deck to its original order, so the title is spectacularly wrong for one famous result.

Also the new result is cool! (14 semi bad riffle shuffles are sufficient to mix)

zahlman2• 1 month ago

It requires eight perfect riffle shuffles, not seven. (I just checked at the Python REPL.) And actually it depends on whether the riffles are done "in" or "out" (i.e. which half of the deck the new top card comes from).

I had understood that seven "typical" riffle shuffles produce good randomness.

WillAdams2• 1 month ago

Yeah, my father could do that consistently, and used it to teach me "Always cut cards" having also memorized the card order for each shuffle --- I guess being on a troop ship for weeks on end had to have some sort of up-side.

Nick Scarne is an interesting name to look up, and his writings are almost on a level with his facility to manipulate cards.

PaulHoule4• 1 month ago

Well I'd imagine a "perfect" shuffling procedure would have an equal probability of all 52! possible outputs which includes the original input and one would expect the sequence that gets you there would be highly symmetric.

possiblywrong• 1 month ago

The article deserves several clarifications:

The deck has to be cut more or less in half before shuffling.

"More or less" is doing some heavy lifting here. The original GSR shuffling model cuts the deck at a point that is binomially distributed, so that for example about one-fifth of the time the cut may be at least as asymmetric as a 21-31 card split, which I think most would agree is nowhere near "the precision of a professional magician."

Also note that the theorem in the paper really focuses only on relaxing the cutting model; the model of subsequent interleaving of the resulting piles is the same, dropping a card from a pile with probability proportional to the size of the pile. (Equivalently but perhaps less intuitively, for the original GSR model with the binomial cut, imagine flipping a fair coin for each card in the deck, then "de-interleaving" by sliding the "heads" cards out, preserving their relative order, and placing that pile on top of the remaining "tails" cards.)

But with that seventh shuffle, the deck suddenly tips into a highly unstructured state.

More accurately, the total variation distance from a uniform distribution first drops below 0.5 at seven shuffles[0]. The actual cutoff phenomenon's asymptotic result would suggest 3/2 lg n shuffles for a deck with n cards, which for n=52 would be closer to nine shuffles.

[0] https://possiblywrong.wordpress.com/2018/09/02/arbitrary-pre...

pmarreck2• 1 month ago

If, instead of doing a binomially-distributed cut, we do a 2-step cut which consists of first a random cut (then take the bottom portion and place it on the top, which in essence shifts all the cards by a random amount), and THEN do a perfect half-and-half split to perform the actual shuffle with?

Would that get us closer to "random enough", quicker?

cadamsdotcom• 1 month ago

Why does it matter that only the cut is imperfect? Isn’t an imperfect interleaving also expressible as a modification to the unevenness of that left/right pile split? It seems the “0110” system doesn’t track relative order of cards but only the landing of each card, which allows each left/right landing to be treated as an independent event. If there’s no dependency on card order, modifying the cut split is a simple way to express both an uneven cut and imperfect interleaving with one variable.

But that assumes the model is only tracking cards’ arrivals in the left and right piles, not their ordering relative to one another. I only got that from the article.

Am I missing something? Is it that the left/right split is actually only informative about the amount of mixing that has occurred under the assumption that the interleaving was perfect, and therefore if imperfect interleaving is possible then one must weaken that guarantee - which then requires a more complex tracking system?

vanderZwan• 1 month ago

Does this new proof have any practical consequences for determining if a PRNG algorithm is any good?

esalman• 1 month ago

Unrelated but the animated photo of the magician performing a shuffle really shows how advanced, efficient, and deliberate our limbs are.

I randomly came across a 1979 bbc documentary on "Word Processors" on YouTube yesterday. Even though I wrangle terabytes of data using AI agents everyday now, it still felt like magic to imagine myself seeing the documentary for the first time in 1979.

fisian2• 1 month ago

Numberphile has a video with Persi Diaconis about the seven perfect shuffles. It gives a lot of insight and anectdotes in addition to the central theorem: https://youtu.be/AxJubaijQbI?si=ED7ufY4oPZnNxCbd

kittikitti• 1 month ago

I thought about this for a very long time. For any given deck of cards, there is a coefficient of the average magnitude of displacement from their current order after a shuffle. This can be positive or negative, but we take the absolute value. Then, divide the total number of cards in a deck by this number to get the number of times you need to shuffle. A total of 7 shuffles implies an average displacement of about 7 cards. I think it's reasonable if you alternate the cuts in between shuffles.

Sloppy shuffles have a much lower average displacement and thus need more shuffles to get to a random state.

iainctduncan• 1 month ago

This is a weird title, because what is a perfect shuffle? The word shuffle does not necessary imply a rifle shuffle, and does perfect mean "it mixes the cards well" or is it meaning "cards go down left-right-left-right". It the later, then "perfect" shuffles don't randomize at all, as evidenced by the fact that 8 faros return the deck to order consistently.

I can do this when in shape, but like most mortal sleight-of-hand practitioners, only with in-hand faros. Actual table faros, what most people are thinking of with a rifle, are the domain of very very few, and even fewer can get that to a point of consistency. In hand faros are not impossible given fresh cards and enough practice.

Seven Perfect Shuffles Randomize a Deck of Cards. But How Many Sloppy Ones?

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