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)
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?
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.
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.
"...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.
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)
> 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.
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?
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.
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
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.
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.
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[ 2.9 ms ] story [ 29.1 ms ] threadYou would need sloppy ones to introduce randomness.
... 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)
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?
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.
Also the new result is cool! (14 semi bad riffle shuffles are sufficient to mix)
> 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...
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?
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.
Sloppy shuffles have a much lower average displacement and thus need more shuffles to get to a random state.
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.