Shuffling Extra Footage (1/3) - Persi Diaconis

[@Galva94a]

Amazing to see in the comments how many mtg players there are here. As a both mathematician and mtg player i'm so proud of the numberphile community!

[@CrystalblueMage]

Sounds like a "Battle of Wits" "Magic the Gathering" Deck

[@Saxie81]

I love these extra footage videos, you pick up more of the professors quirks and personality. Great stuff Brady!

[@elainelorimer818]

6:04 "it's fighting back but were making progress" If anyone wonders about the hesitation in his voice its because we never know if were 'making progress' or 'going down the rabbit hole'

[@renerpho]

Yes! It fought back for another 18 months. (The paper is "Shuffling Large Decks of Cards and the Bernoulli-Laplace Urn Model" by Evita Nestoridi and Graham White, two of Peter Draconis's grad students.)

[@CraaaabPeople]

I had this guy's thesis advisor in college. Love the guy.

[@DavidGreeneMtgJudge]

You heard it here, they're working out how to mathematically randomize Battle of Wit as efficiently as possible! :)

[@cmdrpickles]

If the cards are sleeved, you can get a nearly riffle-like shuffle quality out of a mash shuffle. Makes it significantly easier to randomize very large decks.

[@keiyakins]

Still not easy though. Source: am currently shuffling a large Star Realms trade deck while watching.

[@daniamaya]

I can listen to Diaconis talk the rest of my life.

[@justsignmeup911]

Penn mentioned the "7 shuffles" on Penn & Teller Fool Us. Great to see who figured it out!

[@Pringlesman]

Guy playing a "dungeons and dragons" deck with 250 cards. Yep he's playing battle of witts in mtg.

[@DOSTalks]

I really like Professor Persi, he has such a relaxed and calm way of speaking with a very transparent approach. I'd love to see more videos from him! EDIT: And I have watched through the whole playlist already :D

[@renerpho]

I don't know if someone else has already commented on this, but if you're looking for the answer how to shuffle large decks, it took another 18 months to solve it. The paper, which was put on the Arxiv in October 2016, is "Shuffling Large Decks of Cards and the Bernoulli-Laplace Urn Model" by Evita Nestoridi and Graham White, two of Peter Draconis's grad students.

[@smnh776]

Surely a piecemeal riffle (cut the full deck, then repeatedly take a handfuls of cards from each half and riffle them together and put them on top of the result stack) adequately approximates a single whole-deck riffle, so you could use the same analysis as for a 52-card deck. The casino method shown at the end seems to be broadly equivalent to this but with the steps in a different order.

[@Vendavalez]

Does anyone know if there has been progress with what he mentions in the video for large deck sizes? I can't properly riffle shuffle my MTG Commander decks so I spend way too much time over-thinking how I should do it in a way that it is sufficient.

[@CraftQueenJr]

Vendavalez smoosh.

[@renerpho]

Look up "Shuffling Large Decks of Cards and the Bernoulli-Laplace Urn Model" by Evita Nestoridi and Graham White, two of Peter Draconis's grad students. The paper was put on the Arxiv in October 2016. Draconis is giving an overview of the state of current research in his book "The mathematics of shuffling cards", published in 2023.

[@ericvilas]

Man, if you can prove exactly what scenarios contain the cutoff phenomenon and which ones don't, that's gotta be huge for chemistry and matter physics!

[@muhammadumerzain4726]

what if we riffle shuffle then overhand shuffle a few times but put the cards up and down randomly and the again riffle shuffle and the few alternating overhand shuffle again.this will allow the bottom and top cards to fully mix up in the deck and because of the overhand shuffles they can also end up back at the bottom or any other place in the deck.

[@dudenesshis1687]

I am a Las Vegas table games dealer and we do not shuffle a 52 card deck seven times, the most typical shuffling procedure in Vegas is shuffle-shuffle-strip-shuffle.

[@3geek14]

I often play blackjack with 6 decks shuffled together. In blackjack, the suit doesn't matter. With 6 decks, there are 24 copies each of 13 distinct cards. Randomising that should be far simpler than the problem of randomising 312 distinct cards. In Magic: the Gathering, it is common to have a deck of 60 cards, split into 4 copies each of 15 distinct cards. However, it's also possible to have other partitions and other sizes of decks, such as 8 of one, 9 of another, and 23 distinct cards. With a cheap card shuffler, it's pretty easy to perform a riffle shuffle on decks with 300 cards, so I'll assume that it's always possible to perform a riffle shuffle. The problem I really want to see solved is: for a deck of N cards, and for a given partition of N, how many riffle shuffles does it take to randomise the deck?

[@kashgarinn]

It is interesting seeing the likelyhood of the king of hearts staying at the bottom, but what is the furthest up a king of hearts can go with that shuffle? Could it end up on top of the pile? How many times would you have to shuffle for the king of hearts to (theoretically quickest way) to get to the top position? - I'd believe that the quickest is 13 shuffles, if you always split the deck exactly in half for each shuffle, and thus 13 shuffles will theoretically allow the king to be anywhere within the deck, thus any other card can theoretically be anywhere, thus you have true randomness

[@kanklez]

When you split the deck if the king happened to be near the middle it can end up the top card in one of the piles, the pile formed from the bottom half of the deck, which means it can jump from the middle of the deck to the top in one shuffle if its made it nearish to the middle.

[@shadyblue1000]

I wonder what the statistical model is for randomness for 52 card pickup lol

[@stationshelter]

clever thumbnails

[@janlettens3092]

I don't understand how the deck would be completely random if you just follow the bottom card. Because it doesn't seem obvious to me that it would be uniformly distributed over all positions, after we stop when the chance it would be at the bottom is 1/52. Even more, I think it would be likeliest to be several positions from where it started (depending on the number of shuffles), in some kind of peaked distribution. I must be missing something obvious, can someone help to explain it?

[@agmessier]

When I shuffle, I deliberately force the first cards to drop from my right hand (from the top of the deck) first, and the last cards in my left hand to drop last. If that represents 2 or 3 cards at each end of the deck every time, how does that change the number of shuffles required?

[@Bodyknock]

I do the same things, when I riffle shuffle I do it so that a card from the middle of the deck goes to the bottom of the deck and another card goes to the top. I do that because I figure otherwise I'll have a natural bias that will keep either the top or bottom card on the top or bottom of the deck longer than normal. I then shuffle eight or nine times since I figure that, while seven is the number for an ideal riffle shuffle, my actual riffle shuffle isn't quite perfectly random, I probably get a little extra clumping at the top or bottom of the deck when I do it.

[@XPimKossibleX]

with infinite shuffles the first two and last two cards will not not have moved at all... just do left first sometimes and it'll completely sort it out.

[@agmessier]

michael benzur No, you missed my point. When I shuffle, the top of the deck is in my right hand, so I'm guaranteeing that the last two cards move every time.

[@hobbified]

agmessier anything you "force" is guaranteed to make the result more predictable than if you had left it up to chance, even if you're forcing movement. So the number of required shuffles will go up, if only by a small amount.

[@agmessier]

hobbified I'm skeptical in this case. Have you seen any proof of this point?

[@javierbg1995]

I'm pretty sure the D&D-like card game he's talking about is Munchkin, can relate :D

[@Theatheosis]

(Or maybe a Battle of Wits Deck in Magic)

[@arachnophilia427]

The Atheosis it's pretty much got to be battle of wits.

[@Falcrist]

Arachno Philia Battle of Wits typically runs about 240 cards, though. He was very specific about the number 250. It could still be a Wits deck, but I'm not 100% sure.

[@fullmetalpwn]

The Atheosis Haha, this was my thought when he mentioned that.

[@enricoalessio5434]

The Atheosis I was thinking prismatic MTG

[@gavs928]

I'd be interested to know what degree you have to pile shuffle- dealing the cards into x piles one card at a time and then stacking the piles back together- in order to have a truly randomized deck.

[@madking2]

+Gavin Mac it's the same as a riffle shuffle (7times), but only if you follow two "rules": ONE: you need to use at least 7 piles, and an odd number of piles. TWO: when you are picking up the piles to form your deck, stack the piles in a random order. Thing is, 7 riffles should take all of 30 seconds, 7 pile shuffles might take closer to 4 minutes. I know my choice!

[@rayunseitig6367]

I guess I better just stay out of those casinos. Glad to hear, the laws in Nevada have changed. and please how do I get those ladies to stop giving me drinks. :-) Gotta shuffle off now.

[@j0nthegreat]

i while back i was curious about how long it would take to get back to the original order if you shuffled 'perfectly' (split the deck in half and alternate one from the left, one from the right, then repeat). i think after 26 shuffles you get back to the starting point

[@j0nthegreat]

i just watched the second extra video.. apparently it was 8 times you get back to the beginning!

[@AntoshaPushkin]

j0nthegreat i shouldnt have told that, but it could be used for magic tricks

[@j0nthegreat]

Антоша Пушкин i think that was why i was curious. i never gave it any real effort though

[@Syncopator]

j0nthegreat Actually, it makes a huge difference which side drops first. In one case, it takes 8 times to get back to the same order (when bottom drops first). And after 26 shuffles if the top side drops first, the deck is exactly reversed in order-- so it would take 52 shuffles to return to the original order.

[@subjectt.change6599]

Professor Diaconis is channeling (Ha!) Martin Gardner in these videos. Who knows, maybe Gardner was right to believe in immortality? (Actually, the kind of immortality that I have suggested above is NOT the sort that Gardner believed in. See his book "The Whys of a Philosophical Scrivener" for details.)

[@topilinkala1594]

I play Magic the Gathering and mostly commander. It's funny that 100 sleeved cards is too much to riffle shuffle in my hands but when I take the commander out and it's 99 it is easy. I also do the deal seven piles pick up randomly and do again and again and again...

[@Falcrist]

Pile shuffling isn't actually random, so while it might help, it isn't sufficient.

[@Ratstail91]

I was gonna say that when I shuffle my Magic: The Gathering decks, I'm usually starting with ~20 nearly identical cards (lands) on the bottom. First, I ensure that these cards are evenly spaced through the deck by "pile shuffling": 1. Put the top 7 cards into 7 separate piles 2. Repeat. I personally alternate shuffling from the top and the bottom, as otherwise you still end up with clumps of lands together. This ensures a near perfect spread of cards through the deck. From here, normal ruffling works just fine. It should be noted that most magic players store their cards in individual sleeves, which are designed to make handling and shuffling easy. I can easily shuffle 100 card decks without a problem.

[@albertoracca6308]

Hi fellow magic player! Why stacking your deck to a determinate order if the end result is to have randomized deck? There is no point in doing such a thing, and it can look quite suspicious in competitive tournaments. Here is the method we judges encourage players in shuffling their decks, which involves no stacking at all, which it is very similar in what you do: Rifle shuffle the deck 7 times; Pile split your deck in 7 piles; Rifle shuffle the deck 7 time; Don't split your deck in even piles and don't let your opponents do that: it could look random, but if you start with a particular deck configuration you could easily end up with an even distribution of land and spell which is NOT random

[@danieljryba]

Alberto Racca My local judge asks us to do this before the first match in the tourney in what he calls the RP56R: Riffle 5 times Pile shuffle into 5 piles Pile shuffle into 6 piles Riffle 5 times Then allow opponents to cut before play, all the while keeping your deck face down. This seems to stop us from getting mana screwed a bit more often than not. I do like the '777'ness of your method though!

[@Linvael]

Daniel R Alberto Racca Are these pile shuffles in the recommendations for some social reason (like appeasing the masses to avoid verbal confrontation about shuffling), or are they expected somehow to improve the randomness?

[@danieljryba]

Linvael Usually, it helps to make sure that you don't end up with a block of cards that have no mana or that you draw a ton of mana and nothing to spend it on. Due to the nature of the game play (I assume you don't play, I'm sorry if you do), your mana tends to end up in groups that are hard to break up with just Riffling (kind of like the bottom card always staying near the bottom) and sometimes even if you do manage it, it can all end up in the bottom or top half of your library. Pile shuffling moves it up and down and breaks apart chunks of cards to make the distribution of mana more even. Then you Riffle a few times to make the draw random and let the other player cut your deck for fairness. As far as the social aspect goes, I have no idea. Due to the fact that you have to watch what you are doing while pile shuffling, I have personally seen some of the more 'anti-social' players use it as a way to make the other players leave them alone during the 3 minute gaps between rounds.

[@kanklez]

Daniel R Shuffling in a way that makes mana screw/flood less likely than properly randomizing your deck, 10 riffles according to this video, is cheating. Mana problems are annoying but they are a part of the nature of the game and shuffling in a way that minimizes them compared to random is against the rules.

[@kingcobra0128]

I am actually interested in the higher number shufflers I am going to Create a shuffler

[@leo17921]

After 1 shuffle, it's... a half, or 2, you know, it's 4, 6 and and 2 to the 2 to the 6 is 64 and 2 to the 7 is 128 so [...] Uhh, makes, you can start to see, you know, uhh, you know, cause each time something, now, but of course, I'm saying that our theorem shows that

[@AnAnonymousObject]

I haven't yet watched all 3 parts, but what about the way magic players shuffle? In order to prevent damaging cards and to keep them pristine, there is a method almost like smooshing and poking combined. You place five cards down, and then pile the cards onto them randomly (as much as a human can). Then you pick up the five piles, and stack them randomly. Say you did this method with 52 cards, how many times would you have to shuffle for it to be 'random'.

[@arachnophilia427]

pretty much any form of pile shuffling is not random, and the slightly randomized way you are describing is way slower than mashing. the best technique is to cut the deck approximately in half, and mash/riffle the two halves together. repeat a large number of times. i can generally do this 14+ times -- and sufficiently randomize my deck -- in the time it takes someone to pile shuffle once.

[@AnAnonymousObject]

Arachno Philia I get the benefit of the riffle, i watched the video. Both riffle and smooshing, will bend the cards, and roll the edges. Not to mention the difficulty of doing these while all the cards are in cases. A magic player can shuffle this way in a minute or so. About the same time it takes to shuffle a deck 7 times or smoosh properly.

[@Linvael]

***** For that to work you would need to randomly pick a card to put next which would take a considerable amount of time to get that pesky 29th card when it's turn comes. And require reliable random-generating machinery in human brain, which is not there. And even then it's not evident from the outside that what you did was randomization (as all the information about where goes which card is available during the process), so any competitive player will require you to shuffle in a normal way.

[@XPimKossibleX]

no matter how much you shuffle, if you start with precisely 5 cards each time you'll have groups of 5 random within eachother, but not in the deck. i'll give you an example 24,23,21,22,25,10,12,12,11,14,2,4,3,1,5,9,8,7,5,6,2,16,18,19,17,20.

[@Falcrist]

***** What you're describing is "pile shuffling". Typically, M:tG players won't place the cards randomly, but will go in order around each of 7 piles. *Pile shuffling does NOT randomize the cards*, except in conjunction with one of the other schemes mentioned in the video. Typically, high level players will use a slight variation of the riffle shuffle where they split the deck into two piles and then ram them together. This is done several times along with one or more pile shuffles. The two players in a game will also swap decks and shuffle their opponent's deck. In high level play, this procedure can take upwards of two minutes. I'm not a mathematician, but it seems to me that combining the two methods should lead to more randomness in less shuffles. However, pile shuffling by itself does not actually randomize the deck.

[@sk8rdman]

For MTG (Magic the Gathering, a trading card game) It's possible to have very large decks of cards. Even with the minimum deck size (60) most decks will have about 1/3 of the same card or type of card (mana cards). This means that it's quite likely, especially in larger decks to get what some call mana stuffed (drawing lots of mana cards repeatedly) or mana starved (drawing no mana several times in a row). This is especially true if the deck is not properly shuffled, because a poorly shuffled deck will more likely have "clumps" of mana, or sets of consecutive mana cards that never got split up by other cards in the shuffling process. To prevent this mana clumping, I'll often use a technique a friend showed me that we call the "super shuffle". One card at a time, drawing from the top, you deal the cards into 4 or more even piles, dealing to them in a different order each round and never dealing to the same pile twice in a row. Then you just stack those 4 piles in a random order. This is best preceded and followed by a few riffle shuffles to better randomize their order, but the point of this is that any cards that were adjacent to each other before the "super shuffle" are not any more, so any clumping as a result of poor shuffling or cards just sticking together is not a factor. Mana clumps can still exist, obviously, but only as a result of random chance, rather than human error or card quality.

[@alejanbdroechezuria]

but if you shuffle 7 time, people will know that the last card of the deck will be between the position 45 and 52 . if we play poker you will know that this card will not appear on the table.

[@fiver-hoo]

thats why you cut

[@charmetroldendk]

Hi

[@XPimKossibleX]

tell the kid to smoosh the cards

[@XPimKossibleX]

linas3001 hey i'm curious what i meant by that, can you tell me?

[@MDAnisAnis-xm2jx]

This is proved

[@qwertz12345654321]

doesnt the king of heart only move up like 1-3 spots per shuffle, so that it will always stay on the bottom half even after 7-8 shuffles?

[@ZER0--]

52!

[@SgtScourge]

What if due to quantum theory the randomisation cutoff is tied to human brains and our perception of the cards. Miller's law states a human can only keep seven active things in working memory, plus or minus two. Each shuffle the cards are potentially in all positions with, a few restrictions, until observed. If you shuffled once at looked at them you'd recognize the non-random pattern, same as second, third, and so forth, but the seventh time you can no longer store the states and when observing after that they are truly random. ;-)

[@poiewhfopiewhf]

that isn't surprising at all... if it's true for the bottom card it clearly would be true for every card...

[@ZER0--]

I do not believe this theory that a set of cards hasn't been shuffled the same since cards were invented or ever will be. I understand its a big number 52! (!=Facrorial, ie52x51x50x49x48 etc and it equals 80658175170943878571660636856403766975289505440883277824000000000000 The Isreali lottery had the exact same numbers drawn out just 3 weeks apart.