Question is

How many times do you have to shuffle the cards – a deck of cards – to mix it up

I mean by shuffling probably what you mean by shuffling

Cut ’em about in half; you go like that; you push ’em together

Right and how many times do you have to do that til the cards get all mixed up

and uhh..

There’s a practical answer

The answer’s about 7

And uh.. And it’s not “I think it” or “it feels that way”

It’s a theorem

In contrast

There’s another way that people shuffle cards

Uh.. They shuffle cards this way

You’ve seen people shuffle cards that way

(Brady: That’s how I shuffle cards)

Ok well uh..

And in India, they do it this way

It’s the same – you can see it’s the same

Little – dropping little clumps one after another

And so lots of – some – people shuffle that way

And the answer to “how many of those shuffles does it take to mix up cards?”

is about 10,000

So

It makes a difference

Uh..

(Brady: yes) It makes a difference

There’s a 3rd way of shuffling cards that

is used in tournament poker games

And is used in Monte Carlo

I call that “Smooshing”

So that’s

This method of shuffling cards

You’ve probably seen somebody do that

You might have done it yourself

And then you gather them up and hope for the best

If you smoosh for a minute

uhh..

It passes all the tests we’ve ever thrown at it

um.. If you smoosh for 30 seconds

It’s sort of on the edge

But seems ok

And less than that uh..

It starts failing tests

And somebody could make money against you

Or guess cards right

Suppose you had a scheme for shuffling cards

One of these schemes

And you wanted to think about

“Is it working? Is it random? Or What am I talking about?”

If it was a few cards

Suppose you had 4 cards

And you had some scheme for shuffling them

I don’t know – some specific scheme

You could just try it a lot

With 4 cards there are only 24 possibilities

The top card could be any of 4 cards

The next could be any of 3

That’s 4 times 3 is 12

Times 2 is 24

And then this is forced to go

So you could just do it 1000 times

And see

Do all 24 possibilities occur about equally likely

But with 52 cards

There are about 10^68 arrangements of a deck of 52 cards

It’s more than the number of particles in the universe

Ok so one way of defining randomness is to say all arrangements should be about equally likely

I’ll say a sort of more practical version of it

Suppose that you had a scheme for shuffling cards

And then we were playing a card game

And you had to guess at the cards 1 at a a time as I turned them over

So for example

Y’know

Take a guess – what do you think the top card is

(Brady: 6 of spades)

Uh.. maybe upside down

Not so bad, Brady, not so bad

Uh.. y’know 6, 9, could be

Ok, but suppose

As in a card game or in a casino

The cards were turned up 1 at a time

And somebody tried to guess what they were

Now

We know now

That the 9 of spades is out of the deck

So you’re not gonna guess that again

What do you think the next card is

(Brady: Jack of diamonds)

mm… not so good ok

So

No, but of course it’s not so good

Your chance of being right on the first card is 1/52

If the cards were perfectly mixed

Your chance of being right on the next card is 1/51

Then 1/50 if you have a good memory

And if you have a really good memory, if the cards were all turned face up

you would know what the last card is for sure

It’s the matching 9

If the cards were perfectly shuffled

um..

you expect to get

1/52 + 1/51 + 1/50 etc.

Right and going through the deck 1 at a time

And that adds up to about 4 and a half

So, if somebody

could remember

and was guessing but the cards were well mixed

you could get about 4 and a half right on average

If you don’t riffle shuffle enough

if you riffle shuffle 4 or 5 times

somebody can get 9 or 10 cards right on average

Anybody would say that’s not random

Riffle shuffle is this guy

You riffle ’em together is the way we say it

They sometimes do it on the table this way

Y’know

Casino dealer will do..will do that

That’s riffle

This is called overhand and the other one I call smooshing

I wanted to say it’s not just experimental

that gives us these numbers I gave you

7 shuffles

It’s math

and I wanted to try to explain

a theorem

and I think

this is one that I can explain without writing anything down

Let’s see if I make it

I’m gonna take

A very simple shuffle

Which is take the top card off

And put it in at random

You might put it back on top

You might put it back 2nd from the top

You might put it on the bottom

It’s intuitively obvious that if you did that alot

The cards would get all mixed up

I mean it’s a silly way of shuffling

But it is like a riffle shuffle where you just happen to have cut off 1 card

Y’know I’m riffle shuffling this 1 card into this big deck

Putting it in at random

The deck starts out in order

Yknow ace of spades, 2 of spades, 3 of spades

whatever

and there’s some card at the bottom

I’m gonna suppose it’s the king of hearts

so the deck is in order, you know the order of the deck

It’s in order

(Brady: that’s how it came from the pack)

yes, yes or you might have written them down

the guys in casinos, y’know, record them as they come off right

how it came from the pack

And now, you’re taking the top card off the and poking it in at random

and

just do that a lot

ok, uh huh, poke poke poke

now eventually

because of the rules

some card will go underneath that original bottom card

the king of hearts

there it is on the botom

eventually if you wait long enough, a card goes under the bottom card

how long does that take?

well, the chance of a card going underneath the bottom card is 1/52

Because there are 52 places it can go

So the chance of that then happening is 1/52

Therefore it takes about 52 pokes

on the average to have that happen

ok

and now , keep going like an idiot

and eventually a 2nd card goes underneath the king of hearts

when I put that 2nd card

underneath the king of hearts

so there are 2 card there

even if I told you “hey it’s on poke 503”

I just poked

the 2nd card underneath the king of hearts

it’s equally likely that the 2 cards underneath the king of hearts are in order low-high or high-low

because I’m poking the card in at random

I could have put it above the card that’s on the bottom or below it

I’m poking them in at random

now, keep poking, eventually a 3rd card goes underneath

the king of hearts

there are 2 cards previously

this one

there are 2 cards

this one goes in here, here, or here at random

so all 6 orders are equally likely

every time you put a card in

given, as long as you’re putting it in at random

the cards underneath the king of hearts are in a completely random order

I hope that’s intuitive and it’s true (Brady: that’s make sense) it makes sense

and now look at what happens; keep poking

the king of hearts slowly moves up

it never goes down

it might stay where it is

if I put a card above it

it stays where it is

but if i put a card below, it moves up 1

so eventually the king of hearts comes up to the top

by induction, by the argument we’ve been doing

all the 51 cards are in random order

when i put the king of hearts in at random, the whole deck is random

exactly random

at that moment

every arrangement is equally likely

so that is not only intuitive it is an exact mathematical fact

i hope it makes sense, i think it makes sense

and now, you can just ask

“how long does that take?”

well, it takes 52 pokes for the 1st card to go under

there’s the king of hearts on the bottom

then i put a card underneath it

so now there are 2 places where the next card can go

so it’s it it’s 2/52

So it takes 52/2 pokes

and then it takes 52/3 pokes for the 3rd card to go under

so it’s 52 +52/2 +52/3

and that answer is well it’s – what is it – it’s 52 times log of 52

which is around 200 or so

so it takes about 200 of these pokes

if you shuffle fewer times

if you shuffle 5 or 6 times

it really somebody could really make money against you

in a card guessing experiment

if you shuffle 10 or 11 times

it’s not worth the wear and tear on your shoe leather standing there in the casino

if you’re counting cards

it’s just as close to random as it could be

of course it’s never perfectly random

y’know we’d have to shuffle infinitely often to make it

there’s still some trace of the original order but it vanishes exponentially fast

So wonderful question; let me try to match you with an answer

the model that the 7 shuffle is based on

i’m gonna say it as a slightly more mathy thing

this made me wanna play to shape of my heart!

what if you mixed riffle shuffling and the overhead shuffling? Does that change anything and how?

very cool

52 factorial is eighty unvigintillion six hundred fifty-eight vigintillion one hundred seventy-five novemdecillion one hundred seventy octodecillion nine hundred forty-three septendecillion eight hundred seventy-eight sexdecillion five hundred seventy-one quindecillion six hundred sixty quattuordecillion six hundred thirty-six tredecillion eight hundred fifty-six duodecillion four hundred three undecillion seven hundred sixty-six decillion nine hundred seventy-five nonillion two hundred eighty-nine octillion five hundred five septillion four hundred forty sextillion eight hundred eighty-three quintillion two hundred seventy-seven quadrillion eight hundred twenty-four trillion

More than the number of particles in the universe ?!?

Don't believe you.

How many Rifle shuffles does it take on average to have the deck return to its original state? For example, where all the cards arrive from the factory in order. How many shuffles does it take (besides 0, haa ha) does it take to return the deck to its factory in order state?

If your poking it in you never put a card an the bottom! IN

I've learned to riffle the cards and then raise roughly third of the deck from the middle to top after each riffle (which is pretty simple to do simply by pinching from the sides of the deck and pulling the middle part away) before halving the deck again for the next riffle. I think that it improves the changes that the top and the bottom of the deck get mixed during the suffle, but I have no mathematical explanation if it really matters.

This explains the terrible results I see from the lazy dealers at my local casinos.

I use the smooching method for my tarot readings

I have a question?

Do the amount of players play into the correctness of the shuffle??

How would alternating between riffling and stripping the deck (rearranging 5-7 chunks of cards) change the number of shuffles necessary? In casinos they often use systems like riffle-strip-riffle-strip-riffle. How random is that?

What about the sleeve shuffle? Divide deck in half and push together?

2:14 More than the number of particles in the universe!? Where did you hear that?

I do overhand and underhand at the same time. The whole calculation as to what randomness means kinda goes weird when you get into how dealing out the cards destroys the chance of old patterns openly appearing.

52 × log(52) ≈ 89

52

Σ (52/x) ≈ 236

x=1

Hmm, 7 may be the optimal number of shuffles for a regular deck of cards, but what about a deck of 78, plus orientation, which makes the total amount of possibilities 156?

10,000 – 7 is indeed quite the difference ; )

All of these only apply to traditional playing cards. And a TCG you can’t shuffle in ways that bend the cards because those cards can be worth a bit of money. He didn’t even mention pile shuffling.

Do people even watch poker or other traditional playing card games anymore?l(joke)

How about mixing shuffling styles?

Hmm. I alternate a couple riffle shuffles with a couple overhands, and back and forth. I've found that that helps to mix it up better, but maybe that's just because I'm not riffling enough times!

Does this guy realise his eyes are closed?

10'000 ??? I always shuffled my cards wrong lol.. omg i need to relearn everything.

In your 'overhand' method, you have missed the point of that shuffle. It is not place groups from the top to the bottom pile… It is a card or several, from the top AND from the bottom. So, top and bottom one or several immediately combined and moved to the bottom of the second pile. This dramatically increases the shuffle rate to better than riffle. The 'bottom' hand pulls cards from bottom and top as the hands come together, though it may be difficult to see.

That one at a time shuffling he talked so long about is not actually random if a human does it. A human doesnt put cards at random in a deck. Most of the cards just go randomly close to the middle unless the person conscious puts the card at the bottom or at the top.

So me and my parents play a game that uses 3 decks shuffled together h as te to see thst math equations

5:50 that would be 51 possible places it can go, not 52, because you cant put the top card back onto the top.

What about pile shuffling (as used by some MTG players)?

2:09 that's less than the estimated amount of atoms in the known universe, which is 10^80, which is more then 10^68.

Learning some useful techniques for college 😜

Ever saw Magic: The Gathering Professionals use "pile shuffle"?

Never thought George Washington would teach me how to shuffle cards well. My world is shook.

I'm a computer science student, and as a way of teaching sorting algorithms, my professors often use playing cards as an example. Never learned the best way to shuffle cards so far. I suppose it wouldn't be that hard though when you have some kind of random number generator method

smooshing is similar to the ribbon shuffle where your spread. I noticed overhand didn't randomize, didn't know 10000 times it takes, glad I got rid of doing that solely.

I have no idea why this is in my recommendeds but I ain't even mad

Then there's 52-pickup.

Unless you are a math guy you can't pull off that hairstyle

I prefer to use a combination of riffle and overhand. In my experience if you just riffle, the bottom few cards never move up the deck. Similar to the king of hearts in his poke example. My preferred method is riffle + overhand alternating 5 times.

Sugma 5:00

golden english johnnie

Fun fact: Riffle is

notacceptable as a shuffling method since if you can do it perfectly (i.e. split the pack exactly in half and alternate cards) you aren't actually randomising anything – in fact, done properly, you can return the deck to its starting order with only eight riffles.Mash and wash shuffling are basically the only truly random methods that cannot be cheated.

I love how my smooshing technique turned out to be one of the best ways to do it!

0:36 Did he just do a Hindu force and say in India people shuffle like that?

But what about using a combination of chop shuffle and arch shuffle?

I believe that is Professor Persi Diaconis, a professor of Mathematics at Stanford University, who is also very knowledgeable about card magic, and other forms of performance magic.

At 2'02'': 4x3x2x1=24 come on… x is not the multiplication sign ×, and this is hard to mix them up with a serif font ><

It is interesting the way he talks with his eyes closed

4:36 is he a ventriloquist?

Son (10) thinks there are 53 possible places to put the top card in a fulltime deck…

I usually separate my deck of 96 into two piles and then put those into seven piles.

at least around my region we typically call the 1 card riffle shuffle burying a card when playing a card game.

2:45 Yes I completely understand ( ͡° ͜ʖ ͡°)

4:58 This one card to this big ( ͡° ͜ʖ ͡°)

Ugh, its killing me…..I would bet this gentleman knows some slight of hand and I want to see.

7 riffle shuffles to make a single deck random. How many for say an 8 deck shoe? Same? If it is not fully random, then is it predictable?

What about Pile shuffles and mash shuffles? often uses in Magic the gathering.

10,000. 300 is the largest number that I have counted to.

I just do random piles and add the piles in random way.

Glad to see Bernie Madoff focused on stopping cheating now.

How about 52 card pick em that’s cool the true best shuffle

Ok,

BeethovenI don't like Riffling, by itself, because the top and bottom few fares of the deck rarely move far. Mixing riffles, and say a wash, or even some cuts, I feel is one of the quickest ways to shuffle, yet do so well.

Ebaniy rot etogo kazino!

Gas

0:44 I do this, but at the same time I also take some cards from the bottom (got long fingers – women can confirm)

How does this affect the ratio? cuts it in half?

Faro shuffle is the best shuffle

If you were my math professor in university, I think I could have really loved math!

Youtube suggested me this vid cause of yugioh, I wanted to prevent bricking hard so I clicked. Loved it.

2:45 what a legend

As a coder, I leave cards unshuffled. Instead, I 'chuffle' (my own word)

Pick a number between 1->52 and use it, swap the last card into its place. Then pick a number between 1->51, and again, swap the last card into its place. The swaps keep the cards contiguous, allowing a nice 1->x choice. So… the best, most complete and fastest shuffle – is NOT to shuffle ; ) To start a new game, just join the discards pile with the deck in their existing order and chuffle deal a new game.

So… 'chuffling' … a mixture of the words choose and shuffle ; ) It's a mathematically perfect shuffle in 0 time.

Actually, since it's actually a dealing method more than a shuffling method, feel free to call it 'duffling' ; )))) But if you see anyone doing it at a live game… leave ; )

As a card dealer this makes me cringe

Ohmygod! Subscribed!!!!! Love this brain content

So smooshing is the best. Got it

so as a guy who plays magic the gathering and plays with decks over 1000 bucks who do i not "riffle shuffle" (the first one) and mix up in a proper way for 60 cards or 100 cards as you normally do not go beyond that

There is also the table shuffling method where you make 4+ piles of cards on the table randomly from the top. I do that plus a few overhand shuffles while cycling my deck in dominion

2:10 "its more than the number of particles in the universe"

Nah

Cool, video. Thanks for making it. I found it very enjoyable

Putting aside the argument of computer generated random numbers not necessarily being random, the standard way computers shuffe a deck of cards is an interesting means. Essentially, it takes the deck, and then one at a time, goes through each position, first card, second card, third, etc, and picks a position to swap places with.

So the first card has a 1/52 chance to go right back to the start, but each of the other cards has a 1/52 chance to swap positions with it. And the same goes for all the cards. No slot is off limits, no matter how many times it gets traded, but every position in the deck is guaranteed at least one 51/52 chance to go somewhere else in the deck.

Anyone know if there's been a study of cradle shuffling?

"That's a silly way to shuffle"

Him describing placing the top card in the deck at random.

That is really close to how I shuffle. I switch to do it with the cards on the bottom after a minute though.

Doesn't that give the same result as just taking a card from the remaining unshuffled pile and putting it at a random position to the shuffled pile? It's the same method but skipping all steps where cards would be put ahead of the king of hearts. This results in 52 steps which is a log factor less.

If I'm bored or having a conversation and we're not looking to go again quickly I'll typically start making piles, grabbing cards from the top of the deck and decide "randomly" if I want to put them in an existing pile (at "random") or if I want to make a new pile. Typically by the end I have anywhere from 5-10 piles of cards with varying numbers of cards. I then "randomly" decide what order to grab the stacks in, after which I might do a few clump shuffles, and then repeat the first step.

Is this guy from the 1700s?

I thought I invented the smoosh

1/52 I GUESSED THE TOP CARD, I swear that's probably a one time thing

George Washington"More than the number of particles in the universe." —- No.

There are more possible combinations in a deck of cards than there are atoms in all of earth. When you shuffle a deck that combination has probably never existed before and will probably never happen again.

2:07 whoa….deep.

YO I was just shuffling around a deck while watching this and when he said 6 of spades I flipped the top one and it was the 6 of spades

editThe next one was the Jack of diamonds what is this???Poker dealers at casinos only do 3 riffle shuffles, a "box" shuffle (where you basically just cut about a quarter off the deck, place it down, and repeat 4 times total until the deck is full placed on the table), and then cut the cards roughly in half. This means that the poker cards dealt are very non-random.

What about pile shuffling? Mtg players want to know.

But what about pile shuffling?!

0:06 "what you mean by shuffling" — are we all croupiers here? Can we all riffle shuffle as he does there, with the cards in the air? or at all? Seven times, without any cards ever falling out? Best of luck with that one — we're not all as expert as you.

So I play Magic the Gathering, a card game in wich you usually have your own deck with 60 cards, how many times do I have to shuffle to make it random?

Also, some people I know shuffle the decks by taking it and making 5 piles of 12 cards each and then just picking them up in a random order, is that mathematically efficient?

wtf he's lying. 2:10 the number of particles in the universe is around 10^86

I always shuffle cut the deck and then shuffle again

I love how this video answers the question and shows all the methods in the first minute and thirty seconds. THANK YOU!

Perfect riffle shuffling will take infinitely long to be random, because perfect riffle shuffling fails to randomize.

I like to mix the truffle with the overhand shuffle