The Best (and Worst) Ways to Shuffle Cards - Numberphile

Persi Diaconis (Stanford University) on card shuffling. See full description for links to additional videos and detailed papers.
More links & stuff in full description below ↓↓↓
EXTRAS FROM THIS INTERVIEW -- 1/3: • Shuffling Extra Footag... -- 2/3: • Shuffling Extra Footag... --- 3/3: • Shuffling Extra Footag...
Cards and shuffling videos: bit.ly/Cards_Sh...
PAPERS
Trailing the Dovetail to its Lair (7 shuffles)): bit.ly/1xU6uPI
Shuffling Cards and Stopping Times: bit.ly/1FR0ca1
Overhand Shuffles: bit.ly/1LNBSuU
Discuss this video on Reddit: redd.it/300rkr
BOOK
Magical Mathematics by Persi Diaconis and On Graham: bit.ly/MagicalMath
COIN TOSSING
• How random is a coin t...
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[@altejoh]

It hurts me that the method of shuffling I took a year to learn and have been using for years is less effective than the one i could do since I was 5.

[@Crazd22]

To which methods are you referring?

[@mythicaldata6297]

He's probably talking about the Hindu shuffle

[@Covertfun]

Haha, well I guess you can do 7 riffle shuffles faster than you can do a full minute of smooshing, so there's a small time saving. But you did say you spent a year learning the riffle... So you make a tiny bit of that time back every time you play a card game and do 7riffles in under a minute. If you don't want to have wasted that time, you'd better host a few poker nights ;-)

[@acmramon]

altejoh Einstein is crying now😂😂

[@ombra30]

@@mythicaldata6297 nope, he's talking about smoosh vs riffle shuffle

[@slientkillli7433]

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

[@mindartis4081]

Ha 😂

[@yosselgarcia3100]

Benjamin Franklin

[@KPIBM]

Larry David

[@giulio19user]

Beethoven is unpleased to disagree with you

[@lucasvandijck3047]

Or a crazy scientist

[@ヒラガナ-e5d]

"Now, keep going like an idiot" LOL

[@sfreer6044]

6:06 Best moment, hahaha! xD

[@since1876]

Yeah I laughed at that. To which, my dog woke up from a nap, looked at me like I'm the worst asshole since Hitler, and went back to sleep. One day, I am certain I will wake up to those jaws around my neck. I know this from how he stares at me when he eats. I think he's just waiting for me to buy an automated feeder and fill it with a twenty year supply of food and water.

[@abhaytyagi6179]

@@since1876 haha!

[@MrNikolidas]

If anyone watches Vsauce, there was a wonderful visual analogy as to how long it would take to shuffle a deck of cards into all of its permutations: Stand on the equator. Shuffle a deck of cards once every minute. After a year has passed, take one step around the equator. Once you've walked around the earth, take one drop of water out of the ocean. Continue walking one step a year, taking a single drop out of the ocean after every revolution, shuffling the cards once every minute. Once the oceans of Earth are dry, lay a sheet of paper down on the ground once every revolution. When the stack of paper reaches the moon, refill the oceans and take the paper down and repeat. 250 more times.

[@happydolphin1432]

Jonathan Charles Hold my Beer for a minute

[@cameroncoates4863]

Imagine how much longer it would take with Tarot cards...

[@Lukoro1357]

@@cameroncoates4863 or with magic the gathering. Funny thing is, you could accidentally shuffle a working Turing machine. (search on KZbin :D)

[@blakebell8533]

@@Lukoro1357 wonder how many times you have to shuffle an EDH deck until it's completely randomized

[@nyxcal]

When vampires get bored

[@kg583]

I love these kinds of videos. They show more practical and interesting ways to apply math to games and such.

[@numberphile]

kg583 glad you enjoyed it - more shuffling to come this week!

[@AdrenalineL1fe]

true mathematician doesn's look for a practical way

[@evilplaguedoctor5158]

AdrenalineL1fe but we are not all mathematicians, a lot of us are just people who like math. :)

[@razielhamalakh9813]

Yes, because pure math is so boring... Wait. Why are you here again?

[@kg583]

Raziel Qwazar Because I love math just every other Numberphile subscriber. And when did I say pure math was boring?

[@ElArboool]

did he say "take the top card off" without moving his lips?? 4:36

[@lukesheridan301]

+ElArboool Eagle Eyes

[@Rosi_in_space]

+ElArboool The Matrix is broken, call Microsoft support...

[@mta4965]

laught so hard 😂

[@raghavchutani8634]

Is He an ALIEN? I saw that too.

[@Quicksilver_Cookie]

I'm sure it was just voiced over. There's an audio cut there :)

[@anoukfleur2513]

Aha! So my method of shuffling IS a valid method! Always smooshing from now on.

[@shamrockwishes]

It's valid, but damaging.

[@PotatoMC1]

I do it and everyone makes fun of me. Little do they know... it's effective.

[@keiyakins]

So is riffling.

[@bgm-1961]

Amber Perry No, it's not. It's only damaging when done by people who don't know how to do it.

[@acruzp]

Riffling is better

[@MindYourDecisions]

Great video! This is the best explanation for why you need "7 shuffles" to randomize a deck. I look forward to more vides on Card Shuffling Week.

[@ashmanideep6253]

How come nobody noticed you in these comments? Or even the video isn't popular, that's crazy

[@akankshasharma7498]

Hi presh Talwakar from 6 years hence 🤣

[@o5-1-formerlycalvinlucien60]

yo presh talwalkar! It seems like it's become a tradition for someone to comment here once a year.

[@rohitdoijad7424]

​@@o5-1-formerlycalvinlucien605r55

[@rohitdoijad7424]

​@@akankshasharma74984r⁵t

[@TheGamerSays]

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

[@alexanderperez4440]

😂😂

[@shadowfreak707]

Bruh 💀💀

[@victorclark6030]

KZbin is a wild place

[@lukamrsa8237]

:D

[@АнонимныйКомментатор-б5д]

Yeah, Hans Zimmer's carreer went odd after retirement

[@NekuZX]

I was literally just looking for videos on how to shuffle cards. All of the recommendations were about how to shuffle cards in a cool way, but the only video that actually bothered to explain which method is the most random was this one. Not only I learned what I wanted to learn, but I also got very fascinating facts about random chance! Thank you very much sir Persi Diaconis.

[@mkaali]

The problem with riffle shuffling is that the order on the bigger scale stays relatively same. The bottom cards stay near the bottom and the top cards stay near the top. They can traverse the deck only very slowly, while the cards towards the middle get more randomness. 7 passes might be enough, but what I like to do is to throw a few of those overhand shuffles (0:36) in there or even just cut the deck at some points to alter the macro scale order more.

[@as71318]

I do this as well.

[@vanessaxoax7646]

Ive just done 10 shuffles with 5 riffles each (yeah ik, its a small sample size) and i found the top and bottom most cards in the top or bottom 5 exactly one time each.

[@shaniyadav33]

@@vanessaxoax7646 bro tell me which is more random? Overhand 10 times Riffle 10 times Smooshing for let's say 30 seconds or less

[@Traqr]

​@@shaniyadav33Answered in the first 1:30 of the video - Riffle 7x is essentially "random" if it's done well, so 10 means you've shuffled three random decks. 30s is barely enough smooshing, and overhand needs about a half-hour of shuffling if you're quick. I like to cut the deck ~¼-⅓ before shuffling and after a few riffles to bury the top & bottom of the deck, followed by a few more riffles - don't know how it impacts statistical randomness, but I don't see indications of previous hands in the new deck.

[@lordomacron3719]

Well shuffling shuffling methods while shuffling would be optional.

[@andyhuang8010]

For anyone wondering what the randomization formula is: For a deck of n cards, the number of riffles needed to sufficiently randomize a deck is the smallest prime number greater than n/10. So 52/10=5.2, and 7 is the next prime number. For numbers that get really close to the next prime (like 49/10=4.9), skipping up another prime (so up to 7 in this example) is best.

[@bonesofeao3968]

6:07 the name of my autobiography

[@subjectiveperspective1730]

As a casino dealer I mainly do a poker riffle. It goes like this: 1. Take the unsorted deck. Cut the top third and move it to the bottom. Riffle and combine. 2. Cut the middle third and move it to the bottom, riffle and combine. 3. Cut the bottom third and move it to the top, riffle and combine. 4. Cut the top third and move it to the bottom. Riffle and combine. 5. Cut the middle third and move it to the bottom. Riffle and combine. As you can see, this is only 5 riffles. However, assuming you do a normal riffle without cutting, if you do it card for card with 26 cards in each riffle stack you are putting Ah next to As and so on. I wonder what effect the cutting would have on the riffle. I assume quite a considerable difference as the each cut and riffle moves the "to be riffled" cards to a different part of the deck. In short, initial riffle makes a much bigger difference, a few cards cut differently on each third makes an enormous difference. Thoughts?

[@DashRetro]

One weird thing about the riffle shuffle is if you were to "Perfectly" riffle shuffle, as in you split the deck perfectly in half and your riffle perfectly alternates the two sides like a robot, the deck will actually be comepletely back to its starting position after 7 shuffles. This only applies to a 52-card deck, but it's true!

[@zenithquasar9623]

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

[@358Disaster]

I like what he said at the end. "There is still a trace of the original order, but it diminishes exponentially as you shuffle." Its like mathematical music to my ears.

[@leebard9335]

I agree!

[@billnye1825]

r/iamverysmart

[@soulextinguisher]

Does this mean Yugi was cheating?

[@Abdega]

Undead Wizard So the heart of the cards is just… bad shuffling?

[@Zorgdub]

@@Abdega Nah. He didn't just know what was coming. The card that was coming would be exactly the one he needed. It's his magical power to alter fate with the Millenium Puzzle.

[@Ausar0]

He literally was cheating, yes, but not because of poor shuffling.

[@IcarusAtreides]

Stacking, yes.

[@Kenny-Blankenship]

No. I mean, he was cheating, but not because of what is in this video

[@XxearthxX844]

its important to cut after each riffle shuffle or the top card is always on top

[@samimas4343]

i guess he is banned from playing in most casinos by now

[@hetakusoda2977]

lol

[@nessy3098]

FIND THIS ONE TRICK THIS MAN USED TO SHUFFLE! *CASINOS HATE HIM!*

[@EironPage]

This is the guy casinos call to check they are shuffling effectively, and helped design some of the machinery and systems they use to shuffle quickly and efficiently.

[@XoIoRouge]

Actually, he's not. For multiple reasons. (1) Card Counting is basically null and void in most CASINO card games due to the use of two 52 card decks (yielding 108! possibilities) and a machine shuffler as well as (2) Poker, Hold'em and other 52 card games are competitive because of probability and card counting. That's how the game works. If you watch a professional "finals" for Tournament, you'll see the screen keeps math stats shown for the viewers as to showcase an average % for a certain turnout, for many pairs of cards -- all of that math is being done by the players at that moment. So no, he's not banned. He's just able to be very competitive and will find other mathematicians like him.

[@veitcall]

@@XoIoRouge 108! possibilities? I don't think so... Probably you meant 104! but it should be less - each card is doubled so a lot of the 104! possibilities are identical.

[@sammy0boi]

How tidy are those bookshelves!?

[@thulyblu5486]

sammy0boi well, they don't really reflect randomness, that's for sure ...*or DO they?* woooOOOOooooooOOOOOOoooo

[@Bruno3605]

Mathematically speaking...

[@TheMisteranimal]

School library tidy

[@anuragshukla4118]

Beacause he reads them

[@TheMohawkNinja]

This video was perfectly timed. I just started getting back into Magic: The Gathering, and I would always do the "overhand" shuffle. I didn't know that it was so ineffective.

[@kevinslater4126]

I went to go grab a deck of cards while watching this and accidentally dropped two identical decks together all over the floor. What are the chances that I'm an idiot?

[@Ulkomaalainen]

Okay, let's start. There's 52 cards. Times 2 decks makes 104 cards. Of course you need to take into consideration the 4 suits and 13 cards per suit, so divide by 4*13. The result needs to be halved, because there's two possibilities of the outcome (you are an idiot or you are not). The result should be the answer :)

[@MarioFanaticXV]

Clumsiness isn't really an indicator of stupidity. Granted, the two aren't mutually exclusive, but I wouldn't even say this was an instance of you doing something idiotic, just unfortunate. Now if you did it on purpose, that might be different. Then again, I consider every human being alive (myself included) to be an idiot, so...

[@VuotoHSR]

100%

[@gavin5410]

@@Ulkomaalainen take the number of cards in your deck and divide by the number of cards in a deck.

[@yrtomin]

1 in 52

[@The77Game]

The numbers when it comes to cards has fascinated me for some time now. Like the fact that if you smoosh the cards properly you will likely have put them in an order that they have never been put in before in the history of cards. That is just mindblowing to me.

[@baskoning9896]

In some card games, you DONT want a completely shuffled deck. Dutch 'kingen' comes to mind, where you are supposed to only split the deck 2 times, then deal again. This distributes each play of four cards of (mostly) the same color to all players, so no player ends up with most of one color.

[@rogerwilco2]

I remember reading in a popular science magazine as a kid 25 years ago about how random the various shuffling methods were. Overhand vs Riffle vs Smooshing shuffles were al mentioned and the conclusion I remember is that the Smooshing shuffle was the most time effective and hardest to mess up. So I've been using it ever since.

[@jamesryken5492]

All I got from this video is tat I need to "smoosh" my cards

[@juicyclaws]

riffle shuffle 7 or 8 times

[@Henryguitar95]

I guess you weren’t listening very well then

[@johnnytickler]

Gotta love a smoosh

[@leebard9335]

Probably faster to riffle seven times than wash the cards for a minute.

[@jjs8426]

*S Q U I S H T H A T C A T*

[@shepherdsgamingrun]

Can we request a version of this video using Magic: the Gathering cards? I'm pretty sure I know people who need this video.

[@SaltyKeldion]

No wonder Magic the Gathering feels rigged in competitive play. I don't see many people willing to riffle shuffle their expensive cards.

[@Frosty-oj6hw]

Love videos like this.

[@numberphile]

***** cheers

[@valen8560]

likes of the reply received is just the half of that of the comment.... it stops me from giving like to any of the two....

[@juicyclaws]

same here, aced my probability courses because i find it so intuitive.

[@claudetteshearman3067]

Frosty hat are. The uses of Bluetooth elephane

[@EuanMunro]

"Keep going, like an idiot", is maybe the best line I've ever heard on Numberphile!

[@yellowmeerkat97]

Oh my goodness. THE Persi Diaconis? The man who traveled with, was taught by, lived with, and consistently impressed Dai Vernon? Brady, you've got to ask him about more than shuffles. This man knows more about magic than anyone you'll ever meet.

[@helloiamenergyman]

I KNOW! I could talk to this guy for hours if I knew him. Though I must admit I probably wouldn't be able to teach barely anything to this guy *'cause he's a fricking GENIUS*

[@porcospino289]

Alex Obviously false, if one ever meets PD. Possibly he knows more than anyone ELSE you’ll ever meet.

[@Headhunter_212]

Yes. The same PD. See the Ricky Jay biopic Deceptive Practice.

[@peaudest]

Wow, I explained all the article about card shuffling for a math class (France). I would have never hoped to see the author explain himself his theory. Thank you Numberphile.

[@josephmariani-mezera4171]

I prefer the Hindu (overhand) shuffle. When you do it real fast, the mechanistic and casual feel to it just makes it look cool.

[@ahmetgrcn2663]

It's also very easy to do controls in Hindu

[@IcarusAtreides]

So you are a cheater, unless you shuffle 10,000 times.

[@chaimlukasmaier335]

This is funny, I will present an youth science project dealing with exactly this topic. Our conclusion was that the best way to shuffle is by doing both Strippings and riffels. We did stripped 3 times for one riffel.

[@jeffdawson7841]

Nicely done Sir

[@lambda494]

"Keep going, like an idiot" -induction in a nutshell

[@nashtrojan]

Finally a video that validates my shuffle method. I alternate riffle shuffles, and overhand shuffles (I consider overhand shuffles as a cut) then I repeat that sequence 13 times and I leave the last cut for an opponent.

[@logitechpanasonic3]

3:20 + '..and if you have a really good memory' you get thrown out of the Casino.

[@karlsidney3218]

Just wanted to learn how to shuffle cards l, now I'm majoring in mathematics and trying to understand the cardinality of the set of rational numbers

[@philipnelson5]

Am I the only one that didn't get how 200 pokes turns into about 7 shuffles? I also saw 3/2 log_2(52) = 8.55 shuffles but that wasn't talked about either...

[@mrs.robichaud8823]

I get overwhelmed listening to this but I love it. I am of the simple peasant among a family of science, medicine and engineering and those with this kind of talent take my breath away, like when seeing a rainbow or a lightening strike for the first time. Thank you for the videos

[@ThisNameIsBanned]

Awesome week, i like this stuff and theres plenty of card games for any range of people that will really enjoy this, even people that normally dont care for math at all, will like to know what method of shuffling is actual usefull. For me i have some hard times explaining people that pile shuffling isnt randomizing a deck at all, as you can perfectly place the cards at exactly the spot you want it to be, its only really good for counting cards.

[@goldenratio_phhia]

2:37 “Take a guess: what do you think the top card is?” Me: “hm, 6 of spades” “6 of spades” Me: 0.0 *spits out tea*

[@SinanAkkoyun]

4:36 wtf

[@TheTK-Gaming]

yep

[@yusuf-5531]

You never heard of ventriloquism?

[@Darshao]

i m laughing so hard

[@johnkapp849]

do you know what adr is?

[@naterk9460]

Illuminati confirmed

[@mendelsonja]

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.

[@torgo_]

I'm surprised he didn't mention this fact: the riffle shuffle doesn't mix up the cards as much as you'd think. If you have a 52-pack in store bought order, and you complete 8 perfect riffle shuffles (or 8 perfect out-faro shuffles) then the pack will be restored to its original order! Very hard to do, though some expert card handlers can do it.

[@anticorncob6]

7j557j7j5 That's interesting. I've known that eight perfect riffle shuffles bring the cards back in order but I've never known that there are people who can actually do it.

[@gJonii]

anticorncob6 There are no people who have done perfect riffle shuffles three times in a row. I'm not sure if even two times. It is kinda Holy Grail for sleight of hand artists, but anyone who has tried it has had to give up, it's too inconsistent. Out-faro shuffles however are easier, and with moderate amount of practice, I managed to do a couple of perfect faro shuffles.

[@torgo_]

Joni Hanski Sorry Joni, but you are quite mistaken. There are plenty of cardists who can do a perfect riffle. For anyone who knows how to handle cards, it's not particularly difficult with a bit of practice. It's about 20% technique and 80% card quality. Some of my friends can do endless perfect riffles, I can usually do about 5 or 6 perfect riffles consecutively. For more info google "Joe Barry", he is known for developing one of the common perfect riffle techniques. Note: For beginners, it's *much* easier to learn the faro shuffle (compared to perfect riffle).

[@edsimnett]

so in 8 shuffles where does the first card go? does it stay where it is (which is one definition of a perfect shuffle, but seems to me to be bogus), or does it move through the deck? If it moves through the deck 52 is not factorable by 2 alone, so I don't see how that would work after the 6th shuffle (it goes 2, 4, 8, 16, 32, then (I think) 12, 24, 48, 44 which is not 1. So this 8 riffles only works if you assume you are never moving the first card (i.e. you always shuffle the same way)- is that right?

[@torgo_]

ed simnett Yes, the 8 riffles will need the top card to stay on top for each shuffle. The top card never moves. Also known as an "out-faro shuffle".

[@axxenmardok]

This is already one of my favourite people featured on Numberphile

[@Funyaripka]

4:37 He shows off his ventriloquist skills. Cool video though! 10/10 would watch again.

[@Nattakorps]

my grandfather, who was an avid cribbage player in his spare time, said no more than 7 shuffles. I never knew why, I eventually got some spare time and learned never the same shuffle more than twice in a row. especially riffling, as theres a chance you can undo your shuffle, for the mostpart. so usually id riffle > overhand x2 > riffle > overhand x2 > riffle > deal.

[@coldclearkt]

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

[@toasToaTs]

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

[@Peeja]

In the poking method, it seems like poking the top card anywhere *above* the King of Hearts doesn't accomplish anything. Wouldn't it be just as random, and faster, to always poke the top card somewhere below the King of Hearts? Or, equivalently, to randomly poke each card in the deck into a *new* pile, which begins empty and gradually fills with cards in a random order? That would give you a fully shuffled deck in exactly 52 pokes (though, like the longer poking method, it depends on a truly random poke).

[@seigeengine]

Sure, but one problem is that by randomly placing it anywhere you make it harder to trace the path. If I knew the order the cards were in, and watched you do it your way, I would then also know the order the newly shuffled deck was in.

[@MatthewBakerghj]

I thought about this too. I think it might not be complete random. Take the first card for example, in a complete random deck, it has 1/52 chance of being on top. With the new pile method, the first card has 1*1/2*2/3*3/4.....*51/52 chance of being on top. Is this logic correct? edit: nvm, that's still 1/52

[@seigeengine]

Matthew Baker You're wrong. Draw out all possibilities. In iteration 1, you have only 1 possibility, the first card being on top. Second iteration you have two possibilities, it on top once. Third iteration, there's 6 possibility (for each possibility in iteration 2, you can place the new card on top, in the middle, or underneath),and it's on top twice. Or the options are... iter 1 - A iter 2 - AB, BA iter 3 - CAB, ACB, ABC, CBA, BCA, BAC Meaning in iter 3, there's a 1/3 chance of it being on top.

[@MatthewBakerghj]

seigeengine I think you are recounting. order of AB and BA was already determined in iter 2. So you are doing 1/2*2/3 in iter 3, which is counting the 1/2 twice. Therefore by iter 52, the chance is 1*1/2*2/3*3/4.....*51/52=1/52 chance of being on top edit: i think we are on the same page now. I think you replied to my previous version.

[@seigeengine]

Matthew Baker I am not "recounting" anything. There is one possible outcome after we move the first card. There are two possible outcomes after we move the second card. When we move the third card, there are three places we can put it, on top, in the middle, or on the bottom, but there are two possibilities for how the cards can be ordered, either they're in AB or BA format before we move the third card, so there's 2*3 possible orderings of the new deck at this point. You might try to say "well the deck is already either in AB or BA format, so the other possibility isn't relevant!" But it is, because we don't know whether it's in AB or BA format. If it was in BA format, the probability of A being on top is already zero, if it's in AB format, the probability of A being on top after we move the next card is 2/3, because there's only 1 in 3 chances it'll go on top, or 1*(1-1/3). The math you're trying to use is wrong. The correct math is.... 1 * (1-1/2) * (1-1/3) * ... * (1-1/52)

[@beyse101]

Cliff Stoll is much more calm in this video than usual

[@squidcaps4308]

Riffle shuffle is not enough for some game types. If you know that ace of hearts that was in previous play is very likely to be at the top half of the cards, you can count on the fact that one player is most likely having that particular card. When the game requires more than 50% of the deck to be used and enough cards are revealed, you can make money of from it. I did.. For those game types you have to split the deck between riffle shuffles a few times or you deliberately let one side of the "riffle" to get ahead of the other before they are joined (you let a block of cards to be at the bottom before the rest are shuffled) That takes about 10 shuffles in total ( 7 riffles + 3 other ) It is the blocks you have get rid of and those happen a LOT.. In 7 cards stud, you get 3 of a kinds a lot, when the cards are collected from the table, they are within the vicinity of each other and that can totally kill the randomness. Ans since 7 card stud often uses most of the cards on deck.. well, you can guess that i kept my eyes on the deck when it was collected from the table and shuffled.. ;) If you know that the last ace is at the bottom of the deck... It was these blocks that gave me edge when there was a lazy shuffler but when i shuffled, i always used two types: blocks and riffles... If you mush them before riffle, that is one effective way too, players can "force" a mush by just putting the cards after the game in a wide pile that the dealer is forced to gather, it mixes those "blocks" pretty well too.

[@ykl1277]

a quicker way to do it, take top card, put it in an empty stack, take a 2nd card, put it in a random spot (above or below the 1st card), repeat for all 52 cards (3rd card can be on top, middle or bottom etc, 52nd card can be in any of the positions). It saves the redundant pokes.

[@greglott4977]

What about pile shuffling? Mtg players want to know.

[@jeffreymackay4343]

No way a Magic player is ever going to 'smoosh' shuffle his deck.

[@ENCHANTMEN_]

@@jeffreymackay4343 if they're in sleeves, you can take half in each hand, hold them loosely, and push the two piles together. the edges of the sleeves create a taper to let them smoothly merge

[@tysongrogan1636]

@@ENCHANTMEN_ I think he means the one where you throw them all in a pile and wave them around, which would ruin your deck in a hurry

[@ChibiDarksai]

I double sleeve and I wouldn't dare smoosh shuffle haha.

[@Frisbieinstein]

Prestidigitators tell us that eight perfect riffle shuffles returns a deck to its original order. His shuffle is pretty crude which I think is why he needs so many (seven). I think four alternating better riffle shuffles and random cuts would work pretty well.

[@Jack-Lack]

I had to downvote this pretty much instantly, and here's why: I'm a mathematician. I've been playing Magic: The Gathering for 22 years and shuffled decks of cards some tens of thousands of times. But beyond all that, I've actually conducted statistical studies on card shuffling. I'm also familiar with the "theorem" that this guy is referencing, which is not really a theorem, but a series of papers by Persi Diaconis. So here are my findings... The paper is not sufficiently based on the actual variance of card interweaving that happens in the real world with different decks of cards (i.e. varying degrees of wear) as shuffled by varying people. It is theoretical, and not at all informed by collecting real world statistics. So, here are some probabilistic and statistical facts related to card shuffling: -Probability says that if you number the cards of a deck from 1 to 52, and then perfectly randomize a deck, and then look through the deck and see how many times a card is directly followed by a card 1 number higher, then the count should be approximately 0.98. This is because you are making 51 such comparisons, and each comparison has a (51/52)*(1/51) chance of coming-up positive. -Statistics says that, because there is considerable clumping in a typical riffle shuffle, you actually need to riffle shuffle closer to 100 times before you achieve an approximately random state. Try writing the numbers 1 through 52 on a deck of cards with a pen, putting them in order, riffle shuffling them once, and then counting-up the number of instances in which a card is directly followed by a card 1 number higher. Most likely you will see a large number of such instances, perhaps greater than 10. So then keep repeating this exercise in which you shuffle and then count. Identify how many shuffles it takes before you see 2 or fewer occurrences of a card being 1-higher. Most likely, it should take you around 30 - 120 shuffles. This entire exercise counts as 1 trial. Continue repeating these trials until you have an adequate sample size. I've seen the paper he is referencing be repeated endlessly for decades. Its conclusions are incorrect. But more importantly, the repetitions of this paper are being used to influence shuffling rules and norms in various games like Magic (i.e. inappropriately prohibiting pile shuffling in contexts where it would make sense to allow it). The problem here generally is that these arguments seek to diminish the value and allowance of pile shuffling with respect to riffle shuffling. I believe that a statistically-literate understanding of shuffling algorithms can deliver faster shuffling algorithms, better randomization, and less cheating; but these advances are being shouted down by popular pseudo-math (and this video is only going to exacerbate that problem). My prescription for fastest achievable randomization in a card game like Magic would be 2ish riffle shuffles, followed by a pile shuffle, followed by a small number of additional riffle shuffles (2-3ish). A shuffling algorithm like this will fulfill all of the statistical hallmarks of randomness better than 50+ riffle shuffles while being faster to execute. A note relevant to poker vs Magic: in poker, cards are dealt in a round robin fashion, which somewhat diminishes the harm of cards clumping (i.e. sticking together throughout numerous shuffles). But in a game like Magic, each player draws sequentially from their own personal deck, which exacerbates the importance of clumping. In poker, even if you only shuffle a deck only 3 times, if a player's previous 5-card hand is still relatively clumped together, once you distribute to players in a round robin fashion, its effect on game play will be relatively minimal.

[@diebydeath]

Maybe the paper is incorrect, but if it is theoretical as you say, is that not simply a misguided application of the paper, then? The problem with Magic you're describing sounds more like the general mistake of expecting the real world to behave theoretically. But I don't think it's a problem for the paper to stay within theoretical bounds. I also don't think the professor denies that a person's "skill" in riffle shuffling will affect the outcome. There seems to be a threshold of clumping that one must cross in order for the shuffle to be considered a riffle shuffle. After all, I could do a riffle shuffle where all of Pile A's cards are riffled before any of Pile B's cards, which is the biggest clumping possible, and is actually just a cut. Of course it would take more than 7 of those to achieve randomness, but it's obviously outside the scope of the paper.

[@TRAGEDYM4SK]

A little off topic, but thinking about riffle shuffling expensive magic cards hurts my soul

[@wiktorzmarz]

nerd

[@johnc.4871]

In star wars lcg I felt that rifle shuffle was a waste and allowed the deck to be stacked. I used a off-topic, off-bottom method and every other time was 3 off top with one from bottom and finished a last time with off top, off bottom. The cards appeared very random then.

[@ChibiDarksai]

What's your preferred number of piles for pile shuffling in MtG?

[@bansheedearg]

I think of it this way. The riffle shuffle mixes the top and bottom half of the deck. But the bottoms stay at the bottom, and the tops stay at the top, it's super hard to get a bottom card to the top using riffle shuffle, at best it gets near the middle. The overhand shuffle reverses top and bottom, it inverts the deck. Combine riffle and overhand and you mix the deck almost as well as 52 card pickup. That said a perfect shuffle was promulgated by Knuth. For each object "k" in ordered set "M", exchange "k" with a random member of "M". For 52 cards, that's 52 exchanges, and you have a perfect shuffle. The key is the exchange is random.

[@Pmrpablico]

But if you alternate Overhand and Riffle it's better, because in the Riffle, you pick the top cards, if you alternate, you're picking up ''random place cards'' Is my theory true?

[@NSisFTW]

You get more variety more quickly from doing both, so yes you are correct.

[@danearmstrong5995]

This is how it is done as a casino dealer (sans shuffle machine). Wash (smoosh) > riffle > overhand > riffle x2

[@christopherbondoc9402]

Penn Jillette mentioned Persi Diaconis on an episode of "Fool Us" in reference to shuffling decks. Nice to see him speak.

[@BluefanNL]

I use the overhand shuffle with a twist, instead of always inserting a number of cards on top of the newly formed deck I alternate between adding on top and adding to the bottom. This way each card can easily end up on the other side of the deck which is probably the problem of the normal overhand shuffle. But I haven't tested this mathematically.

[@ZachariasEinstand]

A nice side note for 7 cycles as the most efficient count of shuffeling is the fact that banktransfer encryption is bild out of 7 algorythms over eachother.

[@Manic97LP]

Really? The most of them use AES which has nothing to do with 7 algorithms over each other...

[@TacoMasterMTG]

Could you do something comparing the futility of pile shuffling vs. riffle shuffling?

[@NoActuallyGo-KCUF-Yourself]

Pile "shuffling" isn't futile, but it isn't shuffling either; it is stacking the deck. If the cards are sticky and clumping a lot, piling before or after riffles or mashes/washes can help.

[@balaam_7087]

This was fascinating. As someone who enjoys playing M:tG I’ll be shuffling the middle way from now on...

[@ssjAnnaPaquin]

Don't rifle shuffle mtg cards, sleeve them and mash shuffle

[@KingOfNormans]

"And then, keep going like an idiot..." My favorite line of this video :P

[@JiveDadson]

This has nothing to do with math, but for practical purposes, it is important to throw in an overhand shuffle (strip), then turn half the cards end-for-end before finishing with a riffle shuffle. That will thwart someone who is "edge sorting."

[@JCar85]

I thought dude was wearing a powdered wig there in the thumbnail

[@DaviddeKloet]

It also depends on what you do with the deck after shuffling. If you proceed to deal out the cards one by one into four 13 card hands, fewer than 10,000 shuffles may be enough.

[@Banestalk]

I'd also like to hear about the math behind pile shuffling. In particular, I always make the number of piles a prime number (either 5 or 7 usually). Many people don't believe that makes a difference. Are they right? Also, what about non-standard card decks? 5 card draw is often played with 32 cards, whereas trading card games often use decks of 60-90 cards that almost always contain multiples of the same card (and where these multiples "bunching up" is often a problem), but maybe only half those cards are actually drawn overthe course of the game.

[@vuvuzelaelaela]

+Ann O. Nymus I'm pretty sure pile shuffling isn't shuffling at all, it's deterministic. There's no variance in how you split the halfs, or how many cards you drop when riffling. Cards bunching up isn't a problem, it's pretty likely to happen when you have actual randomness and multiple copies of the cards. Theres a slide in this video that says log2(n) * (3/2) gives you approximately the number of shuffles. 52 cards = 8.55 shuffle, 60 cards gives you 8.86, 90 gives you 9.73.

[@possumverde]

+vuvuzelaelaela You're right. While pile shuffling can mix things up quite well, it is not a truly random method. It's been a long time since I played any of the trading card games but I remember that, due to pile shuffling lacking true randomization, the tournament rules usually allowed it as part of the shuffling process but not as the final method employed.

[@Banestalk]

You're right, of course. In retrospect, that should have been obvious to me.

[@SethalaTheGamer]

+vuvuzelaelaela The benefit to pile shuffling, at least to my understanding, is that it breaks up pre-existing clumps in the deck, and is much less wear and tear on the cards (which is important for games with special cards, or trading card games where an individual card's value can be extremely high). For instance, let's take a game with a standard deck of cards. Each player draws one card from the deck on their turn, then if they have at least 4 cards of the same suit, they discard those cards and score a point. After the deck runs out, the discard pile is reshuffled and the game continues. Now, while doing several riffle shuffles is sufficient to randomize them, let's say you're not willing to bend your cards that way. Chances are, most other methods of shuffling are going to be insufficient to break apart the clumps in the discard pile that form from people discarding similar cards in groups of 4. A pile shuffle, followed by normal shuffling, is going to be enough to break up those clumps, assuming you do it with at least 4 piles. Some normal shuffling after to actually randomize the cards may end up with more clumps forming, but now that the clumps that were created deliberately are broken up, it doesn't matter so much.

[@Nalianna]

Use the plastic card protectors, and shuffle properly. Breaking up "pre-existing clumps" isn't shuffling, it's making a deck homogeneous. You're supposed to be making the deck random.

[@Clymaxx]

I always do a combination of one or more methods when I shuffle cards. Would this not be ideal? I feel like it maximizes randomness from the initial order. I also have a method that I devised that starts not so random but gets more and more divergent. It is basically taking the top card and placing it, then taking the next card and placing it beneath it, then taking the next card and placing it on top of the pile, and then the next is beneath the pile, etc until all the cards are in the new deck. I've found that it is fairly deterministic, but combined with other methods, certain patterns of cards diverge very well by the end of this method.

[@cookieaddictions]

I can't shuffle cards. I'm such a failure.

[@billymarkovsky1495]

yo check out reddit.com/r/cardistry and maybe you'll get it from the tutorials!

[@benthomason3307]

+cookieaddictionsOh come on. Even a toddler could do the wash/ corgi/ chimmey/ smoosh method.

[@benthomason3307]

***** yeah. Since posting that comment, I have also since realized that this method takes up the most time and space on the table. but the way I see it, ranking the three methods presented here goes as follows: Cutting the deck takes thousands of tries to shuffle the deck even decently, so it's right out. riffling the deck takes a mere seven tries, and takes up very little space, but the ordinary method of riffling takes a lot of practice, and I speak fro mexperience when I say that trying to squeeze the two deck halves together can seriously damage the cards if you aren't careful. corgi takes up a lot of space on the table and also takes much longer compared to the other methods, but if you have enough space and a mere minutes worth of patience, you can produce a deck that is *mathematically proven* to be *perfectly* random.

[@kindlin]

First off, I would love to point out that at no point did anyone bring up squishing decks together, especially with regard to riffling. That's a very bad idea that immediately damages the cards. @OP The shuffle is all about letting it do it itself, and you just set it up. The riffle is the easiest part; cut your deck, fold them together like playing a controlled version of 52 card pickup. For the bridge, you have to hold the far edges together firmly (the bases of the 'bridge') with your fingers and another finger or two higher up can help stabilize the very bottom cards. Then you try and let it all go by removing any 'stabilizing' fingers and letting the base go, while still firmly keeping your hands in the arch position so the cards don't go flying.

[@casperes0912]

Do smooching! It's easy and apparently effective.

[@jeremyday9056]

This guy is brilliant.

[@KixMusaid]

Me watching this as a Yugioh player

[@celaldogangunes]

Me too, but i can not do riffle aaaah, my money

[@peterilse4955]

One flaw, I have found with this idea, is that the bottom card and the top card usually isn't moved far away from their original place. You might not be able to know exactly where it goes, but if about half of the cards are given out, and you have in interest knowing whether the previous top card or the previous bottom card is given out, you often can say that the bottom card probably isn't and the top card is. That's why I combine the effective shuffle and Top-to-bottom. A rough shuffle and a fine shuffle.

[@codybuse7407]

I was doing a magic trick on a friend that required 3 specific cards to stay right next to each other. I told him he could shuffle the cards however he wanted to. He smushed the cards for about 15 seconds. The three cards were there, and I finished the trick.

[@jordibear]

I shuffle by dealing out into 8 piles (non equal, of course), and then riffle shuffling them into each other. I think that should be random enough (it's how I shuffle my MTG decks ;))

[@mushkamusic]

+Jordan Gilbert I guess that depends on the order in which you put the piles back together ;)

[@jordibear]

+mushkamusic Usually if I split it into piles (call them A-H) I mix AE, BF, CG, DH. then I mix AECG, BFDH and then all together.

[@mushkamusic]

+Jordan Gilbert Well that certainly sounds reasonable to me, mind you, I'm neither a mathematician nor a card player :D

[@evelynfinegan4687]

+Jordan Gilbert So, assuming you pile shuffle from the top of the deck, we know that the top card (let's say it's a Storm Crow) is at the bottom of pile A. We riffle shuffle pile A and D together, and now we aren't sure if Storm Crow is on the bottom or not, but we do know he's not at the top. So then we mix AE and CG together, and he's still not at the top, but he's still towards the bottom. Finally you mix AECG and BFDH together, and again you don't know where he is exactly, but you know he's not on the top. Doing it just once, as you can see, isn't truly random, because while we don't know the exact positions of the cards, we can assume that the cards at the bottom of the piles (or the cards at the top of the deck when you shuffled) are never going to be on the top. We can reasonably assume that one of the last 8 cards in the old order is on top, as they were on the tops of the piles. We can do it a second time, and now we're still not sure exactly where Storm Crow is, since we was near the bottom of the deck after the first shuffle, we can now assume he's near the top. Probably not at the top of the deck, but near it. Maybe between around the 10-15th card. Do it again and now he's probably about the 20th card from the bottom, and so on. So, essentially, each time we do that process we're only switching the tops and bottom of the deck, and leaving the card in the middle in the middle. Now cards do slowly shift up and down and will make their way to the middle and push those cards up and down. You'd probably have to do this around 7 times like with a riffle shuffle to ensure you had no idea where Storm Crow was in the deck. I'd wager that that method (which is the predominant shuffling method among Magic players) is roughly the same as just riffle shuffling the deck several times. I am unfortunately not that advanced with probabilities, so I can't do the math to verify that, but I would guess it's close to the same, maybe slightly more random. EDIT: After some thinking, I realized that Storm Crow will be in the bottom 8 positions after the first shuffle, the top 16 after the second shuffle, bottom 24 after the third shuffle, and so on. So you would need to shuffle a deck using that method about 8 times to ensure that Storm Crow has at least a 1/60 chance of being in any position in the deck, unless my logic and math are off.

[@SethalaTheGamer]

+Evelyn Finegan Note that he's not pile shuffling, just breaking the deck into piles of roughly similar size. And for your example, all you really have to do to completely lose a card is to perform a cut on each riffle-shuffled pile to completely lose cards even if you know their position when shuffling started.

[@fanicia42]

as a computer science student and a poker player i found this extremely interesting :) keep it up!

[@tempestandacomputer6951]

Could one say that Mash shuffling produces the same results as Riffle?

[@PinkSparklyGamer]

Tempest and a Computer yes. The idea is that every card is below an average of one card from the other half before being below another one from the same half. It's identical in terms of math between the 2 so it's the same. I prefer mass shuffling

[@kenburke4111]

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!

[@cladgreenhero5576]

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.

[@TheZotmeister]

What I do is four riffles, then cut the deck myself about 25% of the way down, then four more riffles before I offer it to an opponent to cut. That 25% cut takes the cards that were at the top and bottom of the deck and puts them smack dab in the middle of the next riffle.

[@Gcrowan]

I like doing a mix between the overhand and riffle shuffle. So I rearrange the general clumps and then riffle shuffle them. And then alternate between them until I've done maybe 3-4 of each. Someone who wants can make sure that the bottom card always remains in place, by releasing that side first every time with riffle shuffles.

[@HansenSWE]

Time to change my shuffle.

[@ando_rei]

As someone who likes card plays and plays quite some with friends (Doppelkopf), this is very interesting. We usually use the overhand shuffle for it is predominant in Germany, I suppose. I always thought the way of shuffling does not significantly influence the cards' distribution -- my eyes were opened. Riffle shuffle then from now on! Thanks Numberphile for the great videos on game theory! It is a very interesting field of mathematics.

[@SerieusFrank]

what is the best way to shuffle 60 cards, with sleeves (so you don't want to smoosh (all cards on table making circular random movements) or bending the card, (poker 2 equal stacks, bending it and then one after one to one new stack))??

[@SerieusFrank]

+Serieus Frank also not sideway shuffling

[@leefisher6366]

+Serieus Frank : Are you an MtG player, by any chance?

[@SerieusFrank]

+Lee Fisher yes

[@GruntXIII]

+Serieus Frank : Best shuffle method is... MTGO :)

[@niking987]

2:15 From what I know there are around 10^80 particles in the universe? So, maybe comparable, but still a lot more than permutations of a deck of cards.

[@GelidGanef]

Now I'm wondering if the "reverse riffle shuffle" used by over-protective card-game-collectors everywhere is just as mathematically rigorous as this one. For the uninitiated, one takes a deck of cards, and card by card, semi-randomly distributes it onto a few smaller piles (usually 3-6). Once the deck is exhausted, the piles are stacked on top of each other, and the process is repeated, at least once or twice more (although rarely to the full seven or eight it probably needs). Plenty of examples in the first five minutes of any MTG or Yu-Gi-Oh tournament video.

[@TheOriginalBoman]

+GelidGanef Table-shuffling (The term I use for the shuffling you have described there) is often not random, as many players will not put the cards in random piles (if there are 3 piles, they will put a card on pile A, then one on pile B, then one on pile C, then repeat). If the placement of the cards into the piles is not random, players can easily manipulate the deck for their desired outcome. That being said, I remember being told that the optimum number of piles for a table shuffle that is being performed with random pile distribution is 7, although there was no mention on how many times this needed to be performed.

[@Eckendenker]

Pile shuffle doesn't really help the randomness. It only makes it easier to distribute different card types across the deck. Usually you riffle shuffle between pile shuffling to really randomize it. It's a grey area shotcut for people that riffle shuffle well (or don't want to, like you said: card collectors).

[@charlesmiller7433]

But the pile method makes it so cards that were used in comboes (ie, right next to each other in order) are almost guaranteed to be farther apart. The problem with riffle and overhand shuffling is that often times comboed cards (cards played consecutively) may remain next to each other.

[@acexszg]

I love this guys voice

[@roberteospeedwagon3708]

8:23 did he mean natural log?

[@mantislazuli]

Yup

[@snbeast9545]

When real mathematicians say "log", they're referring to the natural log, because it's so much more useful than the common log.

[@mythicaldata6297]

Log without a base bro just a natural log as in what the log is in a natural state untouched

[@JxT1957]

i like the card shuffling machine method

[@Sauspreme]

2:09 The commonly accepted answer for the number of particles in the observable universe is 10^80 which is > 10^68 :-P

[@dangriff12]

+DJGerritProductions Thats the theorised amount for the number of atoms. Adding the number of sub atomic particles would be way bigger.

[@Sauspreme]

dangriff12 k. that would still be bigger than 10^68 :-P

[@non-inertialobserver946]

10^68 is more like the number of particles in our universe

[@chewii8427]

ViperDaniel What are atoms made of? Elemental particles... it would be way more

[@probatelaw]

The casino term for "smooshing" is "washing". On a poker table they first wash the cards for about 5-10 seconds, then riffle the cards 3-4 times. I wonder how random they are after that?

[@SeanSkyhawk]

Just to comment once again; I mentioned to Diaconis in person that the way I handle it is break the sequence of riffle shuffles with a Hindu and/or Overhand shuffle, that way if top/bottom were kept there then the shuffling action would disrupt them both.

[@DavenH]

around the six minute mark Persi says the number of 1-card riffles to expect that card under the bottom card, is 52, but surely it's not 52 but the root of (1-1/52)^n=0.5 which is about 36, meaning you'd expect it to happen by the 36th riffle on average.

[@ThisNameIsBanned]

If Magic the Gathering is the reason you got here, vote this up !

[@ThisNameIsBanned]

I know a lot of "no one".

[@TheRyry97]

Yes! Long live MTG and Math!

[@ThisNameIsBanned]

***** Well you can Riffle without bending the cards like crazy. Thats fine for Poker cards or anything which is more or less worthless, but for expensive cards, you should still Riffle, but you will only very very slightly bend the cards on the edges to let them fall into each other. Also more important, you want to keep the cards face-down all the time, if you bend them too much, you or the opponent can see the bottom cards, and thats not really what a shuffle should be doing ; all cards have to be face-down in the process, anything else is bad shuffling.

[@kolyaschaeffer5760]

ThisNameIsBanned I got here both from Blogatog and from following Numberphile. Does this mean I win?

[@ThisNameIsBanned]

Yes you advanced to level 2 nerd. Congratz !

[@LivingLegacy77]

I've always done the overhand shuffle. I might learn the riffle shuffle.

[@sulaimanbahiss7010]

6:07 and keep going like an idiot Lol

[@adrianlongley-preston5190]

I was taught to alternate riffle shuffling with scarne shuffling. (though scarne "shuffling" is really more like fancy cutting) I'd be curious to know how many times one needs to do THAT before one has decent entropy.

[@TheZotmeister]

You were taught well. What I typically do is four riffle-shuffles, then cut the deck myself about 25% of the way down, then four more riffle-shuffles. The 25% cut takes the cards that were at the top and bottom of the deck and puts them smack dab in the middle of the next shuffle. I could probably get by with just three shuffles before and after the cut, but I'm not about to do the math on that XD

[@ashleysmall2718]

Throw the deck up in the air then pick them all back up off the floor. Done.

[@someguy2135]

How many times should you shuffle before playing "Fifty two card pick up?"

[@Rickety3263]

We used to call that “Wanna play cowboys and indians? Ok! You play cowboys I’ll play Indians!” [Throws deck in the air while yelling and dancing]

[@EliWhite]

I'd LOVE to hear him also talk about the two most common methods of shuffling used in CCG games like Magic the Gathering so that the cards aren't hurt: Pile Shuffling, and Mash Shuffling. Especially interesting in hearing how well Mash Shuffling works in relation to Riffle.

[@TheZotmeister]

Mash shuffling IS riffle shuffling, just without the physical "riffle". The only thing the riffle itself does is gives the cards space to interleave so that they don't just smash into one another. Sleeved cards are effectively tapered and therefore have less of a problem getting themselves interleaved when simply shoved at each other, so those who haven't taken the time to learn how to riffle-shuffle sleeved cards (which is definitely non-trivial but quite rewarding and does a LOT less damage than just cramming them together) can get by with mashing... just as long as they don't do so lazily and not actually randomize the top and bottom cards, and on the flipside DON'T TAKE THE DECK OFF THE TABLE TO DO IT. Those who hold one half of their deck aloft in each hand and try to work them into one another is typically revealing a LOT more than just the bottom card to their opponent in the process. Pile shuffling isn't shuffling at all. It's a deterministic reordering of the deck. It's typically done to specifically unstack clumps of cards of the same type resulting from end-of-game scooping (to use Magic as an example, all the lands end up together). That isn't randomization-that is the OPPOSITE of randomization. If someone only does pile SORTING (as THAT is what it should be called) and tries to pass that off as their shuffle, what you need to do is SHUFFLE IT YOURSELF when offered to you for a cut. Yes, that's allowed. (Magic in particular spells it out: Comprehensive Rules, line 103.1.)