[[Official]] Shuffling: The Truth and Maths (Primer) - Magic General - Magic Fundamentals - MTG Salvation Forums - MTG Salvation

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I've seen numerous inquiries cropping up around the various sections of the forum (and on other forums that I frequent 'round the web, and among my friends in real life) regarding shuffling, and in response to them I've been posting/saying the same information over and over again, so I figured I'd just make a thread here so I could simply refer people to it.

Firstly, a few notes:

1. The formulas and maths I'm going to be mentioning/linking to in this thread were developed by statisticians over the past forty or fifty years who were looking at decks of cards that featured numbers and suits. Though our MTG cards don't have numbers and suits, they still adhere to the same principles of mathematical randomness as standard cards, and the formulas still apply. That said, I'm going to be presenting a simplified explanation; I'll provide links to the original articles if you'd like to review my sources, but they're pretty complicated.

2. This thread discusses shuffling in the context of achieving randomness--a complete lack of pattern or predictability--in your cards. The first question to ask yourself, though, is whether you really WANT your cards to be random. Sometimes people are complacent enough to have their decks only mostly random, since randomization can take a while to achieve. Other times people intentionally try to avoid randomness for the purposes of stacking their deck; for more info on this read Mike Flores' blog post here and Tim Pillards's article here--they're from 2009 and 2003, respectively, but the concepts they're getting at are all still valid today. My goal in this thread is not to say whether you want to be random--that's for you to decide--but rather only to clarify exactly how to achieve randomness if you wanted to.

1. What is the optimal shuffling technique?

Of the most common techniques able to be performed by hand (ie not a computer algorithm, but a method you can actually execute in real life), the riffle shuffle achieves mathematical randomness more quickly, consistently, and safely (i.e. with less room for cheating) than any other shuffle, usually by a very wide margin.

The riffle shuffle is what most people use to shuffle a standard deck of 52 cards, including most casinos. You split your deck into two piles, approximately even, and then drop them one onto the other imperfectly, letting the cards cascade from your hands.

By imperfectly I mean that sometimes, as a natural consequence of the speed of the shuffle, your finger slips and you drop two cards or even three at once from one pile. In fact, according to the mathematical interpretation of a riffle used in a groundbreaking stats article described below, the probability of dropping a "clump" from one pile increases depending on the current size of that pile (due to the weight of the cards in your hand and other numerous factors). I underline the words "approximately" and "imperfectly" because, as I'll describe later, these imperfections are vitally important to this technique's efficacy.

Quote from Typical MTG Player »

But won't riffle shuffling DAMAGE MY CARDS???

Unfortunately yet understandably, many people are reluctant to riffle shuffle their decks because it involves bending and therefore potentially damaging the cards. This can be minimized by only riffling the edges together and then sliding the two halves into place; some people even go so far as to double-sleeve their cards to prevent frayed edges while edge-riffling. But some people will still hesitate to riffle even just the edges, especially if they're playing with really, really expensive cards as in Vintage.

Despite this, it really is the fastest, most effective way to shuffle your cards by hand that leaves minimal room for you or your opponents to cheat--whether intentionally or unintentionally (again, more on this below). Accept no substitutes, not when you're shuffling your opponents' decks, not when you're shuffling your own deck.

2. How many riffle shuffles does it take to achieve randomness?

In 1992 three statisticians, Bayer, Aldous, and Diaconis, building on the work of many statisticians before them reaching back to the '50s, published this research paper which has since received tremendous praise and become wildly popular. They came up with an extremely complicated but highly accurate equation for the number of riffle shuffles it takes to randomize a deck of n cards.

I will post a simplified version below in the form of a mathematical limit adapted from this graph. This is an equation you can all punch into your handy-dandy TI-eightywhatever calculators you may or may not have saved from school:

Let:
n = #cards in the deck
s = #riffles to achieve randomness
Then:
The limit of s as n approaches infinity is equal to

1.5 * (log n)/(log 2)

As a limit with stipulation n -> infinity, this equation becomes increasingly true as the number of cards in the deck gets larger (approaches infinity). At lower values of n, this simplification tends to overestimate, but never underestimate, the number of riffles by a little bit.

For example, with 52 cards in the deck, the simplified formula will give you a result of 8.55. However using the full, complicated formula that's much too difficult to remember by heart or to store in your calculator's memory, the number of riffles required is actually around 7.

The good news is that the formula will only overestimate and only by a little bit, and it never hurts to riffle a deck one or two times more than the mathematically demanded number; the only argument against doing so is that it's wasted time, but if you do a riffle properly it shouldn't take you more than a few seconds anyway.

So if you're playing a 60-card Modern deck, riffle at least eight times (though you could probably get away with 7, still). If you're playing a 100-card EDH deck, riffle at least ten times. If you're playing a 500,000-card monster deck, riffle at least 28 times. And so forth.

3. If I shuffle half as many times as I'm supposed to, will the deck be half-randomized?

The relationship between #shuffles and randomness is not linear but rather exponential. The way the math works out (see the graph I linked earlier), the randomness of your deck doesn't increase very much at all for the first few riffles. When you reach the magic number of riffles particular to your deck size, though, the randomness increases sharply and then tapers off...any further riffles still offer some additional randomness, but not enough to care about.

In other words, no, you can't riffle a 100-card EDH deck, which should be shuffled around 10 times, only 5 times and then say "OK, now my deck is half-randomized."

4. What are some bad shuffling methods to be avoided?

Anything but the riffle. No seriously. The tl;dr of this section is that all the other common shuffling techniques either demand an unreasonably large number of shuffles to achieve randomness or otherwise never achieve randomness, though they might seem like it.

4a. Pile shuffling, where you deal the cards into some number of piles and then stack them back up again. This is probably the most common shuffling technique in MTG alongside riffle shuffling because it carries significantly less potential to bend cards, and so I will discuss it more than the others.

Firstly, have you ever seen that famous 21-card magic trick? There's a youtube video here showing how it's done and a wikipedia article here explaining it in words.

The trick involves the contestant selecting a card, any card, from a group of 21, and not showing it to the magician. The magician then pile shuffles face-up three times, each time asking the contestant to identify which pile his card is in--this pile is scooped up second out of the three each time. Then the magician deals ten cards, and the eleventh card, in the very middle of the 21-card packet, is the contestant's card.

Unlike other magic tricks, this one involves no sleight of hand. There is a mathematical pattern intrinsic to the pile shuffling that puts the contestant's card in the very middle of the deck every single time.

This is a perfect example of the truth that people often fail to realize: pile shuffling is not random. Any form of pile shuffling. Not just "mana-weaving," any kind. It's actually one of the least random ways to prepare your deck; in fact, as Flores and Pillards discuss in the articles I linked in one of my disclaimers (also here and here), pile shuffling allows savvy players to actually stack their deck!

Some people on this thread have asserted that pile shuffling does indeed have its benefits, as it allows players to ensure the quality of their sleeves and to ensure their deck has the appropriate number of cards prior to a match. There is nothing heinous about using pile shuffling in addition to random methods for quality control only; the issue emerges when people use pile shuffling instead of random methods, as seen in the widespread fallacy that you can just pile shuffle a few times to effectively "emulate" riffles with less damage.

Quote from David Green, L2 Judge »

There's another method called "pile shuffling," which you can use, but you need to use it in conjunction with another method after piling your deck after piling your cards. The reason for that is that it's really easy to manipulate your deck when you pile shuffle.

The main reason behind pile shuffling is really just to count the cards to ensure you have the appropriate number of cards before you begin your game. (source)

If you're pile shuffling for quality control, you should not need to do it more than once, and you should not need to make more piles than necessary for you to count to 60 (or 40, or 100); in the Flores article he refers to a video describing a pile shuffling method using 5 or 7 piles because those are "mersenne prime numbers aka really random," and as Flores correctly notes, the whole thing is a load of BS.

Mersenne primes have nothing to do with randomness. There's a random number generator known as the "Mersenne twister" in which very large Mersenne prime numbers were chosen arbitrarily by the system's creator as the period length--but there is no connection to MTG and pile shuffling. No matter how many piles you make, your deck will not be random.

To be fair, pile shuffling can be made slightly more random if you drop cards into random piles instead sequential ones...like if you have three piles 1, 2, 3, instead of dropping cards 1-2-3-1-2-3, you should drop them 1-3-2-3-1-2, or what have you. But even then this technique is only marginally random, nowhere near as effective as a riffle shuffle, and it's replete with cheating opportunities to boot.

4b. Overhand shuffling, aka "strip shuffling" where you hold the deck in your right hand and take little packets of cards off the top and put them into your left hand so your deck order is reversed in clumps.

As discussed in this article (Pemantle 1989), overhand shuffling is an extremely inefficient randomization strategy. Depending on variations in technique it would take anywhere from 1000 to 3000 overhand shuffles(!!) to randomize a 52-card deck to the same extent as riffle shuffling that deck only 7 times.

4c. "Perfect" shuffling, aka Faro shuffling (wikipedia here), where the deck is split into exactly two halves and the cards are laid exactly one upon another.

Above I mentioned that the imperfections of a good, quick riffle shuffle are what makes it so effective. This is why: a perfect riffle shuffle simply introduces a nonrandom mathematical sequence, sort of like a more complex variation of pile shuffling. There are two kinds: in-shuffling, where the top card of the deck becomes the second-top; and out-shuffling, where the top card of the deck remains on top. Long story short, Faro shuffling is far from random; magicians use it all the time, as with pile shuffling, to control the placement of certain cards within the deck. Indeed, eight out-shuffles performed sequentially will leave the deck in the same order as it was initially, as demonstrated here in this youtube video.

5. What about mash shuffling?

Mash shuffling, where the deck is split into approximately halves and then "mashed" together, can go either way depending on how skillful you are at mashing cards. If your mashes are tight enough to be almost one-card-upon-another but with some minor imperfections, then you essentially replicate a riffle shuffle. However, this is not the easiest to achieve; if your mashes are generally hasty such that your clumps are rather large, your results will be nowhere near random. Additionally, as shown in the youtube video above, the mash technique can be used to perform the Faro shuffle, too--if you are an expert card handler such that you can reliably cut your deck exactly in half and mash the cards exactly one atop another, then you have just performed a Faro shuffle...and if you do that seven more times, then your deck's order has been totally preserved as if you never shuffled at all. The Faro can be performed with a riffle, but it's significantly harder to pull off because the performer must manipulate, in addition to everything else, the precise slipping of his thumbs from the edges of the cards to only drop one at a time.

In order to set a goal for yourself in your mash shuffling, I strongly recommend riffling either a 52-card 2-thru-ace deck or, preferably, a 60-card Magic deck (so you can get used to the unique feel of Magic cards as you riffle them; they're slightly thicker and harder to control than 52-cards). If you're sensitive about bending your valuable Magic cards, then make a 60-card stack of your two-cent commons, pretend they're a deck, and riffle them. As you split and riffle, observe the manner in which the cards naturally cascade from your hands; when the riffle is done, before you slide the cards together, hold the deck sideways and note the size of the "clumps" from each split (as shown in the image to the right). Almost all the clumps should be of 1 or 2 cards, with an occasional 3 and rarely 4. Treat that distribution you see as your "gold standard;" you want your mashes to be as close to riffles as possible, so you want the distribution of cards during your mash to look exactly the same as what you observe when you riffle.

In other words, mash shuffling can be the same as riffle shuffling, but you have to consciously endeavor to replicate the mechanism of a riffle as closely as possible. If you mash in large clumps, or if you only mash half your deck together at a time, or some other practice, you are only straying further from the mechanics of a riffle, and more importantly you are straying further from true randomness. Additionally mash shuffling leaves more room for cheating than riffle shuffling, so that's something to watch out for.

In summary, mashing is, at best, a more inconsistent way to riffle, and at worst, nothing like a riffle at all. The most consistent way to randomize a deck--yours or your opponents--is to riffle shuffle.

6. Chaos vs Randomness

This section was added to the FAQ by request.

As Fnord points out below, a deck of 60 cards that has been pile-shuffled three times and then riffled once might look, to the naked eye, similar to decks of 60 that had been riffled a full 8 times. More generally, people tend to make the assumption that pile shuffling is random simply because it looks random. This is one of the great cognitive fallacies of mankind; in fact, what looks random at first glance may be quite far from the statistical definition of randomness. As TheLizard proves on page 4 of this thread (with a conceptual explanation by myself at the bottom of this page), the first series of numbers Fnord lists is in fact significantly less random than the others. Unfortunately, when you start talking about how humans interpret things and cognitive fallacies, you get into psychology, which is beyond the scope of this FAQ.

Even for a deck that seems random, during gameplay, over a long period of time (ie, after many, many hands have been dealt), the statistics dictate (via the Law of Large Numbers) that the difference between simply a "chaotic" method and a truly "random" method must become increasingly obvious. To provide a real-life example, consider the MTGO shuffler, which over the last decade has become infamous for being purportedly rigged or otherwise broken because of how often people get mana screwed.

The fact of the matter, as Wizards and others have repeatedly reminded us, is that the MTGO shuffler uses a highly random computer algorithm, more random than what people are used to with their real-life shuffling practices, and so over time people observed this difference and blamed the computer (because of course, between the human and the computer, the computer's the one doing math wrong :rolleyes:). In fact, true randomness involves some clumping and has some chance of mana screw. Shuffling more times will only more consistently reach the predicted number of clumps, not reduce the number of clumps. The only way to reduce the number of clumps is to move away from true randomness by fixing your deck, such as through mana weaving.

In Fnord's example below, as determined by TheLizard, the first deck, which was riffled only twice, contains a greater number of sequences of lands and spells than the other four decks which were riffled 8 times; that is to say, the big "clumps" of lands and spells that can lead to mana screw have been broken up further than they usually would, as if the deck had been mana-woven. Let me say that again: the deck that had been shuffled less was more akin to a deck that had been mana woven (stacked); more shuffling actually leads to a higher chance of mana screw than less shuffling. In the long run this has a poignant and noticeable effect on gameplay.

Once again, whether you want to achieve true randomness is your decision. Keep in mind, however, that the rules call for a randomized deck, so if you stray from randomness through insufficient shuffling, intentionally or unintentionally, and a player or a judge notices that your deck is looking less-than-random, you can be labeled as a cheater for it in competitive play:

Quote from Lee McLain, Ohio, L3 Judge »

Insufficiently randomizing a deck is something that sends up a warning flag to other players and judges alike. In the Penalty Guidelines the penalty for insufficient deck shuffling is a Warning. That is the penalty for an unintentional infraction. If a judge determines that the infraction was intentional, it will be upgraded to Cheating, which will result in disqualification and an investigation by the DCI.

Point being, in the long run, the chaos with which we have become complacent as calling "randomness" does indeed distinguish itself from true randomness, even during gameplay. As online algorithm shufflers demonstrate, people's eyes might deceive them into telling them that certain techniques create randomness, but when they try the real randomized deck it plays differently in the long run.

7. Conclusion

In summary, riffle shuffling (and to an extent mash shuffling, ONLY when it is performed identically to a riffle shuffle) is the most effective and the fastest way to achieve true, mathematical randomness. Other common shuffling techniques like pile shuffling might seem random, but they generally fail to truly randomize your deck, and this difference becomes noticeable in the long run. More importantly, non-riffle methods provide savvy shufflers with opportunities to stack their decks and violate the rules.

I hope the material I presented in this brief FAQ-primer-thingy answers everyone's questions regarding shuffling. Hopefully people will be able to refer to it in the future.

I also encourage feedback of any kind. If I missed any other big questions, or if there's something here you disagree with, please don't hesitate to let me know. If you found this useful, please say thanks (protip: there's a button for that over to the left). I appreciate your support, too.

So what about pile-shuffling + riffle-shuffling? After a game, I tend to pile-shuffle once just to get the clumped stuff spread out and riffle shuffle afterwards to randomize it.

So what about pile-shuffling + riffle-shuffling? After a game, I tend to pile-shuffle once just to get the clumped stuff spread out and riffle shuffle afterwards to randomize it.

Pile shuffling will never hurt your deck, and it is generally a good idea to do it just because it is also a fast way to count your deck (you don't need to count the cards, just remember what pile the last card should go to) and check for sleave damage. However, if you riffle 8+ times any clumps will be gone (and most likely replaced with new ones) anyway.

OP: I think you forgot the most common way to shuffle a sleaved Magic deck: Mash shuffling.

Take 1/4-1/2 of your deck in one hand, the other part in the other hand. Then mash the two piles together.

Good part: Achives the same result as riffle shuffling without risk of bending the cards.
Bad part: It takes more mashes to get a randomized deck. The shuffler can also easier (compared to riffle) control the top or bottom cards and make sure they are never randomzied.

What about mash-shuffling (when you take two halves of your deck and "mash" the cards between each other)? It seems similar to the riffle-shuffle because the "mashes" are imperfect. I ask because I can't riffle-shuffle sleeved cards for the life of me.

So what about pile-shuffling + riffle-shuffling? After a game, I tend to pile-shuffle once just to get the clumped stuff spread out and riffle shuffle afterwards to randomize it.

The whole point of the groundbreaking 1992 study on card shuffling was that if you riffle shuffle a certain number of times, the cards are, for all intents and purposes, fully randomized.

Whatever randomization that you're accomplishing by pile shuffling can be achieved more quickly, more soundly, and with less opportunity for cheating simply by riffle shuffling more times instead.

As an aside, a fully-randomized deck will have clumps sometimes. If you never got mana-screwed or mana-flooded, it wouldn't be random, it'd be fixed! The point of the mulligan system is to filter through the clumps when they come up.

What about mash-shuffling (when you take two halves of your deck and "mash" the cards between each other)? It seems similar to the riffle-shuffle because the "mashes" are imperfect. I ask because I can't riffle-shuffle sleeved cards for the life of me.

If your mashes are tight enough to be the equivalent of a riffle shuffle (like mostly one-atop-the-other, but some "imperfections," clumps of two or three) but at the same time not quite so tight as to be a perfect Faro shuffle (in the youtube link I listed you can see the guy uses mash shuffling to pull off a faro shuffle), then the results should be the same as a riffle shuffle. But the thing is that you have to make a conscious effort to make it like a riffle.

The safest thing to do is just riffle it up. If your cards are sleeved and slippery and hard to handle, you can still just riffle the edges together while they're down on the table (like they do at casinos) and then slide the cards together.

Generaly good and informative post. There are a couple of points though;

Whilst not a useful tool for achieving randomness, the pile shuffle can be used as a precursor to actual shuffling in order to break up any large clumps of the same type of card that you may be aware of. It then attains marginal usefulness in reducing the workload required of your actual shuffling.

The other thing is that I noticed a lack of commentary on what (from experience) I would consider one of the most common shuffling methods amongst tcg players: the push shuffle.

Performed by holding the deck at ninety degrees in your left hand, separating approximately half the deck into your right hand, and pushing one half down into the other.
It has obvious similarities to the riffle shuffle, plus benefits and drawbacks:..
It doesn't bend the cards; unlike a riffle shuffle, the push shuffle allows gravity to spread the cards in your lower hand slightly, meaning that you don't have to bend the cards when you push the two halves together.
It's rather easy to do with sleeved cards, but extremely difficult when the cards are unsleeved.
It can be possible for you or your opponent to catch glimpses of some of the card faces during the shuffle if you are not mindful of the angle of the cards.

The above considerations make the push shuffle entirely unsuitable for classical card games, but makes the push shuffle a viable alternative to the riffle for tcg's where card sleeves are prevailant and the value of the cards can be significant enough to make riffling them a distasteful option.

So what about pile-shuffling + riffle-shuffling? After a game, I tend to pile-shuffle once just to get the clumped stuff spread out and riffle shuffle afterwards to randomize it.

If you are riffling sufficiently to randomize the deck fully, the pile shuffle is moot regardless of your reason for doing it. If the pile shuffle does make a difference, then not only are you not randomizing sufficiently, but doing the pile shuffle could be construed as a form of cheating.

Whilst not a useful tool for achieving randomness, the pile shuffle can be used as a precursor to actual shuffling in order to break up any large clumps of the same type of card that you may be aware of. It then attains marginal usefulness in reducing the workload required of your actual shuffling.

As Flores points out, in the time it would take you to pile shuffle, you could riffle shuffle many more times, since riffle shuffling is such a quick maneuver. That would separate out the clumps in a more random and less cheating-inclined way.

The other thing is that I noticed a lack of commentary on what (from experience) I would consider one of the most common shuffling methods amongst tcg players: the push shuffle.

I've heard it called mash shuffling and other people on this thread used that name. I added a section for it in the main post. It can go both ways.

I think you overestimate how random the deck actually needs to be.

Example: Here are 5 shuffled decks. 4 were shuffled using 8 riffle shuffles. The fifth was pile shuffled 3 times and then riffled once. Can you tell which is the pile-shuffled one? (The numbers are the initial order the card was in.)

I think you overestimate how random the deck actually needs to be.

Example: Here are 5 shuffled decks. 4 were shuffled using 8 riffle shuffles. The fifth was pile shuffled 3 times and then riffled once. Can you tell which is the pile-shuffled one? (The numbers are the initial order the card was in.)

Let's see... I think it's the fourth pile. How did I do?

You do have a point. Those all look sufficiently random.

Mabye you should make a section on Chaotic vs Random TheTrueNub. Pile shuffling being chaotic looks random to the untrained eye (most people's eyes) but is controllable, which is what enables it to be so useful for cheating.

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Let's see... I think it's the fourth pile. How did I do?

Not well. The first pile was the pile-shuffled one.

You do have a point. Those all look sufficiently random.

Mabye you should make a section on Chaotic vs Random TheTrueNub. Pile shuffling being chaotic looks random to the untrained eye (most people's eyes) but is controllable, which is what enables it to be so useful for cheating.

I don't deny that pile shuffling is useful for cheating, which is why you should be suspicious if your opponent only pile-shuffles. But for the honest player just looking to randomize your deck, my point is that it isn't as bad as you make it out to be. A deck pile-shuffled a few times, plus a single riffle shuffle, is functionally good enough.

Not well. The first pile was the pile-shuffled one.

Dagnabbit! You didn't have to tell me the right answer yet. I would have gotten it by the fifth guess.

I don't deny that pile shuffling is useful for cheating, which is why you should be suspicious if your opponent only pile-shuffles. But for the honest player just looking to randomize your deck, my point is that it isn't as bad as you make it out to be. A deck pile-shuffled a few times, plus a single riffle shuffle, is functionally good enough.

True. Without the actual intent to cheat, a chaotic mix is functionally no different from an actual random one.

However, as a storm player, I enjoy my time between rounds, and a lot of pile shuffling would drive me crazy. I can shuffle and cut in under 60 seconds, so everyone else should too.

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I think you overestimate how random the deck actually needs to be.

Example: Here are 5 shuffled decks. 4 were shuffled using 8 riffle shuffles. The fifth was pile shuffled 3 times and then riffled once. Can you tell which is the pile-shuffled one? (The numbers are the initial order the card was in.)

I think you overestimate how random the deck actually needs to be.

Example: Here are 5 shuffled decks. 4 were shuffled using 8 riffle shuffles. The fifth was pile shuffled 3 times and then riffled once. Can you tell which is the pile-shuffled one? (The numbers are the initial order the card was in.)

Firstly and foremostly, you're approaching the issue from the wrong direction. Like I said in my disclaimer and like Flores hints in his article on cheating, how random the deck "needs to be" or "should be" is up to you and your particular goals. My advice on shuffling, which is itself just a condensation into readable, jargon-less form of the work done by dozens of statisticians over the past several decades, is if you're looking to reach the mathematical definition of randomness, true randomness, the uniform distribution.

That said, I'll still provide a conceptual way of solving your problem just so I can put my money where my mouth is--though unfortunately I won't be able to provide an exact answer.

RETROSPECTIVE EDIT: TheLizard has carried out in full on page 4 of this thread the math I explained here only conceptually, for anyone who's reading this post. He determined that the first deck is improperly randomized, as it possesses a significantly greater number of sequences (that is, a greater "weaving" of mana and spells) than what should have resulted from eight shuffles according to the probability.

In order to "solve" (or, in this case, take an educated guess at, since when we're trying to extrapolate general truths from specific samples, an educated guess is the best we can do), I'd count the number of rising sequences in each of those five decks and then plug them into this equation:

Quote from Bayer and Diaconis 1992 »

If n cards are riffle shuffled m times, then the chance that the deck is in a given arrangement pi is

[(2^m + n - r) nCr (n)]/(2^(mn))

where r is the number of rising sequences in pi.

In your case, n = #cards = 60, m = #riffles = 8. I'd have to count the rising sequences by hand unless I found some program that did it for me, which is an excruciatingly tedious and error-prone process, so I'll keep it all conceptual.

After counting the rising sequences, I'd then find which of those configurations had the least probability to have as many rising sequences as it does (ie the pi with the lowest chance) assuming 8 shuffles, and then I'd have to assume that the least probability of receiving such a result after shuffling 8 times corresponds to the most probability of not actually having been shuffled 8 times.

So like let's say, assuming 8 shuffles, the first four configurations are all 80% likely, and the last one is only 10% likely (I just made those numbers up). Then my best answer would be that the last one was the most likely NOT to have been shuffled 8 times. Even then, since we're using samples, I can only give you the "most likely" answer, not the actual answer.

It's also worth noting that the things Bayer and Diaconis were tallying when they were determining how many shuffles equaled true randomness, that is, rising sequences, are directly applicable to MTG.

In a 52-card deck rising sequences are the number of ascending orders, so for example the set {A, 2, 3, 6, 7, 8, 9, 4, 5} has two rising sequences, {A, 2, 3, 4, 5} and {6, 7, 8, 9}. As you riffle more and more, the number of rising sequences increases (ie the sequences become broken up) until the deck becomes fully randomized, ie whatever sequences still remain mind as well be the product of chance. It's this same "breaking up" that we refer to when we talk about "breaking up clumps of land" or "breaking up clumps of nonland" in our own decks, and it's already been proven by people smarter and more accomplished than I that the riffle shuffle is the fastest and most effective way to do it.

Very interesting point. They all look really well shuffled if you ask me.

This is exactly the kind of misconception I'm trying to get rid of in this thread. You can't just look at it by eye and come to a conclusion in the same way you can't just look at pile shuffling by eye and say "yeah, that looks about random" or look at a Faro shuffle by eye and say "yeah, that looks about random." There's math behind it.

True. Without the actual intent to cheat, a chaotic mix is functionally no different from an actual random one.

Not necessarily. You know how people always complain about the MTGO shuffler being rigged or horrible or biased? That's because the MTGO shuffler is an ideal shuffler. It creates "true randomness" in the mathematical sense of the term, and in the long run, over many, many hand deals, people notice the difference between their regular, "chaotic" practices and the actual random ones.

You've already said it was this one, so I looked for a pattern in it. I see one where a card above 40 hits every 2 or 3 cards with few a exceptions. If those were your lands, that's a pretty good weave. I didn't look at the others.

You've already said it was this one, so I looked for a pattern in it. I see one where a card above 40 hits every 2 or 3 cards with few a exceptions. If those were your lands, that's a pretty good weave. I didn't look at the others.

If you go looking for patterns you will find them. They are the faces in the trees.

This really is going nowhere fast though. Back on topic:

Faro shuffling! How many people can do it, and considering it's likeness to mesh shuffling, how likely would it be to go unnoticed? I riffle shuffle, because mesh shuffling damages my sleeves, and I have the dexterity to riffle without bending cards, but I am confident I could do a Faro with enough practice. Sleeves make it almost too easy. I'm sure a deck could be easily stacked with proper Faros.

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Just going to throw this out there but while the rules demand your cards be random, as a player, you want something else. Personally I want every group of seven cards to have a three four split lands to spells. Obviously achieving the second seems to contradict the first, but if you told me to go from a deck that had just been pile sorted by card type and play after 7 riffles I'd say no thanks personally. Shuffling technique is certainly important in the results seen by high level players. I doubt many of them simply riffle 7 Times and call it good. I'm willing to bet most of us, through much playing have developed shuffling routines we feel mitigate mana screw.

(and I don't mean mana weaving your deck or anything, just a set routine you go through where after you're done piling/mashing/riffling you feel like you've achieved a good land to spell balance and I'm sure you know what I'm saying so don't bother calling me a cheater or anything bc I just won't respond - I've seen how these threads end up)

Point about presentation: the "conclusion" should probably have some sort of summary in it rather than being a fluffy sign-off. In particular, it would be useful to be able to tell at a glance which shuffling methods are good and which ones aren't without reading the entire essay.

Faro shuffling! How many people can do it, and considering it's likeness to mesh shuffling, how likely would it be to go unnoticed?

In theory anyone can do it with enough practice, and if it's done correctly (ie quickly, fluidly, without giving a chance to reveal that the shuffler is actually meticulously choosing exactly half the deck and weaving them exactly together), then it's extremely difficult to be noticed during the shuffling process--that's why it's a magic trick. The only simple way to detect a perfect Faro shuffle (or a pile shuffle cheater, or any other kind of shuffle cheater if he's good at what he does) is when the deed is already done, and that relies on the fact that you or the judge (more importantly the judge) knows what to look for. For example if you're Extirpate-ing your opponent and are searching his library and find that all 24 his lands are organized in 12 neat clumps of two lands each, ie a nigh-perfect mana weave, you have to recognize that that's extremely improbable from a statistical standpoint and call him out for it.

Mash shuffling can be done soo much faster than riffle shuffling. As long as you have a technique to spread the cards apart, reveal them to nobody, and push them together, you can get two effective mashes together in the time it takes for one deck thrashing riffle.

I mash shuffle in such a way that I always make sure that ~10 cards from the top of the bottom pile end up on the top pile. In this way, I have an 'escalator effect' That moves throughout the deck ensuring that no area goes insufficiently shuffled for whatever reason. I tend to shuffle 15-30 times before a game, so it's pretty random (for the past ten years at least).

I consider myself pretty good at shuffling, but I find it hard to imagine someone Faro shuffling via-mash without looking awfully suspicious. They'd have to be a djinn or something, at that point just give them the game.

*come to think of it, I emphasize different portions of the deck at different times, but I change it up to make sure I'm neglecting the top and bottom halves of the deck. I'm not trying to cheat with what I'm doing, but more sufficiently randomize. I consider 7 riffles to be abhorantly few shuffles for a magic game. I can get in 15-30 shuffles before a game/after shuffle effects.

**I average 29 mashes in 60 seconds. I imagine I could probably get 14 riffles in...I'm not great at riffling...but if all my mashes are as good as one half riffle, I should be all set.

Just going to throw this out there but while the rules demand your cards be random, as a player, you want something else. Personally I want every group of seven cards to have a three four split lands to spells. Obviously achieving the second seems to contradict the first, but if you told me to go from a deck that had just been pile sorted by card type and play after 7 riffles I'd say no thanks personally. Shuffling technique is certainly important in the results seen by high level players. I doubt many of them simply riffle 7 Times and call it good. I'm willing to bet most of us, through much playing have developed shuffling routines we feel mitigate mana screw.

I won't call you a cheater; I said in my disclaimer that you may want, for whatever reason, your deck to be more or less random, and that's your choice. But I will stand by my assertion that some mana screw/mana flood is actually part of randomness. If you're going to mana weave, either by pile shuffling or otherwise, I simply ask that you do not perpetuate the misconception that that's what randomness is by saying your deck is "randomized" or "sufficiently randomized" or "mind as well be randomized," because unless you riffle shuffle 7-8 times (60 cards) or 9-10 times (100 cards), your deck is probably not very random at all.

Point about presentation: the "conclusion" should probably have some sort of summary in it rather than being a fluffy sign-off. In particular, it would be useful to be able to tell at a glance which shuffling methods are good and which ones aren't without reading the entire essay.

Noted. Now there's a handy blurb for the lazy people.

I normally mash shuffle the piles instead of stacking them like it's done in pile shuffling. I never riffle shuffle because it bends the cards and I get cancer whenever I see others doing it. I mean, those are expensive cards that are being devalued by that kind of warping. I don't give a single crap about randomization as long as my cards are intact. And I swear I'll put up a fight if someone DARES to riffle shuffle my deck.

I achieve a pretty nice randomization by just distributing the deck in piles and mash shuffling those piles in each other until the deck is whole. I do it to ensure my opponent that I'm not cheating or stacking my deck or anything like that. There's no need to go this low. I'm not Alex Bertoncini.

I never riffle shuffle because it bends the cards and I get cancer whenever I see others doing it.

With sleeves, and some decent hands, it's possible to riffle perfectly without bending the cards. Just make sure you aren't pressing on the actual cardboard when you perform the riffle, and the cards are never actually bending. The sleeves provide enough distance and enough grip to allow this.

I don't trust anyone else to riffle my cards though. That would be like riffling through my cards. And god forbid someone bridges them afterwords. Now THAT bends cards.

Quote from "Asterisk" »

Mash shuffling can be done soo much faster than riffle shuffling.

Yes, but though it may not look it, mashes are nowhere near as efficient as riffles. Mashes of much larger chunks than riffles, so while it may be faster to perform individual mashes, it will take longer to achieve the same level of randomization as proper riffles.

Quote from "TheTrueNub" »

In theory anyone can do it with enough practice, and if it's done correctly (ie quickly, fluidly, without giving a chance to reveal that the shuffler is actually meticulously choosing exactly half the deck and weaving them exactly together), then it's extremely difficult to be noticed during the shuffling process--that's why it's a magic trick. The only simple way to detect a perfect Faro shuffle (or a pile shuffle cheater, or any other kind of shuffle cheater if he's good at what he does) is when the deed is already done, and that relies on the fact that you or the judge (more importantly the judge) knows what to look for. For example if you're Extirpate-ing your opponent and are searching his library and find that all 24 his lands are organized in 12 neat clumps of two lands each, ie a nigh-perfect mana weave, you have to recognize that that's extremely improbable from a statistical standpoint and call him out for it.

Hmm... Faro shuffling seems bad now.

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Firstly, a few notes:

1. The formulas and maths I'm going to be mentioning/linking to in this thread were developed by statisticians over the past forty or fifty years who were looking at decks of cards that featured numbers and suits. Though our MTG cards don't have numbers and suits, they still adhere to the same principles of mathematical randomness as standard cards, and the formulas still apply. That said, I'm going to be presenting a simplified explanation; I'll provide links to the original articles if you'd like to review my sources, but they're pretty complicated.

2. This thread discusses shuffling in the context of achieving

randomness--a complete lack of pattern or predictability--in your cards. The first question to ask yourself, though, is whether you really WANT your cards to be random. Sometimes people are complacent enough to have their decks only mostly random, since randomization can take a while to achieve. Other times people intentionally try toavoidrandomness for the purposes of stacking their deck; for more info on this read Mike Flores' blog post here and Tim Pillards's article here--they're from 2009 and 2003, respectively, but the concepts they're getting at are all still valid today. My goal in this thread is not to say whether youwantto be random--that's for you to decide--but rather only to clarify exactly how to achieve randomness if you wanted to.1. What is the optimal shuffling technique?Of the most common techniques able to be performed by hand (ie not a computer algorithm, but a method you can actually execute in real life),

the, usually by a very wide margin.riffle shuffleachieves mathematical randomness more quickly, consistently, and safely (i.e. with less room for cheating) than any other shuffleThe riffle shuffle is what most people use to shuffle a standard deck of 52 cards, including most casinos. You split your deck into two piles, approximately even, and then drop them one onto the other imperfectly, letting the cards cascade from your hands.

By imperfectly I mean that sometimes, as a natural consequence of the speed of the shuffle, your finger slips and you drop two cards or even three at once from one pile. In fact, according to the mathematical interpretation of a riffle used in a groundbreaking stats article described below, the probability of dropping a "clump" from one pile increases depending on the current size of that pile (due to the weight of the cards in your hand and other numerous factors). I underline the words "approximately" and "imperfectly" because, as I'll describe later, these imperfections are vitally important to this technique's efficacy.

Unfortunately yet understandably, many people are reluctant to riffle shuffle their decks because it involves bending and therefore potentially damaging the cards. This can be minimized by only riffling the edges together and then sliding the two halves into place; some people even go so far as to double-sleeve their cards to prevent frayed edges while edge-riffling. But some people will still hesitate to riffle even just the edges, especially if they're playing with really, really expensive cards as in Vintage.

Despite this, it really is the fastest, most effective way to shuffle your cards by hand that leaves minimal room for you or your opponents to cheat--whether intentionally or unintentionally (again, more on this below).

Accept no substitutes,not when you're shuffling your opponents' decks, not when you're shuffling your own deck.2. How many riffle shuffles does it take to achieve randomness?In 1992 three statisticians, Bayer, Aldous, and Diaconis, building on the work of many statisticians before them reaching back to the '50s, published this research paper which has since received tremendous praise and become wildly popular. They came up with an extremely complicated but highly accurate equation for the number of riffle shuffles it takes to randomize a deck of

ncards.I will post a simplified version below in the form of a mathematical limit adapted from this graph. This is an equation you can all punch into your handy-dandy TI-eightywhatever calculators you may or may not have saved from school:

As a limit with stipulation n -> infinity, this equation becomes increasingly true as the number of cards in the deck gets larger (approaches infinity). At lower values of

n, this simplification tends to overestimate, but never underestimate, the number of riffles by a little bit.For example, with 52 cards in the deck, the simplified formula will give you a result of 8.55. However using the full, complicated formula that's much too difficult to remember by heart or to store in your calculator's memory, the number of riffles required is actually around 7.

The good news is that the formula will only overestimate and only by a little bit, and it never hurts to riffle a deck one or two times more than the mathematically demanded number; the only argument against doing so is that it's wasted time, but if you do a riffle properly it shouldn't take you more than a few seconds anyway.

So if you're playing a 60-card Modern deck, riffle at least eight times (though you could probably get away with 7, still). If you're playing a 100-card EDH deck, riffle at least ten times. If you're playing a 500,000-card monster deck, riffle at least 28 times. And so forth.

3. If I shuffle half as many times as I'm supposed to, will the deck be half-randomized?The relationship between #shuffles and randomness is not linear but rather exponential. The way the math works out (see the graph I linked earlier), the randomness of your deck doesn't increase very much at all for the first few riffles. When you reach the magic number of riffles particular to your deck size, though, the randomness increases sharply and then tapers off...any further riffles still offer some additional randomness, but not enough to care about.

In other words, no, you can't riffle a 100-card EDH deck, which should be shuffled around 10 times, only 5 times and then say "OK, now my deck is half-randomized."

4. What are some bad shuffling methods to be avoided?Anything but the riffle. No seriously.The tl;dr of this section is that all the other common shuffling techniques either demand an unreasonably large number of shuffles to achieve randomness or otherwise never achieve randomness, though they might seem like it.4a. Pile shuffling, where you deal the cards into some number of piles and then stack them back up again. This is probably the most common shuffling technique in MTG alongside riffle shuffling because it carries significantly less potential to bend cards, and so I will discuss it more than the others.Firstly, have you ever seen that famous 21-card magic trick? There's a youtube video here showing how it's done and a wikipedia article here explaining it in words.

The trick involves the contestant selecting a card, any card, from a group of 21, and not showing it to the magician. The magician then pile shuffles face-up three times, each time asking the contestant to identify which pile his card is in--this pile is scooped up second out of the three each time. Then the magician deals ten cards, and the

eleventh card, in the very middle of the 21-card packet, is the contestant's card.Unlike other magic tricks, this one involves no sleight of hand. There is a mathematical pattern intrinsic to the pile shuffling that puts the contestant's card in the very middle of the deck every single time.

This is a perfect example of the truth that people often fail to realize:

pile shuffling is not random. Any form of pile shuffling. Not just "mana-weaving," any kind. It's actually one of theleastrandom ways to prepare your deck; in fact, as Flores and Pillards discuss in the articles I linked in one of my disclaimers (also here and here), pile shuffling allows savvy players to actually stack their deck!Some people on this thread have asserted that pile shuffling does indeed have its benefits, as it allows players to ensure the quality of their sleeves and to ensure their deck has the appropriate number of cards prior to a match. There is nothing heinous about using pile shuffling

in addition torandom methods for quality control only; the issue emerges when people use pile shufflinginstead ofrandom methods, as seen in the widespread fallacy that you can just pile shuffle a few times to effectively "emulate" riffles with less damage.If you're pile shuffling for quality control, you should not need to do it more than once, and you should not need to make more piles than necessary for you to count to 60 (or 40, or 100); in the Flores article he refers to a video describing a pile shuffling method using 5 or 7 piles because those are "mersenne prime numbers aka really random," and as Flores correctly notes, the whole thing is a load of BS.

Mersenne primes have nothing to do with randomness. There's a random number generator known as the "Mersenne twister" in which very large Mersenne prime numbers were chosen arbitrarily by the system's creator as the period length--but there is no connection to MTG and pile shuffling. No matter how many piles you make, your deck will not be random.

To be fair, pile shuffling can be made

slightlymore random if you drop cards into random piles instead sequential ones...like if you have three piles 1, 2, 3, instead of dropping cards 1-2-3-1-2-3, you should drop them 1-3-2-3-1-2, or what have you. But even then this technique is only marginally random, nowhere near as effective as a riffle shuffle, and it's replete with cheating opportunities to boot.4b. Overhand shuffling, aka "strip shuffling"where you hold the deck in your right hand and take little packets of cards off the top and put them into your left hand so your deck order is reversed in clumps.As discussed in this article (Pemantle 1989), overhand shuffling is an extremely inefficient randomization strategy. Depending on variations in technique it would take anywhere from 1000 to 3000 overhand shuffles(!!) to randomize a 52-card deck to the same extent as riffle shuffling that deck only 7 times.

4c. "Perfect" shuffling, aka Faro shuffling(wikipedia here), where the deck is split into exactly two halves and the cards are laid exactly one upon another.Above I mentioned that the imperfections of a good, quick riffle shuffle are what makes it so effective. This is why: a perfect riffle shuffle simply introduces a nonrandom mathematical sequence, sort of like a more complex variation of pile shuffling. There are two kinds: in-shuffling, where the top card of the deck becomes the second-top; and out-shuffling, where the top card of the deck remains on top. Long story short, Faro shuffling is far from random; magicians use it all the time, as with pile shuffling, to control the placement of certain cards within the deck. Indeed, eight out-shuffles performed sequentially will leave the deck in the same order as it was initially, as demonstrated here in this youtube video.

5. What about mash shuffling?Mash shuffling, where the deck is split into approximately halves and then "mashed" together, can go either way depending on how skillful you are at mashing cards. If your mashes are tight enough to be almost one-card-upon-another but with some minor imperfections, then you essentially replicate a riffle shuffle. However, this is not the easiest to achieve; if your mashes are generally hasty such that your clumps are rather large, your results will be nowhere near random. Additionally, as shown in the youtube video above, the mash technique can be used to perform the Faro shuffle, too--if you are an expert card handler such that you can reliably cut your deck exactly in half and mash the cards exactly one atop another, then you have just performed a Faro shuffle...and if you do that seven more times, then your deck's order has been totally preserved as if you never shuffled at all. The Faro can be performed with a riffle, but it's significantly harder to pull off because the performer must manipulate, in addition to everything else, the precise slipping of his thumbs from the edges of the cards to only drop one at a time.

In order to set a goal for yourself in your mash shuffling, I strongly recommend riffling either a 52-card 2-thru-ace deck or, preferably, a 60-card Magic deck (so you can get used to the unique feel of Magic cards as you riffle them; they're slightly thicker and harder to control than 52-cards). If you're sensitive about bending your valuable Magic cards, then make a 60-card stack of your two-cent commons, pretend they're a deck, and riffle them. As you split and riffle, observe the manner in which the cards naturally cascade from your hands; when the riffle is done, before you slide the cards together, hold the deck sideways and note the size of the "clumps" from each split (as shown in the image to the right). Almost all the clumps should be of 1 or 2 cards, with an occasional 3 and rarely 4. Treat that distribution you see as your "gold standard;" you want your mashes to be as close to riffles as possible, so you want the distribution of cards during your mash to look exactly the same as what you observe when you riffle.

In other words, mash shuffling

canbe the same as riffle shuffling, but you have to consciously endeavor to replicate the mechanism of a riffle as closely as possible. If you mash in large clumps, or if you only mash half your deck together at a time, or some other practice, you are only straying further from the mechanics of a riffle, and more importantly you are straying further from true randomness. Additionally mash shuffling leaves more room for cheating than riffle shuffling, so that's something to watch out for.In summary, mashing is, at best, a more inconsistent way to riffle, and at worst, nothing like a riffle at all. The most consistent way to randomize a deck--yours or your opponents--is to riffle shuffle.

6. Chaos vs RandomnessThis section was added to the FAQ by request.

As Fnord points out below, a deck of 60 cards that has been pile-shuffled three times and then riffled once might look, to the naked eye, similar to decks of 60 that had been riffled a full 8 times. More generally, people tend to make the assumption that pile shuffling

israndom simply because itlooksrandom. This is one of the great cognitive fallacies of mankind; in fact, what looks random at first glance may be quite far from the statistical definition of randomness. As TheLizard proves on page 4 of this thread (with a conceptual explanation by myself at the bottom of this page), the first series of numbers Fnord lists is in fact significantly less random than the others. Unfortunately, when you start talking about how humans interpret things and cognitive fallacies, you get into psychology, which is beyond the scope of this FAQ.Even for a deck that seems random, during gameplay, over a long period of time (ie, after many, many hands have been dealt), the statistics dictate (via the Law of Large Numbers) that the difference between simply a "chaotic" method and a truly "random" method must become increasingly obvious. To provide a real-life example, consider the MTGO shuffler, which over the last decade has become infamous for being purportedly rigged or otherwise broken because of how often people get mana screwed.

The fact of the matter, as Wizards and others have repeatedly reminded us, is that the MTGO shuffler uses a highly random computer algorithm, more random than what people are used to with their real-life shuffling practices, and so over time people observed this difference and blamed the computer (because of course, between the human and the computer, the computer's the one doing math wrong :rolleyes:). In fact, true randomness involves some clumping and has some chance of mana screw. Shuffling more times will only more consistently reach the predicted number of clumps, not reduce the number of clumps. The only way to reduce the number of clumps is to

move awayfrom true randomness by fixing your deck, such as through mana weaving.In Fnord's example below, as determined by TheLizard, the first deck, which was riffled only twice, contains a greater number of sequences of lands and spells than the other four decks which were riffled 8 times; that is to say, the big "clumps" of lands and spells that can lead to mana screw have been broken up further than they usually would, as if the deck had been mana-woven. Let me say that again: the deck that had been shuffled

lesswas more akin to a deck that had been mana woven (stacked);more shufflingactually leads to ahigherchance of mana screw than less shuffling. In the long run this has a poignant and noticeable effect on gameplay.Once again, whether you

wantto achieve true randomness is your decision. Keep in mind, however, thatthe rules call for a randomized deck, so if you stray from randomness through insufficient shuffling, intentionally or unintentionally, and a player or a judge notices that your deck is looking less-than-random,you can be labeled as a cheater for it in competitive play:Point being, in the long run, the chaos with which we have become complacent as calling "randomness" does indeed distinguish itself from true randomness, even during gameplay. As online algorithm shufflers demonstrate, people's eyes might deceive them into telling them that certain techniques create randomness, but when they try the real randomized deck it plays differently in the long run.

7. ConclusionIn summary, riffle shuffling (and to an extent mash shuffling, ONLY when it is performed identically to a riffle shuffle) is the most effective and the fastest way to achieve true, mathematical randomness. Other common shuffling techniques like pile shuffling might seem random, but they generally fail to truly randomize your deck, and this difference becomes noticeable in the long run. More importantly, non-riffle methods provide savvy shufflers with opportunities to stack their decks and violate the rules.

I hope the material I presented in this brief FAQ-primer-thingy answers everyone's questions regarding shuffling. Hopefully people will be able to refer to it in the future.

I also encourage feedback of any kind. If I missed any other big questions, or if there's something here you disagree with, please don't hesitate to let me know. If you found this useful, please say thanks (protip: there's a button for that over to the left). I appreciate your support, too.

Legacy: GWR Enchantress <--That's my banner! (lol tinypic removed it)Casual: WB[[Primer]]Clerics Tribal; BU AffinityEDH: ...U[[Primer]]Arcum Dagsson; BG Legal Stax; B Illegal StaxProxy: .WX TriniStaxOther stuff:[[Official]]Shuffling, Truth + MathsIF THERE WERE NO GOD - IT WOULD BE NECESSARY TO INVENT HIMUBWPile shuffling will never hurt your deck, and it is generally a good idea to do it just because it is also a fast way to count your deck (you don't need to count the cards, just remember what pile the last card should go to) and check for sleave damage. However, if you riffle 8+ times any clumps will be gone (and most likely replaced with new ones) anyway.

OP: I think you forgot the most common way to shuffle a sleaved Magic deck: Mash shuffling.

Take 1/4-1/2 of your deck in one hand, the other part in the other hand. Then mash the two piles together.

Good part: Achives the same result as riffle shuffling without risk of bending the cards.

Bad part: It takes more mashes to get a randomized deck. The shuffler can also easier (compared to riffle) control the top or bottom cards and make sure they are never randomzied.

The whole point of the groundbreaking 1992 study on card shuffling was that if you riffle shuffle a certain number of times, the cards are, for all intents and purposes, fully randomized.

Whatever randomization that you're accomplishing by pile shuffling can be achieved more quickly, more soundly, and with less opportunity for cheating simply by riffle shuffling more times instead.

As an aside, a fully-randomized deck will have clumps sometimes. If you never got mana-screwed or mana-flooded, it wouldn't be random, it'd be fixed! The point of the mulligan system is to filter through the clumps when they come up.

If your mashes are tight enough to be the equivalent of a riffle shuffle (like mostly one-atop-the-other, but some "imperfections," clumps of two or three) but at the same time not quite so tight as to be a perfect Faro shuffle (in the youtube link I listed you can see the guy uses mash shuffling to pull off a faro shuffle), then the results should be the same as a riffle shuffle. But the thing is that you have to make a conscious effort to make it like a riffle.

The safest thing to do is just riffle it up. If your cards are sleeved and slippery and hard to handle, you can still just riffle the edges together while they're down on the table (like they do at casinos) and then slide the cards together.

I updated the OP to say this.

Legacy: GWR Enchantress <--That's my banner! (lol tinypic removed it)Casual: WB[[Primer]]Clerics Tribal; BU AffinityEDH: ...U[[Primer]]Arcum Dagsson; BG Legal Stax; B Illegal StaxProxy: .WX TriniStaxOther stuff:[[Official]]Shuffling, Truth + MathsIF THERE WERE NO GOD - IT WOULD BE NECESSARY TO INVENT HIMUBWWhilst not a useful tool for achieving randomness, the pile shuffle can be used as a precursor to actual shuffling in order to break up any large clumps of the same type of card that you may be aware of. It then attains marginal usefulness in reducing the workload required of your actual shuffling.

The other thing is that I noticed a lack of commentary on what (from experience) I would consider one of the most common shuffling methods amongst tcg players: the push shuffle.

Performed by holding the deck at ninety degrees in your left hand, separating approximately half the deck into your right hand, and pushing one half down into the other.

It has obvious similarities to the riffle shuffle, plus benefits and drawbacks:..

It doesn't bend the cards; unlike a riffle shuffle, the push shuffle allows gravity to spread the cards in your lower hand slightly, meaning that you don't have to bend the cards when you push the two halves together.

It's rather easy to do with sleeved cards, but extremely difficult when the cards are unsleeved.

It can be possible for you or your opponent to catch glimpses of some of the card faces during the shuffle if you are not mindful of the angle of the cards.

The above considerations make the push shuffle entirely unsuitable for classical card games, but makes the push shuffle a viable alternative to the riffle for tcg's where card sleeves are prevailant and the value of the cards can be significant enough to make riffling them a distasteful option.

Edit: 'nathed several times over

If you are riffling sufficiently to randomize the deck fully, the pile shuffle is moot regardless of your reason for doing it. If the pile shuffle does make a difference, then not only are you not randomizing sufficiently, but doing the pile shuffle could be construed as a form of cheating.

As Flores points out, in the time it would take you to pile shuffle, you could riffle shuffle many more times, since riffle shuffling is such a quick maneuver. That would separate out the clumps in a more random and less cheating-inclined way.

I've heard it called mash shuffling and other people on this thread used that name. I added a section for it in the main post. It can go both ways.

It's OK. I appreciate any feedback.

Legacy: GWR Enchantress <--That's my banner! (lol tinypic removed it)Casual: WB[[Primer]]Clerics Tribal; BU AffinityEDH: ...U[[Primer]]Arcum Dagsson; BG Legal Stax; B Illegal StaxProxy: .WX TriniStaxOther stuff:[[Official]]Shuffling, Truth + MathsIF THERE WERE NO GOD - IT WOULD BE NECESSARY TO INVENT HIMUBWExample: Here are 5 shuffled decks. 4 were shuffled using 8 riffle shuffles. The fifth was pile shuffled 3 times and then riffled once. Can you tell which is the pile-shuffled one? (The numbers are the initial order the card was in.)

[46, 13, 8, 15, 52, 19, 38, 40, 7, 51, 59, 24, 3, 26, 11, 37, 47, 14, 22, 58, 54, 25, 33, 6, 44, 10, 42, 29, 55, 4, 57, 30, 36, 9, 32, 53, 17, 43, 28, 20, 60, 39, 5, 31, 12, 50, 16, 48, 27, 35, 2, 21, 23, 49, 34, 1, 45, 18, 56, 41]

[2, 15, 25, 36, 17, 30, 21, 22, 13, 39, 59, 53, 19, 1, 29, 6, 26, 35, 5, 51, 27, 12, 54, 60, 31, 49, 48, 57, 45, 16, 52, 40, 32, 50, 55, 20, 44, 38, 8, 24, 4, 42, 7, 9, 56, 18, 28, 58, 33, 14, 10, 34, 37, 47, 43, 3, 41, 11, 23, 46]

[53, 55, 30, 48, 46, 33, 29, 54, 50, 44, 6, 41, 16, 18, 38, 28, 10, 5, 36, 20, 60, 45, 1, 32, 23, 26, 24, 8, 27, 57, 49, 25, 12, 39, 40, 34, 3, 42, 11, 15, 37, 22, 21, 52, 19, 17, 43, 2, 47, 9, 51, 59, 35, 4, 58, 13, 14, 31, 7, 56]

[59, 29, 25, 48, 31, 19, 33, 60, 22, 4, 17, 53, 42, 26, 54, 35, 24, 51, 37, 30, 6, 13, 38, 39, 34, 40, 57, 15, 10, 44, 2, 20, 56, 58, 45, 23, 36, 5, 9, 55, 32, 8, 28, 12, 49, 43, 41, 18, 21, 16, 46, 50, 52, 27, 14, 3, 7, 47, 1, 11]

[42, 48, 54, 18, 6, 43, 8, 49, 44, 52, 56, 31, 38, 30, 35, 51, 4, 16, 40, 14, 17, 12, 13, 9, 25, 37, 15, 26, 20, 1, 21, 22, 7, 19, 57, 45, 2, 29, 3, 46, 59, 28, 58, 60, 5, 55, 50, 33, 41, 24, 39, 27, 23, 34, 47, 11, 10, 32, 36, 53]

Practice for

Khans of TarkirLimited:Draft: (#1) (#2) (#3) (#4) (#5)

Let's see... I think it's the fourth pile. How did I do?

You do have a point. Those all look sufficiently random.

Mabye you should make a section on Chaotic vs Random TheTrueNub. Pile shuffling being chaotic looks random to the untrained eye (most people's eyes) but is controllable, which is what enables it to be so useful for cheating.

Lycanthropy Awareness Day.Hoping for a cure, or at least an outbreak.

Level 1 Judge (yay)

Not well. The first pile was the pile-shuffled one.

I don't deny that pile shuffling is useful for cheating, which is why you should be suspicious if your opponent only pile-shuffles. But for the honest player just looking to randomize your deck, my point is that it isn't as bad as you make it out to be. A deck pile-shuffled a few times, plus a single riffle shuffle, is functionally good enough.

Practice for

Khans of TarkirLimited:Draft: (#1) (#2) (#3) (#4) (#5)

Dagnabbit! You didn't have to tell me the right answer yet. I would have gotten it by the fifth guess.

True. Without the actual intent to cheat, a chaotic mix is functionally no different from an actual random one.

However, as a storm player, I enjoy my time between rounds, and a lot of pile shuffling would drive me crazy. I can shuffle and cut in under 60 seconds, so everyone else should too.

Lycanthropy Awareness Day.Hoping for a cure, or at least an outbreak.

Level 1 Judge (yay)

Very interesting point. They all look really well shuffled if you ask me.

LONG LIVE LEGACYFirstly and foremostly, you're approaching the issue from the wrong direction. Like I said in my disclaimer and like Flores hints in his article on cheating, how random the deck "needs to be" or "should be" is up to you and your particular goals. My advice on shuffling, which is itself just a condensation into readable, jargon-less form of the work done by dozens of statisticians over the past several decades, is if you're looking to reach the

mathematical definition of randomness, true randomness, the uniform distribution.That said, I'll still provide a conceptual way of solving your problem just so I can put my money where my mouth is--though unfortunately I won't be able to provide an exact answer.

RETROSPECTIVE EDIT:TheLizard has carried out in full on page 4 of this thread the math I explained here only conceptually, for anyone who's reading this post.He determined that the first deck is improperly randomized, as it possesses a significantly greater number of sequences (that is, a greater "weaving" of mana and spells) than what should have resulted from eight shuffles according to the probability.In order to "solve" (or, in this case, take an educated guess at, since when we're trying to extrapolate general truths from specific samples, an educated guess is the best we can do), I'd count the number of rising sequences in each of those five decks and then plug them into this equation:

In your case, n = #cards = 60, m = #riffles = 8. I'd have to count the rising sequences by hand unless I found some program that did it for me, which is an excruciatingly tedious and error-prone process, so I'll keep it all conceptual.

After counting the rising sequences, I'd then find which of those configurations had the least probability to have as many rising sequences as it does (ie the pi with the lowest chance) assuming 8 shuffles, and then I'd have to assume that the least probability of receiving such a result after shuffling 8 times corresponds to the most probability of not actually having been shuffled 8 times.

So like let's say, assuming 8 shuffles, the first four configurations are all 80% likely, and the last one is only 10% likely (I just made those numbers up). Then my best answer would be that the last one was the most likely NOT to have been shuffled 8 times. Even then, since we're using samples, I can only give you the "most likely" answer, not the actual answer.

It's also worth noting that the things Bayer and Diaconis were tallying when they were determining how many shuffles equaled true randomness, that is, rising sequences, are directly applicable to MTG.

In a 52-card deck rising sequences are the number of ascending orders, so for example the set {A, 2, 3, 6, 7, 8, 9, 4, 5} has two rising sequences, {A, 2, 3, 4, 5} and {6, 7, 8, 9}. As you riffle more and more, the number of rising sequences increases (ie the sequences become broken up) until the deck becomes fully randomized, ie whatever sequences still remain mind as well be the product of chance. It's this same "breaking up" that we refer to when we talk about "breaking up clumps of land" or "breaking up clumps of nonland" in our own decks, and it's already been proven by people smarter and more accomplished than I that the riffle shuffle is the fastest and most effective way to do it.

This is exactly the kind of misconception I'm trying to get rid of in this thread. You can't just look at it by eye and come to a conclusion in the same way you can't just look at pile shuffling by eye and say "yeah, that looks about random" or look at a Faro shuffle by eye and say "yeah, that looks about random." There's math behind it.

Not necessarily. You know how people always complain about the MTGO shuffler being rigged or horrible or biased? That's because the MTGO shuffler is an ideal shuffler. It creates "true randomness" in the mathematical sense of the term, and in the long run, over many, many hand deals, people notice the difference between their regular, "chaotic" practices and the actual random ones.

Legacy: GWR Enchantress <--That's my banner! (lol tinypic removed it)Casual: WB[[Primer]]Clerics Tribal; BU AffinityEDH: ...U[[Primer]]Arcum Dagsson; BG Legal Stax; B Illegal StaxProxy: .WX TriniStaxOther stuff:[[Official]]Shuffling, Truth + MathsIF THERE WERE NO GOD - IT WOULD BE NECESSARY TO INVENT HIMUBWYou've already said it was this one, so I looked for a pattern in it. I see one where a card above 40 hits every 2 or 3 cards with few a exceptions. If those were your lands, that's a pretty good weave. I didn't look at the others.

If you go looking for patterns you will find them. They are the faces in the trees.

This really is going nowhere fast though. Back on topic:

Faro shuffling! How many people can do it, and considering it's likeness to mesh shuffling, how likely would it be to go unnoticed? I riffle shuffle, because mesh shuffling damages my sleeves, and I have the dexterity to riffle without bending cards, but I am confident I could do a Faro with enough practice. Sleeves make it almost too easy. I'm sure a deck could be easily stacked with proper Faros.

Lycanthropy Awareness Day.Hoping for a cure, or at least an outbreak.

Level 1 Judge (yay)

(and I don't mean mana weaving your deck or anything, just a set routine you go through where after you're done piling/mashing/riffling you feel like you've achieved a good land to spell balance and I'm sure you know what I'm saying so don't bother calling me a cheater or anything bc I just won't respond - I've seen how these threads end up)

And that's unambiguously cheating.

In theory anyone can do it with enough practice, and if it's done correctly (ie quickly, fluidly, without giving a chance to reveal that the shuffler is actually meticulously choosing exactly half the deck and weaving them exactly together), then it's extremely difficult to be noticed during the shuffling process--that's why it's a magic trick. The only simple way to detect a perfect Faro shuffle (or a pile shuffle cheater, or any other kind of shuffle cheater if he's good at what he does) is when the deed is already done, and that relies on the fact that you or the judge (more importantly the judge) knows what to look for. For example if you're Extirpate-ing your opponent and are searching his library and find that all 24 his lands are organized in 12 neat clumps of two lands each, ie a nigh-perfect mana weave, you have to recognize that that's extremely improbable from a statistical standpoint and call him out for it.

Legacy: GWR Enchantress <--That's my banner! (lol tinypic removed it)Casual: WB[[Primer]]Clerics Tribal; BU AffinityEDH: ...U[[Primer]]Arcum Dagsson; BG Legal Stax; B Illegal StaxProxy: .WX TriniStaxOther stuff:[[Official]]Shuffling, Truth + MathsIF THERE WERE NO GOD - IT WOULD BE NECESSARY TO INVENT HIMUBWI mash shuffle in such a way that I always make sure that ~10 cards from the top of the bottom pile end up on the top pile. In this way, I have an 'escalator effect' That moves throughout the deck ensuring that no area goes insufficiently shuffled for whatever reason. I tend to shuffle 15-30 times before a game, so it's pretty random (for the past ten years at least).

I consider myself pretty good at shuffling, but I find it hard to imagine someone Faro shuffling via-mash without looking awfully suspicious. They'd have to be a djinn or something, at that point just give them the game.

*come to think of it, I emphasize different portions of the deck at different times, but I change it up to make sure I'm neglecting the top and bottom halves of the deck. I'm not trying to cheat with what I'm doing, but more sufficiently randomize. I consider 7 riffles to be abhorantly few shuffles for a magic game. I can get in 15-30 shuffles before a game/after shuffle effects.

**I average 29 mashes in 60 seconds. I imagine I could probably get 14 riffles in...I'm not great at riffling...but if all my mashes are as good as one half riffle, I should be all set.

Legacy:

RWBG Goblins

RRR Burn

WBU Affinity

UBR Sac-Land Tendrils!

BBBPox

Next possible deck: D&T, but that just wouldn't be right.

Modern: R Goblins (work in progress)

Standard: I only care about standard when Goblins is a deck.

Limited: I only care about limited when Goblins are in the set.

Pauper:

RGoblins

URCloudpost

other decks

Goblins.

I won't call you a cheater; I said in my disclaimer that you may want, for whatever reason, your deck to be more or less random, and that's your choice. But I will stand by my assertion that some mana screw/mana flood is actually part of randomness. If you're going to mana weave, either by pile shuffling or otherwise, I simply ask that you do not perpetuate the misconception that that's what randomness is by saying your deck is "randomized" or "sufficiently randomized" or "mind as well be randomized," because unless you riffle shuffle 7-8 times (60 cards) or 9-10 times (100 cards), your deck is probably not very random at all.

Noted. Now there's a handy blurb for the lazy people.

Legacy: GWR Enchantress <--That's my banner! (lol tinypic removed it)Casual: WB[[Primer]]Clerics Tribal; BU AffinityEDH: ...U[[Primer]]Arcum Dagsson; BG Legal Stax; B Illegal StaxProxy: .WX TriniStaxOther stuff:[[Official]]Shuffling, Truth + MathsIF THERE WERE NO GOD - IT WOULD BE NECESSARY TO INVENT HIMUBWexpensivecards that are being devalued by that kind of warping. I don't give a single crap about randomization as long as my cards are intact. And I swear I'll put up a fight if someone DARES to riffle shuffle my deck.I achieve a pretty nice randomization by just distributing the deck in piles and mash shuffling those piles in each other until the deck is whole. I do it to ensure my opponent that I'm not cheating or stacking my deck or anything like that. There's no need to go this low. I'm not Alex Bertoncini.

StandardUB Sacrifice

RWU Miracle Control

ModernUB Tezzerator

Clint Cearley ♥

With sleeves, and some decent hands, it's possible to riffle perfectly without bending the cards. Just make sure you aren't pressing on the actual cardboard when you perform the riffle, and the cards are never actually bending. The sleeves provide enough distance and enough grip to allow this.

I don't trust anyone else to riffle my cards though. That would be like riffling through my cards. And god forbid someone bridges them afterwords. Now THAT bends cards.

Yes, but though it may not look it, mashes are nowhere near as efficient as riffles. Mashes of much larger chunks than riffles, so while it may be faster to perform individual mashes, it will take longer to achieve the same level of randomization as proper riffles.

Hmm... Faro shuffling seems bad now.

Lycanthropy Awareness Day.Hoping for a cure, or at least an outbreak.

Level 1 Judge (yay)