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 fromBut won't riffle shuffling DAMAGE MY CARDS???
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 fromThere'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.
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
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.
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I don't have any solid evidence that it's the most popular or not, but I won't deny that it's definitely one of the easier formats to play, and generally the most rewarding to skilled limited players. Obviously it's in large numbers because you don't need a deck to play, you can just show up and play, so it's easier to go to these events if you're a player of any variety.
It's just, I personally don't liked sealed so I'm on the complaining side.
47.8%* is close enough to "half" to call it "half" for the sake of complaining.
*It's 22 sealed events out of 46, not 21.
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This feels like a Yu-Gi-Oh card. Look at all those stars. Lets sacrifice two creatures to summon it!
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You're forgetting about...
• The card has a standard Magic back or is a double-faced card.
You're altering a lot of things just so you can use all those gold bordered cards you own. If the price isn't going to drop (and we agree that it'll likely cause them to rise) then the only reason to do this is so that there is more availability of cards to use, namely those of which you already own and just wish you could use.
If you get to use your gold bordered cards, why can't I use those blank cards that come in the decks, and just write what I want to on them? Why can't I use cards I printed onto REAL magic cards that I blanked myself? Why can't I use proxies I 100% made my self? If you can change all the rules about what can and cannot be used, why can't I change them too to benefit myself as well?
Are you starting to see the slope?
And what about players who don't want to use sleeves? Sure, they'll probably need to exchange the cards in their deck with others as the event goes on, but nothing in the rules is preventing them so long as they have a magic back, and all cards that are playable in events have magic backs. The almost didn't however! A history lesson would teach us that back with Arabian Nights they almost changed the backs to a lavender color scheme.
Ultimately if the reason you want this is to introduce about less than 1% more of those cards into circulation, for a discount of $20 off $180, then why are would we do this again?
Yeah, but that's also magical christmas land thinking. It's not a bad idea, but lets be real here, that's about as likely to happen as them abolishing the reserve list and mass printing the real things in modern border. I'm not saying it won't ever happen, but for that to happen the way you want it seems like something that wouldn't happen for years.
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Don't forget that those life gaining triggers will each resolve one at a time as well and that they'll buff his cats through thrun, buffing his life with the next trigger a little more, then buffing his cats even more, then buffing his life more so then the last, and then buffing his cats yet again even more so (you get the idea).
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I think there are a few cards worthy of an FNM or GP promo, but those typically are dubbed based on their median current popularity.
Maybe Jund Charm
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So in situation one you would assume they were totally guilty, but in situation two you would be willing to accept that perhaps your fakes are in your deck by no fault of your own? I just find that a little comical.
Aside from that, this looks like it will be either a game lose for a deck/decklist problem (with a potential fix) if the judges determine it was an unfortunate mistake, or a disqualification for cheating if the judges feel this was intentional. If you found yourself in this pickle I would urge you to call a judge about it. If they find out later that you were aware of the mistake and hid it from them, things could get a lot worse for you.
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"*****".