Order and Chaos: Shuffling in Magic
Michael Thicke

Rosencrantz and Guildenstern, summoned by the king of Denmark, are eating dinner by the side of the trail. Earlier in the day, Rosencrantz found a coin on the path, and has been flipping it since. It has landed heads each time, much to Guildenstern's consternation.


Rosencrantz, (flipping a coin): Heads... heads... heads... heads... heads... heads... heads...

Guildenstern: I think I have it: time has stopped dead. The single experience of one coin, being spun once, has been repeated...

Rosencrantz (counting): ...156...

Guildenstern: One hundred and fifty six times; on the whole doubtful. Or, a spectacular indication of the principle that each individual coin, spun individually, is as likely to come down heads as tails, and therefore, should cause no surprise each individual time it does.



Land... land... land... land... land. We've all had this happen: time stopped, the single, agonizing experience of drawing one land, repeated over and over again. Sometimes everyone feels like Guildenstern, stuck in a loop of time that seems to defy the laws of chance. What makes it worse is that people's misunderstandings about probability are deeply ingrained.

To get at the problem, I'd like to start with a quiz. These two images are the same size, and both contain 1500 dots. Your challenge is to decide which shows a random distribution of dots, and which has order (Bonus question: what is the rule underlying the ordered picture?).



The two images above model different natural distributions. Image A is supposed to look like the night sky – the dots are stars, randomly distributed. Image B models a colony of glowworms on a cave roof, which looks much like a star-filled sky, but have an underlying organization.

The difference between the images stems from the glowworm's ecology: it cleans the surrounding area of prey, creating a “zone of inhibition” that other glowworms will not enter. This spreads the glowworms evenly over the cave surface.



Stars, on the other hand, are distributed randomly throughout the sky (excepting the Milky Way). There is essentially no interaction between stars that appear nearby to each other, and stars that appear proximate are usually not really close at all.

When most people look at these two images, they pick the glowworms as random, and the stars as ordered. We seem to have an innate tendency to look for patterns in chaos, and expect uniformity from chance. When people look at the sky, they see constellations; there is order: lines, squares, belts, serpents, hunters. When we look through a deck and see land clumps, and spells stuck together, we imagine causes – I picked up all my land together after the game, I didn't shuffle well enough, I didn't pile shuffle, these sleeves are sticky. Wrong on both counts.

This must happen to you all the time: you win a game where your opponent draws poorly. Afterwards, they thumb through their deck, find a land clump or a particular series of cards, and offer it as proof that you got lucky. And then you nod, commiserate on their bad luck, and sign the match slip apologetically. But really, all they have demonstrated is that they didn't cheat.

The essential tension of shuffling is between the wishes of the player shuffling the deck and the goal of the shuffling process. The player wishes their lands and spells to be distributed evenly, while shuffling aims to randomize the order of the cards. The images above illustrate the difference, but it might be easier to see with a one-dimensional picture, as our decks are really one-dimensional in nature.



This is just a linear version of the stars and worms images – each black mark is a land, and the white spaces are spells. Each line contains 60 spots, with 24 land marks. The top line is a dream – no clumps of land, and very few places where a mulligan would be necessary. The bottom is reality – clumps of land, places where there is almost no land, and places where there are few spells. Reality is ugly, uneven and risky, while the dream is safe and dependable.

I've been talking a lot about random distributions versus even distributions. What is random and what constitutes randomization? For our purposes, there are two criteria:

1) The position of each card in the deck does not depend on the position of any other card in the deck.
2) It is impossible to predict anything about the configuration of the deck after randomization from the configuration of the deck before randomization.

Let's try an experiment. Get a deck of cards, set aside everything except the A – 10 of diamonds and the A – 10 of clubs. Now sort them so the order goes: Ad -> 10d, Ac -> 10c., as shown below:



Then pile shuffle the deck three times, into four piles in typewriter fashion rather that back and forth: if the piles are numbered 1 – 4, put a card in pile 1, then 2, 3, 4, and then 1 again (as is shown in the illustration). Between each shuffle, pick up the piles so that pile 1 is on top, followed by pile 2, and so on.



After the three iterations, examine the order of the cards. If everything went properly, they should be completely reversed. The lesson here is that pile shuffling violates criteria #2; in fact, it accomplishes the opposite – you can predict exactly the configuration of the deck after shuffling from the configuration before. The wheels in your head should probably be whirring right now, but we'll get to the cheating implications later. For now just recognize that pile shuffling does not accomplish randomization.

So is there a legitimate reason to pile shuffle your deck? Sort of. In tournaments I pile shuffle my deck exactly once before each game, to count the number of cards in my deck. I do the same to my opponent's if they don't pile shuffle in front of me to let me count along. Beyond that, pile shuffling is usually an admission that you are rotten at riffle-shuffling – you have no faith that cards beside each other won't remain that way through multiple riffles.

This is somewhat understandable with sleeved decks. Sweat often causes cards to stick together, and since you pick your lands and graveyards up more or less in a pile after each game, the stuck together spells will often be of the same type. I personally always use textured sleeves in tournaments because they (among other things that I'll get to in a future article) don't stick together.

Mana weaving seems to have a similar motovation. Rune Horvik, a respected Judge and NetRep for the DCIJUDGE-L mailing list, wrote this about mana weaving:



There is NO legitimate reason for a player to mana weave before shuffling, other than to “improve his draw”, which is not supposed to happen. Random is random, it should not be possible to reliably determine which card (type) the upcoming card is. If mana weaving improves your draw you don't randomize enough (DCIJUDGE-L, March 19, 2003).



This is a clear condemnation of the practice, and I think it speaks against pile-shuffling indirectly as well, as the same logic applies. None of this really matters, if people riffle shuffle sufficiently afterwards, but (as this series will eventually get to) it facilitates stalling. The rules allow three minutes between games to sideboard and shuffle, so there really isn't time to do things that don't accomplish randomization. Although the three-minute rule isn't enforced very often at the lower levels, it should be, because time constraints are even more severe, with 50 minute rounds in most PTQs instead of the 60 minute rounds on the Pro Tour.

Riffle shuffling violates criteria #2 as well. For instance, for any given shuffle, you know that the top card of the deck before shuffling will still be near the top after shuffling. In fact, if you can riffle perfectly – always have the two halves of the deck flawlessly interleaved after the shuffle – you can get the same cycling of configurations as in pile shuffling. But nobody riffles perfectly: it is almost impossible to never have two cards from the same half fall consecutively (I spent days trying to master this; trust me, it was futile). The general guideline is that seven riffles acheives sufficient randomization, and most Magic players shuffle more than this. I've seen Magic articles that go into the detailed math of shuffling, but in my opinion nobody really gains anything useful from those discussions. The more you shuffle, the closer your deck will be to truly randomized; that's all you really need to know.

Since shuffling well isn't necessarily in the player's best interest, it is always a good idea to shuffle your opponent's deck. As Magic is a collectible as well as a strategy game, some people are reluctant to hand their property over to a stranger for manhandling. I'm not very confident in the effectiveness of the “separate and mash together” method of shuffling, so I like to riffle my opponents' decks. I've had a few players (usually at pre-release level events) get angry at me for this. Calling a judge to shuffle for you usually solves this problem.

Since Casey McCarrel was caught stacking Brian Hegstad's deck at US Nationals 2001, players have become less trusting with their decks in other's hands. McCarrel side-shuffled his opponents' decks, repeatedly glancing at the bottom card, and moving it to the top if it was non-land (or the reverse). Since players can only cut their decks after their opponent shuffles, they would either draw very few lands, or way too many. This incident has made it very important to obviously look away from your opponent's deck while you are shuffling it.

The other reason to look away from your opponent's deck is to make sure that in the event you make a mistake and drop some of his/her cards, you don't get an automatic game loss. If you spill their cards and obviously don't see them, you will probably walk away with a warning, but if you do the punishment could be more severe.

Early on in the Pro Tour's history, deck stacking was probably the most well known form of cheating. Players often didn't shuffle their opponents' decks, so stacking your own deck was very profitable. Mike Long, in particular, was often accused of this.

A friend of mine stacked his deck when he played on the Junior Pro Tour (this was a long time ago). I asked him if it involved something like the pile shuffling method I outlined above, and he said it was similar, so let's go into some more detail about how someone would stack their deck by pile shuffling. As a first try, you could simply order your deck as you like, figure out how many times you need to repeat the pile shuffle to return to the normal configuration, and only pile shuffle when you sit down.

The problems here are clear: people are automatically suspicious if you never riffle, and you have to pile shuffle several times to loop through all the configurations. So let's get slightly more sophisticated.

1) Put all your land on top of your spells.
2) Before sitting down, pile shuffle enough times so that you are only a couple iterations from returning to the configuration with all your lands on top of your spells.
3) After sitting down, complete the pile shuffles.
4) Riffle shuffle once – this allows you to spread the land through your deck.
5) Do some lazy side shuffles to make it look like you're randomizing.

So now we have a workable, if crude, method of deck stacking. And it is very easy. Hopefully this makes it clear that you should always shuffle your opponent's deck thoroughly – because it might appear that they are shuffling legitimately when they really aren't. If you shuffle everyone's deck, then no particular person should be offended. Some people think they are being sportsmanlike by just tapping their opponent's deck. They aren't, they're just setting themselves up to be duped.

The purpose of this has been to make you wary, but I should make it clear that most people are honest, and the Pro Tour has, from all reports, been pretty much cleansed of this method of cheating. Don't be a sucker, but don't let paranoia ruin your experience either.

The shuffling rules are very good. I don't see any need to change anything about them. The only thing that needs improving is the general playing population's understanding about it. I hope this has been a step in that direction. In the light of Ryan Fuller's recent disqualification from Venice, my next article will cover sleeves, a topic that I have had much personal experience with. Until then, comments and questions are always welcome.

- Michael Thicke



Notes:
The Rosencrantz and Guildenstern episode comes from “Rosencrantz and Guildenstern are Dead”, the best movie (in my opinion) to ever receive zero stars from Ebert . For the especially challenged, Rosencrantz and Guildenstern were friends of Hamlet.... In Hamlet?... Shakespeare? He wrote plays. Right.

The images at the top of the article are created by a Java program that I wrote based on a similar one written by Ed Purcell, a physicist and friend of Stephen Jay Gould. Each image contains 13,824 cells (144 x 96), of which 1500 are filled. For Image A, a random number between 0 and 13,824 is chosen and the appropriate cell is filled unless it already has been. The procedure is the same for Image B, except that the cell is also not filled if any of the 8 surrounding cells are occupied – representing the “zone of inhibition” created by other glowworms (from “Glow, Big Glowworm” in Bully For Brontosaurus, 1991)

Gould passed away last year. He was one of the best science writers out there, and a well respected paleontologist. I would recommend him to anyone interested in science in general, or natural history in particular.