Randomization

Last weekend at DragonCon, I overheard a player giving advice to his opponent on how to shuffle. The player suggested laying out the cards one at a time into five piles in order to increase randomization. When I explained to him that this method of so-called “pile shuffling” does nothing to add to randomness in the deck, he insisted I was wrong. I was tempted to break out an old card trick my sister showed me over a decade ago, wherein I sort the deck into a number of piles, ask which pile your card is in, pick up all the piles and repeat. That’s all it takes for me to be able to identify which card was chosen.

So let’s start with what it means for a deck to be random. First off, randomization and shuffling are synonyms. One of my opponents at the last pre-release, upon being told that pile shuffling didn’t actually randomize his deck, informed me that he wasn’t randomizing, he was shuffling. I had a very Inigo Montoya moment—I do not think that word means what you think it means.

A truly random deck is one where each card has an equal probability of being in any position, and those probabilities do not change as more information becomes known. For instance, if I draw my Havoc Festival, I should not be able to use that information to conclude that my Wound Reflection is somewhere in the next ten cards. True randomness will reduce but not eliminate the chance of mana flood or mana screw; in a deck with an even distribution of spells and lands, spell-spell-spell-spell, spell-land-spell-land, and land-land-land-land have approximately equal probabilities of occurring. Each individual distribution of cards is exactly as likely to happen as any other; the fact that there are more ways for lands and spells to be distributed in an approximately even manner than for all to be clumped together means that as a whole those types of distributions are more likely to occur.

If I know the location, or approximate location, of any card in the deck, the deck is not random. For instance, if I shuffle my cards face-up, then flip the deck over and do a single face-down shuffle, I know that whatever card was on the bottom is still very close to that position. I’ll also know that said card is still very close to whatever cards it was close to when I turned the deck over, so if I draw one of them I’ll know the others are coming up soon, even if my opponent has cut my deck.

Most people think a random distribution means everything is approximately evenly spaced, like a group of people standing in an elevator. In fact, clumping is an aspect of randomness, because the presence of, say, one land card, has no bearing on the positioning of others. This is a very difficult thing for the human brain to grasp. It seems impossible that if you have twenty-three people in a room, two of them will share a birthday, but in fact there’s a better than fifty percent chance of that being the case. (The math on this is simple. Given two people, there’s a 364/365 chance that they don’t share a birthday. Multiply this by a third person’s 363/365 chance of not sharing a birthday with either of them, since now there are two days already taken. The fourth person is 362/265, and so on. As you multiply all of these numbers together, the probability of no one sharing a birthday shrinks, until it’s less than fifty percent, meaning there’s a greater than fifty percent chance that at least two people do share a birthday.)

Back on the topic of pile counting, there’s actually an infamous cheat known as the double nickel, wherein the player counts out his cards into five piles, stacks the piles, and repeats. This creates a near-perfect distribution of lands versus spells. Since most opponents only cut the deck, rather than shuffle, the distribution is preserved. (If I have land-spell-spell throughout my deck, it doesn’t matter what the top card is, the top seven will have either two or three lands, with another land either the first or second draw.)

In a less blatantly cheaty-face example, say I know the card that’s thirteenth from the top. I count my cards into five piles. The card I know is now in the third pile, third from the bottom. In a ninety-nine card deck, it’s going to wind up twenty-second from the bottom. Any card whose position I knew before the pile count, I can uniquely identify its position afterwards. No knowledge has actually been lost, therefore the deck is no more random than it was before the supposed shuffle.

Does that mean that an opponent who uses this technique is automatically cheating? Not necessarily. Judges refer to it as pile counting for a reason. It’s a great way to make sure you’re presenting a legal deck by ensuring you have all forty, sixty, or ninety-nine cards. It can also be helpful in determining if any cards are stuck together and unsticking them, so those cards don’t remain next to each other throughout the shuffling process.

So how should you shuffle? Riffle shuffling is best—that’s your typical playing card shuffle—but it’s rather difficult with a pile of ninety-nine sleeved Magic cards. One thing I like to do, after a few initial shuffles, is to cut the deck in half and shuffle each half separately. With sleeved cards, it can also be easier to mash shuffle (where you just mash the two piles of cards together, rather than bend them and let them riffle through your fingers), although mash shuffling isn’t as effective a randomization technique.

For a fifty-two card deck of playing cards, or a sixty card Magic deck, seven riffle shuffles will make the deck essentially random. In a Commander deck, you need at least one more, and that’s if you’re shuffling the entire deck at once. If not, first you have to randomize which cards are in the top versus bottom half (and not simply by cutting, since that preserves cards being in the same half, since they’ll move together from the top to bottom or vice versa), then cut it in half and shuffle each half seven times, before then shuffling the two halves back together.

For perfect randomization, it takes eleven or twelve shuffles to make a fifty-two card deck indistinguishable from true random. That number is going to increase as the number of cards in the deck increases. Of course it’s not terribly feasible to shuffle a Commander deck a baker’s dozen times before every game and after every search of the library; games take more than long enough as it is. Especially since it’s a casual format, I’m not really worried about players knowing the relative positions of cards. If you’re cheating in Commander, you’re doing it wrong. The main point of shuffling here is to create a random distribution of lands, which will hopefully provide semi-regular land drops and allow you to play your deck, while also undoing any knowledge you’ve gained of card positions either from looking through your library, or putting the cards on top after a previous game. Seven or eight shuffles—or seven shuffles of each half of the deck—should be sufficient for the purposes of the game. After searching the library, four or five riffle or mash shuffles will undo any knowledge gained, since it’s highly unlikely you’ll be able to memorize the relative positions of ninety cards in the few seconds it takes to find the card you want.

Now go forth and spread the news! And don’t let people tell new players that pile counting randomizes their deck.

 

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