Shuffling, part 1

by MTGViolet

So, for our first foray, we’re going to look at shuffling. We all do it, usually several times a day, but how does it work, exactly? What methods work the best for shuffling, and which don’t work at all?

That last question is interesting: Which methods of “shuffling” get used on a regular basis that don’t actually shuffle your deck. Perhaps you already know the answer, it’s the bane of judges everywhere, the so called “Pile Shuffle”.

Pile shuffling (though, as we will see, it isn’t really shuffling at all), is when a player deals cards from the top of their deck into piles in front of him or her, then picks up those piles one after another. It certainly seems random, but it is in actual fact anything but. I’ll try to illustrate this for you, starting with a simpler example.

Take a deck of 12 cards that we deal into six piles. Say also that the cards are either lands (L) or spells (S), and at the beginning the player sorted them so that they alternate land-spell-land. This is allowed, as long as the deck gets shuffled thoroughly since the order they started in will be unrelated to the order they end up in.

So, to start, our deck looks like this: L S L S L S L S L S L S (counting from the top to the bottom. If you flip them around to look at the card faces, this order will be reversed, but you usually shuffle face down)

Let us pile shuffle this deck now. The first pile will get the first land card, the second will get the first spell, the third will get the second land, and so on, till we end up with this:

L  S  L  S  L  S
L  S  L  S  L  S
1  2  3  4  5  6

When we stack these we have a deck that looks like this:

L L S S L L S S L L S S

You may already see a problem here. While this order isn’t exactly the same as the first one, it still has a very regular pattern about it, and is very predictable. This should already raise red flags, but let’s continue.

The second pile shuffle will look like this:

S  S  L  L  S  S
L  L  S  S  L  L
1  2   3  4 5  6

Remember, the first card we lay down will be at the bottom of the pile, once we lay down the first six piles the next card will be on top of the first. Piling these together gives us:

S L S L L S L S S L S L

Pretty close to our original. Already after two pile shuffles we have a big problem, but what happens after the third?

L  S  S  L  S  L
S  L  S  L  L  S
1  2  3  4  5  6

Which leads to

L S S L S S L L S L L S

Now, you might be thinking, this looks pretty random. Sure, some lands and spells are clumped together, but if you got handed a deck that looks like this, you’d assume that the player had actually shuffled it, right? Well, let’s go on to some more pile shuffles and see if this stays true.

L  L  S  L  L  S
L  S  S  L  S  S
1  2  3  4  5  6

L L L S S S L L L S S S

And one more:

L  L  L  S  S  S
L  L  L  S  S  S
1  2  3  4  5  6

L L L L L L S S S S S S

So, now we’ve gone through five pile shuffle, and we’ve nicely put all our lands on top, and all our spells on bottom. Not a very good order, strategically, but that’s not important. What is important is that we know exactly what the order is and, what’s worse, all we have to do is pile shuffle one more time

S  S  S  S  S  S
L  L  L  L  L  L
1  2  3  4  5  6

S L S L S L S L S L S L

And we have this. In other words, the exact same order that we started with, except with lands and spells reversed. In fact, do another six pile shuffles – and I’ve seen players easily do that many before starting a game – and you’ll have the same order as in the beginning. Don’t believe me? Try it yourself!

This is why pile shuffling isn’t considered randomization. After each shuffle you know exactly where the cards are going to be. Even if you pick up the piles in a random order, you still have the same relative positions of cards in the piles, which can still lead to a very non-random outcome, especially if all you care about is two qualities, such as land or non-land.

That’s it for now. When I continue this topic I’m going to break out the algebra, but next week I’m going to talk a little more about what “random” means.

Exercises:
(It’s a math blog, of course there are exercises!)

  1. What happens when you pick up the piles randomly? What can you say about the order of cards after a given number of pile shuffles?
  2. Try out the pile shuffle with a full deck of thirty lands and thirty non-land spells. How many pile shuffles do you need to get back to a pattern of land-spell-land?
  3. What about if you pile shuffle a deck while tracking three qualities: land, creature, non-creature spell. Assume equal quantities of each.
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