Magic is a game with an often annoying luck component. Sometimes you draw too many lands, sometimes too few, sometimes the same card several turns in a row. One way to help mitigate this problem is to shuffle well. Cards tend to be clumped after a game (notably lands), so if you just plop the cards in play on top of your library afterwards and get on with it, your deck will definitely not be randomized.

There are steps you can take to shuffle well. Note that any form of seeding, or stacking the deck, is obviously strictly frowned upon. Some people think it is OK to intersperse lands every three cards because they half-heartedly shuffle afterwards. If it helps avoid mana problems, it’s seeding. If it doesn’t help avoid mana problems, why do it?

There are several popular ways to shuffle: the riffle shuffle (where you interweave two halves of the deck), stripping (where you take a chunk from the bottom, place it on the top, and repeat — sometimes called overhand shuffle), and pile shuffling (where you deal the cards into piles and reassemble the deck).

Persi Diaconis has studied the problem of shuffling a playing deck in great depth. For a playing deck of 52 cards, he says it takes about 5 riffle shuffles for casual play randomization, about 7 for pure randomization. Since we are playing with 60 cards, lets say we should riffle shuffle 6 or 7 times to randomize our decks.

Persi’s findings indicate just how bad stripping is. To randomize a deck via stripping as well as 5 riffle shuffles, he says it would take 2500 strips. This is because stripping only moves cards around in big clumps. Small regions of cards will still remain ordered. So, stripping is all but useless.

Pile shuffling is not technically a randomization method. You are placing the deck into piles in a definite, but arbitrary order. So, it does get rid of clumps of cards, but you are going to have to be careful about how you do it. If you pile shuffle a lot, I would recommend using a different number of piles each time and moreover, only ever using a prime number of piles. For example, only deal the deck into 2, 3, 5, 7, 11, etc. piles. This way, if you pile shuffle again, you won’t just go back to a previous ordering. For example, if you shuffled into 2 piles and then shuffled into 4 piles, you are undoing the effect of the previous shuffle.

What is particularly interesting is what happens when you pile shuffle using factors of the total size of the deck (assuming you rearrange the deck in a simple left-to-right manner). A Magic deck will have 60 cards, or 2 × 2 × 3 × 5 cards. If you pile shuffle four times, each time using one of those numbers of piles, you will have just reversed the order of the deck! Try it yourself with a smaller deck, say of 10 cards into 2 piles then 5. The order in which you choose the factors doesn’t matter. Pile shuffling all the factors of the deck size just reverses the deck!

If you pile shuffle a deck with 60 cards into piles of 2, then 3, then 5, you will have almost just reversed the cards, coming short by one pile shuffle of 2. Thus, you have the same result as if you had just pile shuffled a reversed deck into 2 piles once. That’s not very random. Using factors of the deck size can only ever shuffle it so much.

If you throw in a pile shuffle of a non-factor, like 7, it looks pretty random. As mentioned above, pile shuffling will never truly randomize, but no obvious patterns will arise from non-factors. So, I recommend sticking to non-factors and using 7, 11, or 13 piles. And obviously, do several pile shuffles (each with a different number of piles) because just doing it once separates cards but doesn’t give them the chance to randomly get back with their previous neighbors. Even better, just riffle shuffle.

The end result is that stripping doesn’t do jack, pile shuffling will barely randomize and then only if you take careful precautions, and riffle shuffling is your friend.

If you have trouble riffle shuffling or are worried about hurting your cards (give me a break, guys), I recommend just using the corners of the cards. Take the two halves, put your thumbs under a corner of both, lift, let them drop in an interweaving pattern, and push the two halves together. I find using the corners does not require as much dexterity as the more typical full riffle. It also doesn’t reveal the cards as much.

Hey thanks for this, I was really wondering about the right way to pile shuffle. Apparently the answer is not to 🙂 Guess I’ll have to practice my rifling before the next FNM so I don’t look like a tool and toss half my deck into a pile on the floor ^^

Thank you MT – best summary of this I’ve seen so far (after having just read about a dozen other articles on number of shuffles needed for real randomness)!

“Guess I’ll have to practice my rifling before the next FNM so I don’t look like a tool and toss half my deck into a pile on the floor ^^”

Heh – that’s a different kind of pile shuffling! But I haven’t seen anyone publish equations on how well that randomises things. I wonder if table height and scatter distance make significant differences in that method. 😉

I did a little analysis of pile shuffling methods and posted here: http://www.reddit.com/r/math/comments/1uw7j7/shuffling_sequence_for_playing_cards/cemdvsi

Two pile shuffles in a row with properly chosen factors can do a pretty good job of randomizing things, as judged by the Kendall Tau distance of the shuffled deck vs. the original deck.