### Randomness, part 3

First off, I’d like to apologize for not posting this week. This Wednesday, which is when I usually post these, happened to be my thirtieth birthday, and I was out of town. So this week’s post is a bit delayed. I’ll try to get back on schedule for next week.

If you read last week’s article, you may despair that you can ever truly randomize your deck. Can your deck ever truly be random, if every shuffle is deterministic?

Well, yes, and no. The answer veers a bit out of mathematics and into philosophy (and a bit of biology).

When we talk about randomization, we might be content with a definition of “the deck is sufficiently random if neither player can be reasonably expected to know the (absolute or relative) location of any card or cards.” In other words, if the human brain isn’t capable of tracking the cards’ positions as you shuffle, that’s good enough for us. But what are the limits of the human brain?

Since I’m not a neurobiologist (shocking, I know), I can’t say for sure, but I’d say that, for a large percentage of the population, a pile shuffle would already meet this definition. For professional players, even a single riffle shuffle might be enough to confound their tracking skills, or at the most two. Then there are the Vegas card counter types – and, before you say anything, they do exist, at every level of play – who can track cards through an arbitrary number of ideal riffle shuffles without much effort. So clearly we can’t rely on the fallibility of the human brain to consider a deck shuffled.

luckily, there is another answer. I mentioned the riffle shuffle above. In an ideal riffle shuffle, we split the deck in half and into two piles (using a 12 card deck as an example)
``` 1 7 2 8 3 9 4 10 5 11 6 12 ```

Then we alternate the cards from one pile and then the other, ending up with the order

`1 7 2 8 3 9 4 10 5 11 6 12`

And the corresponding permutation

`(2 7 4 8 10 11 6 9 5 3)`

1 and 12 are mapped to themselves, so we don’t need to mention the, in the permutation. (already we should be suspicious of this form of shuffling)

Of course, you might be asking yourself, how is this any more random than a pile shuffle? Well… It’s not, honestly. But here we save ourselves through human fallibility.

The average magic player won’t be able to cut a sixty card deck exactly in the middle. When letting the cards fall, allowing one card to alternate with the other exactly over sixty cards is incredibly hard. When we riffle shuffle, we are actually choose a random riffle-ish shuffle from the hundreds of thousands of possible permutations over sixty cards. Granted, most riffle shuffles will still have the problem of keeping the top and the bottom cards in the same place, but that’s why we cut after shuffling, to remove even that regularity. And, last but not least, and this is important. when your opponent presents his or her deck to you, you should always not just cut it, but shuffle it. that way, even if your opponent is a Vegas card shark who has mastered the art of the ideal riffle shuffle (more common than you might think), you can still be assured he or she will have a sufficiently random deck, because you’ll be sure to follow what you’ve learned here to shuffle the deck in a way that even your opponent can’t know the location of any of the cards.

Of course, these cards will still not actually random. They are deterministically distributed in your deck in a non-random way. We just don’t know what the order is until we draw the cards themselves. We could link the cards to a random number generator, bringing true quantum randomness into play, but, as I defined it above, it’s enough for us as magic players to just not know where the cards are, they needn’t be truly random. Once the deck is properly shuffled, we can treat it as though it were truly randomized for the rest of these articles.

Exercises

• What is the permutation for an ideal riffle shuffle over a sixty card deck?
• After how many ideal riffle shuffles will the deck be in the same order as it started?
• What other forms of shuffling do you know? How would you describe their potential for randomness? You don’t have to do so mathematically, just describe it.