Mathematics: The Shuffling

Recently, my brother-in-law Alex has been into the card game Magic: The Gathering. Over the holidays he had the whole family playing, and the question came up of how many times a deck needs to be shuffled to be "random". For an ordinary 52-card deck, the oft-quoted result of 7 was originally given in a New York Times article from 1990. Doing a bit more research, I found a paper following up on that article with a few more details. I wasn't entirely satisfied with that, both because the shuffles described didn't quite match my techniques, and because I found the probability calculations difficult to follow in my first jet-lagged, and now sickened state, so I wrote some code to simulate a series of shuffles. The two types of shuffle I usually do are:

Riffle: Split the deck in two, and interleave the piles, roughly alternating. I modeled this as a split using a Gaussian distribution around the center, and then picking from the two piles with 50/50 chance.

Overhand: Holding the deck in one hand, move packs of cards to the other. I modeled this as splitting the deck into a set number of unequal packs, and reversing the order of those packs.

Now we have to figure out the quality of these shuffles. I've talked about one way to measure randomness before, but that doesn't really apply here, since all the cards (in a standard deck anyway) are unique. For a deck labeled 1-52, the two quality measures I came up with are: the total number of cards in sequences, and the distance of each card from its original position. Over a series of shuffles, here are the means and standard deviations for those two measures on a set of 5000 decks:

For both distance measures, the riffle shuffle appears to reach a stable value in fewer shuffles. Breaking up runs is clearly not a strong suit of the overhand shuffle, but it moves cards from their original position pretty quickly. As far as the minimum number of shuffles though, it appears 7 is a reliable choice, but you could get away with fewer on average.