FOUR POINTS: Math behind shuffling reveals important lessons for card workers

March 31st, 2015 | Joe Hadsall | Filed Under Four Points

Usually when magicians think about shuffling, anything random is FAR from our minds. We’re about CONTROL. We don’t need any random chaos, we just want things to look chaotic. When we shuffle, we can keep a card or a packet of cards right where we want them. Heck, we can keep the whole deck in the same order, if we want to.

Mathematicians are clearly not interested in our version of control.

They are much more fascinated by what’s going on with every shuffle, where every card is going and the mathematical principles that govern it — or how to best ensure that cards are sufficiently randomized. Part of that math is discussed in two videos on Numberphile’s YouTube channel, a channel dedicated to videos about numbers.

In two videos (posted below) featuring mathematicians Persi Diaconis, of Stanford University, and Federico Ardila, of San Francisco State University, the mechanics of shuffling are explained in fascinating detail. The two break down what makes a shuffle random, and how a perfect shuffle ends up in its original order. Once you get past how mind-blowing this information is, you can put it to good use. Mainly these four points:


Seven imperfect riffle shuffles are the best way to ensure a true random order. Note the success rate of those other kinds of shuffles, and you’ll get an idea of what shuffle to use in certain circumstances. It also teaches a lesson about how random shuffles LOOK.


Let’s look at one of those random decks: Grab your favorite cards and shuffle them seven times imperfectly. Get it good and random. Got it? Congratulations! It is statistically likely that the order of the cards you hold is completely unique to you, and has never been held by anyone else in the history of ever!

How is that possible? In short, there are 8.1 x 10(67) different ways a deck of cards can be ordered. Oh, you use the jokers? My bad: Now you’re talking 2.3 x 10(71) different ways. As Ardila suggests in the second video, there are about 7.1 billion people on Earth right now. That’s a 7 with only 9 zeroes behind it. Duplication is not impossible, but unlikely. Despite all the hours practicing a good fair shuffle, you never had the deck end up in the same order twice.


Back to shuffling: As Diaconis explains, perfection in shuffling leads to predictability. We magicians shuffle a lot, and most of us could probably divide the deck into two equal halves by feel alone. But that leads to our downfall, in terms of ensuring a random layout. So, the more orderly and perfect the shuffle, the more you can predict what’s going on, based on…


Most magicians know about what Adila and Diaconis call a “perfect shuffle.” In the second video, Adila breaks down what’s going on with that shuffle, card by card. If cards are, as Aaron Fisher coined them, a paper engine, then Adila’s diagram of where each card goes after every shuffle is a mind-blowing look under the hood. He explains how cards end up in exactly the same place after a few

Note that the top card and bottom card never leave (a property that can also be tweaked by the privy). Between those two cards is a process that looks like a chaotic dance, but is actually an orderly march. Cards in the outer thirds travel back and forth between, while cards in the middle third stay there, for the most part. If you’re interested in controlling cards, that’s something you want to remember.

Time to grab a deck and study. This is Persi Diaconis talking about the best and worst ways to shuffle cards:

And this is Federico Ardila analyzing the mechanics of a perfect shuffle:

FOUR POINTS is a regular feature that celebrates magicians’ favorite number by highlighting four critical bits of importance, awesomeness or otherwise. Send your suggestions to

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