Seven shuffles: how long it takes to mix a deck

What a riffle actually does.

Cut the deck roughly in half, then let the two halves drop together, a few cards from each side at a time. Nothing about that destroys order; it interleaves it. If the deck started as one rising run — 1, 2, 3 … 52 — a single riffle leaves two rising runs woven together. A second riffle leaves at most four, a third at most eight. The deck is being folded like dough, and dough takes a while to mix.

You can watch the folding in the rainbow above. After one riffle the colours are still clearly two gradients zipped together. After three you can still pick out stripes. Somewhere around five or six it starts to look like noise — and the “neighbours still together” count, which starts at 51, tumbles towards the handful you would expect from a genuinely random order.

Why seven.

In 1992 Dave Bayer and Persi Diaconis analysed the mathematics of the riffle and showed that about (3/2) × log₂(n) shuffles are needed to mix n cards — for 52 cards, about seven. With fewer, the original order is still detectable to anyone who knows what to look for; with more, you gain almost nothing. The result made the New York Times under the headline “In Shuffling Cards, 7 Is Winning Number”.

“Mixed” is a matter of definition, and other definitions give other numbers. Measuring the information left in the deck instead, Lloyd Trefethen and his father argued that five riffles do a reasonable job and six make thorough mixing practically certain. Under a stricter yardstick the answer is eleven. Seven is the number that stuck because it answers the question most card players are asking: could an attentive opponent still exploit the order?

And please don’t overhand.

The casual shuffle most of us do — dribbling little packets from one hand into the other — is spectacularly bad at this. Diaconis and Jason Fulman put the number of overhand shuffles needed to mix a deck well at around ten thousand. Three overhands and a cut, which is what happens at most kitchen tables, leaves the deck very nearly in the order the last hand left it.

If that sounds like an abstract worry, consider the size of the space you are failing to explore. A deck of 52 cards can be arranged in 52! ways — about 8 × 10⁶⁷, a number that makes the atoms in the Earth look sparse. A properly shuffled deck is almost certainly in an order that has never existed before. A badly shuffled one is a rerun.

For a list of names rather than cards, there is a shortcut to the fully mixed state: a computer can produce any one of those orderings with equal probability in a single step. That is what the shuffler below does, with no practice and no sore thumbs.