Riffle vs overhand shuffle
Six ways to shuffle, run on the same deck, measured two ways. Press the deck to shuffle it.
Shuffle a deck and measure it
not yet shuffled
The two numbers disagree, and that is the point
Rising sequences count how many runs the deck still breaks into that are in ascending order. A new deck is one. A random deck averages 26.5. It is the standard measure and it is a good one, because a single riffle can only ever produce two — so the count tells you almost exactly how many riffles a deck has had.
Ask it about anything that is not a riffle and it goes strange. Deal the deck into four piles and stack them up and it reports forty, which is higher than a random deck. Give the deck five perfect out-faros and it reports thirty-two. Both of those decks are in a fixed, known order that can be undone exactly, and one number says they are better shuffled than chance.
So the table below counts something else: how many pairs of cards that started next to each other are still touching. Fifty-one to begin with, about two once a deck is properly mixed. That is the one a card worker feels, because a stack surviving a shuffle is exactly clumps surviving a shuffle.
Every shuffle, every number of passes
Is the overhand shuffle worse than the riffle shuffle?
By a long way, on that measure. One riffle leaves about twenty-five of the fifty-one neighbors still touching. One overhand shuffle leaves about thirty-nine — it moves blocks of cards around rather than mixing cards. Ten overhand shuffles still leave roughly six pairs together; seven riffles leave two, which is what random looks like.
Packet size is the whole character of it, which is why the panel lets you change it. Strip eight cards at a time and you are moving four or five blocks; strip two at a time and you are closer to reversing the deck than mixing it.
Does dealing cards into piles shuffle them?
No. A pile deal is one fixed permutation. The same deck in gives the same deck out, every time, so anybody who knows how many piles you dealt can put it back. Nothing in the table's right-hand columns improves, because there is nothing random to accumulate — dealing it twice more just applies the same fixed shuffle twice more.
It is on the page because of how well it scores on the other measure. A deal that can be reversed in ten seconds reads as better than chance, and that is worth seeing once.
What is the best way to shuffle a deck of cards?
A table wash gets there in one go, which is why casinos open a new deck that way. It is slow and it looks like nothing, and for a card table it is the honest answer.
Riffling is the practical one: seven riffle shuffles leave a deck close enough to random for most purposes, and that page has the argument and the simulation behind the seven. The faro shuffle is the opposite of all of this — a shuffle designed to preserve order rather than destroy it.
Questions people ask
Is the overhand shuffle worse than the riffle shuffle?
By a long way, on the measure that matters to a card worker. One riffle leaves about twenty-five of the fifty-one original neighbors still touching. One overhand shuffle leaves about thirty-nine. Ten overhand shuffles still leave roughly six, and seven riffles leave two, which is what a random deck looks like.
Does dealing cards into piles shuffle them?
No. A pile deal is one fixed permutation: the same deck in gives the same deck out every time, so anybody who knows how many piles you dealt can undo it exactly. It scores forty rising sequences, which is higher than a random deck, and that is a fact about the measure rather than about the deal.
What are rising sequences in a deck of cards?
The number of runs the deck breaks into that are still in ascending order. A new deck is one long run. A random deck averages 26.5 for fifty-two cards. One riffle shuffle can only ever produce two, which is why the count is such a good measure of riffling and such a poor measure of anything else.
What is the best way to shuffle a deck of cards?
A table wash gets there in one go, which is why casinos use it. Riffling is the practical answer: seven of them leave a deck close to random. An overhand shuffle needs far more passes than anybody gives it, and a pile deal never gets there however long you keep dealing.
Where the models come from
The riffle is the Gilbert–Shannon–Reeds model: a binomial cut, then cards falling from whichever hand still holds more in proportion. It is the model Bayer and Diaconis used for the seven. The overhand strips packets of a geometric size; the wash is a uniform permutation. All of them, and both measures, live in one file that this page and how many shuffles share, so the two pages cannot quote different numbers at you.
Every figure on this page was computed in your browser as the page loaded, over two hundred fresh decks per cell. Press the button under the table and you get a different set, drawn the same way.
Harrison Alley
Hobbyist · Writes Sleightless
A hobbyist, not a professional. I write up the self-working end of card magic, because it is the door I would point a beginner at, and every trainer here is played through and checked before it ships.
More
Mathematical card tricks is the rest of this: parity, base three, arithmetic that wraps at thirteen, and one number that cancels itself out.