Shuffling

[SS] Stan the Stat declaimed, “In the 2006 movie The Prestige Michael Caine’s character Cutter explains:”

Every great magic trick consists of three parts or acts. The first part is called “The Pledge”. The magician shows you something ordinary: a deck of cards, a bird or a man. He shows you this object. Perhaps he asks you to inspect it to see if it is indeed real, unaltered, normal. But of course… it probably isn’t. The second act is called “The Turn”. The magician takes the ordinary something and makes it do something extraordinary. Now you’re looking for the secret… but you won’t find it, because of course you’re not really looking. You don’t really want to know. You want to be fooled. But you wouldn’t clap yet. Because making something disappear isn’t enough; you have to bring it back. That’s why every magic trick has a third act, the hardest part, the part we call “The Prestige”.

[RR] “Interesting parallel to the flop, the turn, and the river in Texas Hold ‘Em”, Roderick the Rock noted.

[LL] “I’ve never heard anyone claim that poker’s turn comes from the magic term, but it makes some sense, as both are in the middle”, Leroy the Lion continued.

[RR] “Magic tricks can certainly flop if the sleight of hand isn’t performed well or the intended result fails.”

[LL] “And there’s plenty of prestige if the river card wins you a big poker tournament.”

[SS] “Good magic can be entertaining even if you know how it’s done. Great magic though, leaves you wondering how they did it.”

[LL] “What makes really great magic are the incredible skills that the magicians practice countless hours honing. Sleight of hand. Misdirection. Physical manipulations that you might have thought were completely impossible.”

[SS] “Like the perfect shuffle. Can you cut a deck of cards perfectly in half, so each half has exactly 26 cards? Can you riffle shuffle1 those two halves so that the cards are perfectly alternated from each half? Can you then execute each of those two skills eight times in a row without a single card getting out of place?”

[SS] “If you started with a sorted deck and did exactly that, you’d be right back where you started.2 And ready to perform any number of card tricks with your seemingly well-shuffled deck. If you did the first five shuffles ahead of time, you could then start the trick by fanning a seemingly already randomly shuffled deck to start with.”

[RR] “What would that shuffled deck look like, so we could spot it if we saw it?”

[SS] “Better yet, here are all eight permutations of a deck that starts with each suit sorted in order from Ace to King:”

0 1 2 3 4 5 6 7 8
A♣ A♣ A♣ A♣ A♣ A♣ A♣ A♣ A♣
2♣ A♥ A♦ 7♥ 4♦ 9♣ 5♣ 3♣ 2♣
3♣ 2♣ A♥ A♦ 7♥ 4♦ 9♣ 5♣ 3♣
4♣ 2♥ A♠ 7♠ T♠ Q♦ K♣ 7♣ 4♣
5♣ 3♣ 2♣ A♥ A♦ 7♥ 4♦ 9♣ 5♣
6♣ 3♥ 2♦ 8♣ 4♥ 2♠ 8♦ J♣ 6♣
7♣ 4♣ 2♥ A♠ 7♠ T♠ Q♦ K♣ 7♣
8♣ 4♥ 2♠ 8♦ J♣ 6♣ 3♥ 2♦ 8♣
9♣ 5♣ 3♣ 2♣ A♥ A♦ 7♥ 4♦ 9♣
T♣ 5♥ 3♦ 8♥ 4♠ 9♦ J♥ 6♦ T♣
J♣ 6♣ 3♥ 2♦ 8♣ 4♥ 2♠ 8♦ J♣
Q♣ 6♥ 3♠ 8♠ J♦ Q♥ 6♠ T♦ Q♣
K♣ 7♣ 4♣ 2♥ A♠ 7♠ T♠ Q♦ K♣
A♦ 7♥ 4♦ 9♣ 5♣ 3♣ 2♣ A♥ A♦
2♦ 8♣ 4♥ 2♠ 8♦ J♣ 6♣ 3♥ 2♦
3♦ 8♥ 4♠ 9♦ J♥ 6♦ T♣ 5♥ 3♦
4♦ 9♣ 5♣ 3♣ 2♣ A♥ A♦ 7♥ 4♦
5♦ 9♥ 5♦ 9♥ 5♦ 9♥ 5♦ 9♥ 5♦
6♦ T♣ 5♥ 3♦ 8♥ 4♠ 9♦ J♥ 6♦
7♦ T♥ 5♠ 9♠ J♠ Q♠ K♦ K♥ 7♦
8♦ J♣ 6♣ 3♥ 2♦ 8♣ 4♥ 2♠ 8♦
9♦ J♥ 6♦ T♣ 5♥ 3♦ 8♥ 4♠ 9♦
T♦ Q♣ 6♥ 3♠ 8♠ J♦ Q♥ 6♠ T♦
J♦ Q♥ 6♠ T♦ Q♣ 6♥ 3♠ 8♠ J♦
Q♦ K♣ 7♣ 4♣ 2♥ A♠ 7♠ T♠ Q♦
K♦ K♥ 7♦ T♥ 5♠ 9♠ J♠ Q♠ K♦
A♥ A♦ 7♥ 4♦ 9♣ 5♣ 3♣ 2♣ A♥
2♥ A♠ 7♠ T♠ Q♦ K♣ 7♣ 4♣ 2♥
3♥ 2♦ 8♣ 4♥ 2♠ 8♦ J♣ 6♣ 3♥
4♥ 2♠ 8♦ J♣ 6♣ 3♥ 2♦ 8♣ 4♥
5♥ 3♦ 8♥ 4♠ 9♦ J♥ 6♦ T♣ 5♥
6♥ 3♠ 8♠ J♦ Q♥ 6♠ T♦ Q♣ 6♥
7♥ 4♦ 9♣ 5♣ 3♣ 2♣ A♥ A♦ 7♥
8♥ 4♠ 9♦ J♥ 6♦ T♣ 5♥ 3♦ 8♥
9♥ 5♦ 9♥ 5♦ 9♥ 5♦ 9♥ 5♦ 9♥
T♥ 5♠ 9♠ J♠ Q♠ K♦ K♥ 7♦ T♥
J♥ 6♦ T♣ 5♥ 3♦ 8♥ 4♠ 9♦ J♥
Q♥ 6♠ T♦ Q♣ 6♥ 3♠ 8♠ J♦ Q♥
K♥ 7♦ T♥ 5♠ 9♠ J♠ Q♠ K♦ K♥
A♠ 7♠ T♠ Q♦ K♣ 7♣ 4♣ 2♥ A♠
2♠ 8♦ J♣ 6♣ 3♥ 2♦ 8♣ 4♥ 2♠
3♠ 8♠ J♦ Q♥ 6♠ T♦ Q♣ 6♥ 3♠
4♠ 9♦ J♥ 6♦ T♣ 5♥ 3♦ 8♥ 4♠
5♠ 9♠ J♠ Q♠ K♦ K♥ 7♦ T♥ 5♠
6♠ T♦ Q♣ 6♥ 3♠ 8♠ J♦ Q♥ 6♠
7♠ T♠ Q♦ K♣ 7♣ 4♣ 2♥ A♠ 7♠
8♠ J♦ Q♥ 6♠ T♦ Q♣ 6♥ 3♠ 8♠
9♠ J♠ Q♠ K♦ K♥ 7♦ T♥ 5♠ 9♠
T♠ Q♦ K♣ 7♣ 4♣ 2♥ A♠ 7♠ T♠
J♠ Q♠ K♦ K♥ 7♦ T♥ 5♠ 9♠ J♠
Q♠ K♦ K♥ 7♦ T♥ 5♠ 9♠ J♠ Q♠
K♠ K♠ K♠ K♠ K♠ K♠ K♠ K♠ K♠

[SS] “After the first shuffle, the cards are in pairs, with clubs and hearts together and diamonds and spades together.

After the second shuffle, the cards are sorted by denomination.

After the third shuffle, the cards are matched with their opposite denominations (e.g., deuces with eights and nines) and opposite suits.

After the fourth shuffle, the cards are in groups of four, spaced out denominationally, one of each suit.

After the fifth shuffle, the cards are five apart denominationally (decreasing), with suits almost paired.

After the sixth shuffle, the cards are four apart denominationally, with suits in groups of three or four.

After the seventh shuffle, the cards are two apart, with suits together (two batches).

After the eighth shuffle, the cards are back in their original order.”

[LL] “The conventional wisdom is that you should mix the cards using at least seven riffle shuffles,3 but apparently you shouldn’t do it too perfectly!”

[SS] “That advice is certainly useful for people who might only shuffle a couple times otherwise, but twelve or more shuffles should really be your target if you want a truly randomized deck.”

[RR] “Most people just want to play cards, not shuffle all day.”

[LL] “Well, at least cut the deck before dealing. The top card could still be on top no matter how many times the deck was shuffled.”

[SS] “That’s why professional croupiers throw in a strip shuffle after every couple riffle shuffles.4 The strip shuffle also adds randomness, reducing the number of riffles you need to do. My new shuffle routine is seven riffles with a strip after every even riffle.”5

Footnotes:

  1. The riffle shuffle is definitely the best of the six common shuffling methods. The Overhand Shuffle (pulling cards out from the side of the deck), the Hindu Shuffle (pulling cards out from the end of the deck), and the Strip Shuffle (basically just repeated cuts) are very inefficient. The Weave Shuffle or Faro Shuffle (forcing two half decks together along their edges) is too good and not good for the cards.
  2. Specifically, this is an out shuffle, where the top and bottom cards never move. If the top card of the bottom half of the deck becomes the new top card each time instead, it’s an in shuffle. While the out shuffle takes eight iterations to return the deck to its original order, the in shuffle inverts the deck after 26 shuffles and restores it after 52.
  3. Note: there are two main flavors of the riffle shuffle. The common method lifts the short edges of the halves high and merges them together, with or without a subsequent bridge. The casino method merges the corners of the halves together. This is better for the cards and much less likely to expose the card faces.
  4. See the beginning of this shuffling video (longer explanation).
  5. The entire sequence of riffle-riffle-strip-riffle-riffle-strip-riffle-riffle-strip-riffle will produce a shuffled deck random enough for anyone.

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