About Faro Shuffle
Faro shuffle is a shuffling method where two equal-sized pack are interweaved against each other like a zipper. The performer would cut the deck into two equal packets - each packet contains 26 cards. The packets are squared up before getting it bumped each other. Applying a little pressure on the corners, the cards would interweaved one by one, like that of a zipper. This is a controlled shuffled because you can apply the rules of mathematics with this shuffle. This shuffle can be used as part of a cardistry routine, where once you have the cards interweaved, you can spring it.
There are two ways of Faro Shuffling:
- Out-Faro Shuffle : In this type, the top and bottom card of the deck will retain their position while the rest of cards change their position.
- In-Faro Shuffle : In this type, the top and bottom card will change position with the rest of the cards, as those two cards will be weaved in.
This method of cards shuffling is discovered and studied by Persi Warren Diaconis, in which he wrote in his papers titled "How to Perfectly Shuffle a Deck of Cards in Two Ways". The method is explained thoroughly by magician John Maskelyn. Faro shuffling's two method were coined by magician and mathematician Alex Elmsley.
As mentioned, faro shuffle is a controled shuffle, as it won't randomize the order of the cards. You can easily determined which card landed on any position. To see how each faro shuffling method works out, you can see these snaps below: (source: Wikipedia) [The examples shown are only using 6 cards - observe on the positions on those cards]
In In-Faro Shuffle, the position changes for every card involved. It took 3 shuffles to get it back to its original order:
In Out-Faro Shuffle, the position of the top and bottom card stays the same, while the rest of the cards change positions. It took 4 shuffles to get it back to its original order:
Now, let's see how Faro Shuffle does when we have the standard deck of 52 playing cards.
In-Faro Shuffle a Deck of 52 Cards
- You need a total of 52 shuffles in order to get the deck to return to its original order.
- After 26 shuffles, the deck will be on a reversed order from the original. So, if you start with any USPCC's NDO [New Deck Order] = Ace+King of Spades - Ace+King of Diamonds - King+Ace of Clubs - King+Ace of Hearts; after 26 shuffles, it'll be = Ace+King of Hearts - Ace+King of Clubs - King+Ace of Diamonds - King+Ace of Spades.
- If you try to monitor the position of a card, you will notice that it will land on every position, and it doesn't land on any position twice.
This is the mathematical formula of determining the position of a card in In-Faro Shuffle:
f(k) = 2k; IF k <= (n/2) [for card that is in the top half of the pack]
f(k) = 2k - (n+1); IF k > (n/2) [for card that is in the bottom half of the pack]
k = position ; n = number of cards
Example: Follow the position of Jack of Spades. Initial Position: 13
After the first shuffle, it will be on position 26 [2 x 13]
Example: Follow the position of Seven of Clubs. Initial Position: 33
After the first shuffle, it will be on position 13 [2 x 33 - (1 + 52)]
This is the position sequence of a card in In-Faro Shuffle [following the Ace of Spades]
1 > 2 > 4 > 8 > 16 > 32 > 11 > 22 > 44 > 35 > 17 > 34 > 15 >
30 > 7 > 14 > 28 > 3 > 6 > 12 > 24 > 48 > 43 > 33 > 13 > 26 >
52 > 51 > 49 > 45 > 37 > 21 > 42 > 31 > 9 > 18 > 36 > 19 > 38 >
23 > 46 > 39 > 25 > 50 > 47 > 41 > 29 > 5 > 10 > 20 > 40 > 27 \\
As you can see, the Ace of Spades have traveled in every positions from 1-52 in the course of 52 perfect In-Faro shuffles. If you follow other cards: like Eight of Diamonds or Queen of Hearts, just shift the starting position to the first, and the rest is like clockwork.
Out-Faro Shuffle a Deck of 52 Cards
- You need a total of 8 shuffles in order to get the deck to return to its original order.
- If you add extra 12 cards into the deck, making it a total of 64 cards, you only need 6 shuffles to get it back to its original order.
- This is the most preferable method of Faro Shuffle that everyone loves to use, as it is retaining the top and bottom card.
This is the mathematical formula of determining the position of a card in Out-Faro Shuffle:
f(k) = 2k - 1; IF k <= (n/2) [for card that is in the top half of the pack]
f(k) = 2k - n; IF k > (n/2) [for card that is in the bottom half of the pack]
k = position ; n = number of cards
Example: Follow the position of Queen of Spades. Initial position: 12
After first shuffle, it will be on position 23 [2 x 12 - 1]
After second shuffle, it will be on position 45 [2 x 23 - 1]
After third shuffle, it will be on position 38 [2 x 45 - 52]
This is the position sequence of the cards in Out-Faro Shuffle:
2 > 3 > 5 > 9 > 17 > 33 > 14 > 27 \\
4 > 7 > 13 > 25 > 49 > 46 > 40 > 28 \\
6 > 11 > 21 > 41 > 30 > 8 > 15 > 29 \\
10 > 19 > 37 > 22 > 43 > 34 > 16 > 31 \\
12 > 23 > 45 > 38 > 24 > 47 > 42 > 32 \\
18 > 35 \\
20 > 39 > 26 > 51 > 50 > 48 > 44 > 36 \\
- Position 1 and 52 will always the same, as they never move.
- The rest of the cards are switch position for 8 times before it goes back to its original position before the initial shuffling.
- Card in position 18 will always switch places with card in position 35.
Mathematician and magician Alex Elmsley discovered a trick where you can put the top card in any desired position in a deck just by using the combinations of In-Faro and Out-Faro shuffles. The trick is by expressing the card's desired position in Binary Number, and after the combinations of shuffling, the card will be there.
BINARY : Mathematical numbers expressed in 0 and 1
FORMAT: F E D C B A = A is 2^0=1 - B is 2^1=2 - C is 2^2=4 - D is 2^3=8 - E is 2^4=16 - F is 2^5=32; the basics is simply total up which position has 1 and skipped the position with 0
Example: 13 = 1101 [1+4+8]
Example: 47 = 101111 [1+2+4+8+32]
PS: You can find this method in a few marked decks!
1 = In-Faro Shuffle ; 0 = Out-Faro Shuffle
In-Faro Shuffle = 2X
Out-Faro Shuffle = 2X - 1
In Case Scenario, let's say the top card is Queen of Hearts, and you like to have it on position 13. That means, there will 12 cards above it. Express the number 12 in Binary, which is 1100. You need to shuffle the deck: In-Faro twice, then Out-Faro twice. After that, deal 12 cards off from the top, and the 13th card will be the Queen of Hearts.
In mathematical calculations: In [1x2=2] > In [2x2=4] > Out [2x4-1=7] > Out [2X7-1=13]
Another Case Scenario, let's say the top card is Jack of Spades, and the spectator states a number from 1-52 and the spectator choose 22. That means, there should be 21 cards above the main card. Express the number 21 in Binary, which is 10101. You need to shuffle the deck: In-Faro > Out-Faro > In-Faro > Out-Faro > In-Faro. You ask the spectator to deal 21 cards off from the top, and in the moment of suspense, you singled out the 22nd card, and it is the Jack of Spades!
In mathematical calculations: In [1x2=2] > Out [2x2-1=3] > In [2x3=6] > Out [2x6-1=11] > In [2x11=22]
It doesn't matter if you express the Binary number [i.e: 10=1010] without the zeros on the back  OR with the zeroes on the back  = The zeros [Out-Faro shuffles] will not affect the outcome - even you do pre-liminary Out-Shuffles, it always retain the position of the top card!
Faro Shuffle is a very popular method of card shuffling, and there is a scientific explanations to it. While some of you might think math is boring, this is what you should know [quoted by Chris Ramsay]: some of the best principles in magic are based on mathematics, and some of the best creators in magic are mathematicians. While you learn the practical method of faro shuffling, by knowing the theoritical side of it, you can create cool stuffs with it.
If you would like to try these out, go and try learn the shuffling method in tutorials or do the shuffle in an unconventinal way: manually stacking the cards until it interweaves. If you doubt on losing count during the shuffles, just take note of the first card of the second pack.
EXTRA: During playing cards production, there are two types of cut that will affect the way you Faro Shuffle. Traditional Cut, in which the cards are faced up during the cutting - creating the upward bevel on the edges, makes the faro shuffle easier from the bottom the the top [it is good for table shuffling]. Modern Cut, in which the cards are faced down during the cutting - creating the downward bevel on the edges, makes the faro shuffle easier from the top to the bottom. Legends Playing Cards have the Diamond Cut, which creates a very smooth edges on the cards, making the cards easily faro-ed either way.