The Wonders of Shuffling Cards with Mathematical Precision
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 interweave one by one, like that of a zipper. This is a controlled shuffle because you can apply the rules of mathematics with this shuffle. This shuffle is a basic move in which conjurers and cardists should know. For conjurers, they are able to shuffle the cards in style along with doing self-working tricks. For cardists, they are able to build up their moves into something spectacular, like the giant fan, bloom, and other moves.
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 the cards change their position.
- "In-Faro" Shuffle: In this type, the top and bottom card will change position along with the rest of the cards, as those two cards will be weaved in.
- The two types of Faro Shuffling. Left is Out-Faro, Right is In-Faro
Perfect Shuffling
As mentioned, faro shuffle is a controlled shuffle, as it won't randomize the order of the cards. You can easily determine which card landed on any position. To see how each faro shuffling method works out, you can see these examples below [source: Wikipedia]. The examples below show the faro shuffle only using 6 cards - observe on the position of these 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-Faro Shuffle Schematic
- infaro.png (35.59 KiB) Viewed 801 times
In "Out-Faro" Shuffle, the position of the top and bottom card stays the same, while the rest of the cards changes positions. It took 4 shuffles to get it back to its original order:
- Out-Faro Shuffle Schematic
- outfaro.png (43.55 KiB) Viewed 801 times
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 deck NDO (New Deck Order) = A>K of Spades - A>K of Diamonds - K>A of Clubs - K>A of Hearts; after 26 shuffles, it'll be = A>K of Hearts - A>K of Clubs - K>A of Diamonds - K>A of Spades.
- If you try to monitor the position of a card along your journey to in-faro shuffling, you will notice that it will land on every position, and it doesn't land on any position twice.
This is the position sequence of a card in "In-Faro" Shuffle [follow the Ace of Spades throughout the deck in the series of 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 of the card / n = number of cards
Example: Follow the position of Jack of Spades in "In-Faro". Initial Position: 13After the first shuffle, it will be on position 26 [2 x 13]
Example: Follow the position of Seven of Clubs in "In-Faro". Initial Position: 33After the first shuffle, it will be on position 13 [2 x 33 - (1 + 52)]
As you can see, the Ace of Spades have travelled in every position from 1 to 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. None of the cards in the deck will land on the same position!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 \\
"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 at least 6 shuffles to get it back to its original order.
- This is the most prefereable method of Faro Shuffle that everyone loves to use, as it retains the top and bottom card - for conjurers, this is one of the advantages.
This is the position sequence of the cards 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 of the card / n = number of cards
Example: Follow the position of Queen of Spades in "Out-Faro". Initial Position: 12After the first shuffle, it will be on position 23 [2 x 12 - 1]After the second shuffle, it will be on position 45 [2 x 23 - 1]After the third shuffle, it will be on position 38 [2 x 45 - 52]
1 \\
52 \\
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 be the same, as they never move.
- The rest of the cards will switch position for 7 times before it goes back to its original position before the deck shuffling.
- Card in position 18 will always switch places with card in position 35.
Deck Manipulation
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!
The Basic of Binary Number
Binary is a mathematical number expressed with only two digits of 0 and 1. In computer language, binary expressed 'true' and 'false' in programming codes, where '0' is 'false/off' and '1' is 'true/on'. Binary number can be converted into Decimal number by analysing the set of 0s and 1s in the expression.
This is how the two types of Faro Shuffle expressed in Binary Number:FORMAT: F E D C B AA is 2^o = 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 = 32The basics of converting is adding up the amount where 1s are and skipping the 0s
Example: 13 = 1101 [1 + 4 + 8]Example: 47 = 101111 [1 + 2 + 4 + 8 + 32]PS: You can find this binary system in some of the marked decks available - where you only pay attention to only 4 slots
1 = "In-Faro" Shuffle / 0 = "Out-Faro" Shuffle
"In-Faro" Shuffle = 2X
"Out-Faro" Shuffle = 2X - 1X = Position of the card
In Case Scenario - let's say the top card is Queen of Hearts, and you like to have it in position 13. That means, there will be 12 cards above it. Express the number 12 in Binary, which is '1100'. You need to shuffle the deck: "In-Faro" shuffles twice, then "Out-Faro" shuffles twice. After that, deal 12 cards off from the top, and the 13th card will be the Queen of Hearts!
In mathematical calcualtions:
In [1 x 2 = 2] > In [2 x 2 = 4] > Out [2 x 4 - 1 = 7] > Out [2 x 7 - 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 as the spectator reveals the chosen card, you flipped it and shows the Jack of Spades!
In mathematical calculations:
In [1 x 2 = 2] > Out [2 x 2 - 1 = 3] > In [2 x 3 = 6] > Out [2 x 6 - 1 = 11] > In [2 x 11 = 22]
It doesn't matter it you express the Binary number [i.e.: 10 = '1010] without the zeroes on the back [1010] OR with the zeroes on the back [001010] = The zeroes ("Out Faro" Shuffles) will not affect the outcome - even you do those pre-liminary "Out-Faro" Shuffles; it always retains the position of the top card! Only when you have done the first "In-Faro" Shuffle, then you have to do those "Out-Faro" shuffles.
Conclusion
Faro Shuffle is a very popular method of card shuffling, and there is a scientific explanation to it. While some of you might think math is boring, this is what you should know [quoted by Chris Ramsay]:
While you learn the practical method of faro shuffling, by knowing the theoretical side of it, you can create cool stuffs with it."Some of the best principles in magic are based on mathematics, and some of the best creators in magic are mathematicians."
If you would like to try these out, go and try learn the shuffling method in tutorials or do the shuffle in an unconventional way: manually stacking the cards until it interweaves. If you doubt on losing count during 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 to 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-Shuffled either way.
