Mathematical Facts & Curiosities

142 entries in Cards / Sleights / Shuffles (non-riffle) / Faro Shuffle / Mathematical Facts & Curiosities
Creators Title Comments & References Year Source Page AA Categories
Charles T. Jordan Perfect Riffle Shuffle without details, four shuffles for sixteen cards to recycle, "Mr. Downs, however, can handle a full pack of 52 cards with the degree of dexterity necessary to restore its original order."
 		1919/1920 Thirty Card Mysteries 32
Fred Black The Shuffle faro tables
Related to 1940 Expert Card Technique 145
Fred Black The Endless Belts
Related to 1940 Expert Card Technique 147
Fred Black Chart of Seventeen
Related toVariations 1940 Expert Card Technique 147
The Eighteenth Card using 18-35-faro-principle with honest shuffle, risky
 		1940 Expert Card Technique 150
Alex Elmsley In and Out Definition
1958 The Faro Shuffle 1
Edward Marlo Half and Half Shuffle basically the stay stack principle applied to two cards
 		1958 The Faro Shuffle 29
Edward Marlo "Half Plus One" bringing a key card next to a certain card with faro shuffle
 		1958 The Faro Shuffle 30
Edward Marlo Observations faro as a false shuffle and other comments
 		1958 The Faro Shuffle 34
Alex Elmsley In and Out Shuffle Definition
1958 Faro Notes 1
Edward Marlo A Correction commentary on ECT tables, see also new hardcover edition for further commentary
Inspired by 1958 Faro Notes 8
Edward Marlo The Chain Calculator how to calculate position of any card after faro shuffles, memorized deck
 		1958 Faro Notes 12
Russell "Rusduck" Duck Faro Favorites Elmsley's Restacking Pack
Related to 1958 The Cardiste (Issue 10) 14
Russell "Rusduck" Duck Perma-Stack based on Elmsley's Restacking Pack idea
Related to 1958 The Cardiste (Issue 10) 15
Edward Marlo On the Re-Stacking Pack two spectators decide for numbers and remember the cards at their number four times with faros in between, each has a four of a kind
Inspired by 1964 Faro Controlled Miracles 18
18-35 Principle
1964 Faro Controlled Miracles 19
Karl Fulves Faro-Shuffling Machines examining the problem of finding an algorithm to find a faro combination to shuffle from position x to position y, discussed with a 6-card deck
 		Nov. 1967 Epilogue (Issue 1) 7
Roy Walton A Faro Tree examining the problem of finding an algorithm to find a faro combination to shuffle from position x to position y
Also published here Nov. 1967 Epilogue (Issue 1) 8
Karl Fulves Q & A a deck is given a known sequence of faro shuffles (e.g. IOIIOOOIOIIOOIO), problem: how to recycle to get original order with faro shuffling
 		Mar. 1968 Epilogue (Issue 2) 15
Edward Marlo Marlo Re-Stacking Pack two spectators decide for numbers and remember the cards at their number four times with faros in between, each has a four of a kind
Inspired by 1969 Expert Card Mysteries 175
Karl Fulves Faro Transforms discussing properties of the faro to exchange two cards within the deck and to recycle the order
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 2
Karl Fulves Faro Rings notation to illustrate behavior of cards during faro shuffles, see also Addenda on page 60
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 2
Karl Fulves Position Determination following a card's position during in and out faros
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 3
Karl Fulves The Triple Faro Ring
1969 Faro & Riffle Technique (Issue Faro Techniques) 4
Karl Fulves General Transform Characteristics discussing how the order is affected through faro shuffling in a 2n deck
1. Reversibility
2. The Recycling Corollary
3. Commutative Property
4. Additive Property
5. Position Equivalency
6. Substitutions
7. Non-Symmetric Transforms
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 7
Karl Fulves The Fourth-Order Deck discussing transpositions of two cards within the deck
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 12
Karl Fulves Three Way Transposition note
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 15
Karl Fulves Fractional Transforms note
 		1969 Faro & Riffle Technique (Issue Faro Techniques) 15
Karl Fulves The Recycling Problem "The general solution is somewhat more involved and will not be discussed here.", see references for more on that
Related to 1969 Faro & Riffle Technique (Issue Faro Techniques) 16
Karl Fulves The 3n Deck properties related to the Triple Faro
 		1969 Faro & Riffle Technique (Issue The Triple Faro) 46
Karl Fulves Recycling The 3n Deck with the Triple Faro
 		1969 Faro & Riffle Technique (Issue The Triple Faro) 47
Karl Fulves Inverse Shuffles properties of the Triple Faro
 		1969 Faro & Riffle Technique (Issue The Triple Faro) 47
Karl Fulves Adjacencies problem of bringing two cards at random position together with faro shuffles
 		1970 Faro & Riffle Technique (Issue First Supplement) 53
Jean Marc Bujard La Magie du nombre cyclique 142857 using cards and dice, number is multiplied and performer places result on table using cards, oddity of the number 142857
 		1970 Hokus Pokus (Vol. 31 No. 3) 42
S. Brent Morris 1. The Ordinary Faro Shuffle mathematical properties
 		1974 Permutations by Cutting and Shuffling 4
S. Brent Morris 2. The Generalized Out Faro Shuffle
1974 Permutations by Cutting and Shuffling 10
S. Brent Morris 2. The Generalized In-Faro Shuffle
1974 Permutations by Cutting and Shuffling 22
S. Brent Morris 4. A Permutation Theorem
1974 Permutations by Cutting and Shuffling 27
S. Brent Morris 5. A Number Theoretic Result
1974 Permutations by Cutting and Shuffling 31
S. Brent Morris 6. Definitions and Examples
1974 Permutations by Cutting and Shuffling 38
S. Brent Morris 7. The Case N = 2
1974 Permutations by Cutting and Shuffling 41
S. Brent Morris 8. Decks With All k Equal
1974 Permutations by Cutting and Shuffling 49
S. Brent Morris 9. The General Case
1974 Permutations by Cutting and Shuffling 55
Edward Marlo 1835 Prediction card at chosen number is predicted, using 18-35 faro principle, three methods (duplicate card, equivoque, ..)
 		1975 Hierophant (Issue 7 Resurrection Issue) 55
Murray Bonfeld Novel Faro Relationships introducing mathematical language and some properties
  • Basic Terminology and Operations
  • For A 52 Card Deck Only
  • For A 51 Card Deck Only
 		1977 Faro Concepts 2
Murray Bonfeld Faro Functions further notations and properties
 		1977 Faro Concepts 8
Murray Bonfeld Even Number Of Cards relationships for decks with 2n cards
 		1977 Faro Concepts 8
Murray Bonfeld Faro Shuffle Recycling Table required number of in and out shuffles listed for a deck with two to 52 cards
 		1977 Faro Concepts 10
Murray Bonfeld Up And Down Faro System turning one half over before faro shuffling them together and how it affects the recycling properties
 		1977 Faro Concepts 11
Murray Bonfeld Unit Shuffles
Related to 1977 Faro Concepts 11
Murray Bonfeld Multiples Of Four relationships for decks with 4n cards
 		1977 Faro Concepts 12
Murray Bonfeld Odd Numbers Of Cards relationships for decks with 2n-1 cards
 		1977 Faro Concepts 13
Murray Bonfeld Unit Restorations
Related to 1977 Faro Concepts 16
Murray Bonfeld The 32-Card Deck: An Analysis twenty properties and relationships for a deck with 32 cards, some things also hold for a deck with 2n cards
 		1977 Faro Concepts 18
Murray Bonfeld The Principle of Internal Shuffling following groups and belts within a 52-card deck and how they behave under variations of in- and out-shuffles
  • Controlling 16 Cards Among 52
  • Controlling 10 Cards Among 52
  • Controlling 8 Cards Among 52
  • Inshuffle Groups
  • Odd Deck Technique
Related to 1977 Faro Concepts 27
Murray Bonfeld Any Card, Any Number - The First System shuffling card from position x to the top in odd deck, modified in-faro for even deck that ignored bottom card, reverse method for Alex Elmsley's Binary Translocation No. 1
Inspired byRelated to 1977 Faro Concepts 41
Murray Bonfeld Any Card, Any Number - The Second System bringing a card from position x to y with faro shuffling, odd deck, with even deck modified in-faro is required that ignores top card, generalization of Alex Elmsley's Binary Translocations
Related to 1977 Faro Concepts 42
Murray Bonfeld, Alex Elmsley Principles and Routines applications
Inspired by 1977 Faro Concepts 48
Murray Bonfeld More Theorems relationships when faros are combined with cuts in even deck
  • Cuts And Faros Combined
  • Shuffle Theorems
 		1977 Faro Concepts 52
Edward Marlo The 49 Control five cards
 		1979 Marlo's Magazine — Volume 3 363
Karl Fulves The Null Anti-Faro Restacking pack concept
 		1979 Faro Possibilities 4
Karl Fulves The Theoretical Faro definition of IO and OI as an entity and properties of IO- and OI-sequences
  • The Conjugate Pair Faro
  • The Inverted Conjugate Pair Faro
Related to 1979 Faro Possibilities 6
Karl Fulves The Null Faro an idea similar to Alex Elmsley's Restacking concept
Related to 1979 Faro Possibilities 8
Karl Fulves Utter Chaos some properties for decks with and odd number of cards
 		1979 Faro Possibilities 8
Karl Fulves, Steve Shimm Faro Shuffle Machines examining the problem of finding an algorithm to find a faro combination to shuffle from position x to position y, discussed with a 6-card deck
Related to 1979 Faro Possibilities 9
Roy Walton A Faro Tree examining the problem of finding an algorithm to find a faro combination to shuffle from position x to position y
Also published here 1979 Faro Possibilities 13
Karl Fulves The Tracking Faro stay stack type principle with two separate odd decks
 		1979 Faro Possibilities 17
Karl Fulves Solution to a Problem how to return to original order if a known sequence of in and out faros was performed
 		1979 Faro Possibilities 19
Karl Fulves The General Recycling Problem how to return to original order if an unknown sequence of in and out faros was performed
Related to 1979 Faro Possibilities 20
Karl Fulves The Missing Link relation of Milk Build Shuffle to faro
 		1979 Faro Possibilities 25
Karl Fulves (2) Primitive Cycles maintaining sequences that are repeated
Related to 1979 Faro Possibilities 27
Karl Fulves (3) The Half Faro faro applied to long-short deck, double faro
 		1979 Faro Possibilities 27
Karl Fulves (4) Faro/Stebbins bringing a thirteen-cards deck into Si Stebbins order with faros
 		1979 Faro Possibilities 28
Karl Fulves Interrogating the Deck bringing a card to top with faro shuffles
Related to 1979 Faro Possibilities 29
Murray Bonfeld Morray Bonfeld's Faro Program program for programmable calculator to find how many faros are required for recycling the order
Related to 1979 Interlocutor (Issue 29) 112
Karl Fulves Fake Shuffles fake faro shuffle and fake false shuffle with gaffed red/blue decks
 		1981 Octet 38
Roy Walton General Notes on Weave Shuffles
1981 The Complete Walton — Volume 1 194
T. Nelson Downs No Shuffle eight perfect shuffle recycle a deck
 		1985 The Fred Braue Notebooks (Issue 2) 14
Karl Fulves Least Totals six-card deck solution for problem in introduction
 		1986 The Return Trip 2
Karl Fulves Flotation Device another solution for problem in introduction
 		1986 The Return Trip 4
Karl Fulves Ring Diagrams
Related to 1986 The Return Trip 5
Karl Fulves A Catalog of Shuffles another solution for problem in introduction
 		1986 The Return Trip 6
Karl Fulves The Uniqueness Theory on the uniqueness of the order after a random in/out faro shuffle sequence
 		1986 The Return Trip 9
Karl Fulves Transpoker two poker hands, each Ace through Five in red and black, spectator names one of the values, performer shuffles the hands together and deals, named value is only odd-backed card in both hands, "transposition shuffle"
Related toVariations 1986 The Return Trip 11
Karl Fulves Time Bent Back what one knows about the last shuffle of an in/out faro shuffle sequence
 		1986 The Return Trip 13
Karl Fulves Separation Shuffles faro shuffle sequences that mix each half within itself, keeping them separated
Related to 1986 The Return Trip 14
Karl Fulves Singleton Shuffles "separation shuffles" that allow one card from both halves to transpose
 		1986 The Return Trip 16
Karl Fulves If Known another solution for problem in introduction if total number of shuffles is known
 		1986 The Return Trip 22
Karl Fulves Shuffle Diagrams
Related to 1986 The Return Trip 23
Karl Fulves The Stay Stak Constraint as stay stack features applies to problem in introduction
 		1986 The Return Trip 25
Karl Fulves Ring Subset
1986 The Return Trip 26
Karl Fulves How Many States?
1986 The Return Trip 27
Karl Fulves Basic Shuffle Equations how many shuffles it takes to get a deck back to original order
 		1986 The Return Trip 29
Karl Fulves Position Equations notation for faro shuffling
 		1986 The Return Trip 30
Karl Fulves Mix Relativity faro type from the point of view of the card
 		1986 The Return Trip 31
Karl Fulves Expanded Decks notation for faro shuffling
 		1986 The Return Trip 31
Karl Fulves Not in Descartes futile method of Cartesian notation
 		1986 The Return Trip 32
Karl Fulves Faro Trees "The faro tree gives a clear, unambiguous picture of what happens to the deck as it is shuffled."
 		1986 The Return Trip 33
Juan Tamariz Notes on the Faro and other Shuffles 1. On the supposed difficulty of the Faro
2. On the effects that can be performed with the Faro
3. On other uses
4. On subtleties, variations and new ideas
 		1989/91 Sonata 82
Juan Tamariz 1. To correct small errors
1989/91 Sonata 83
Peter Duffie A Far Out Faro Chart For Faro Fantasizers table in which can be seen which cards transpose in a single faro shuffle with packets from eight to fifty-two cards (like 18 <-> 35 in a full deck)
Inspired by
  • "Countdown to Purgatory" (Rod Ethtie, Al Smith's Abacus, Vol. 1 No. 11)
 1993 Card Selection 11
Alex Elmsley The Mathematics of the Weave Shuffle long article for "mathematicians" with the following subchapters
 		1994 The Collected Works of Alex Elmsley — Volume 2 302
Alex Elmsley The Odd Pack and Weave
1994 The Collected Works of Alex Elmsley — Volume 2 304
Alex Elmsley Equivalent Odd Pack
1994 The Collected Works of Alex Elmsley — Volume 2 304
Alex Elmsley Returning a Pack to the Same Order mathematical discussion
 		1994 The Collected Works of Alex Elmsley — Volume 2 305
Alex Elmsley Solving the Shuffle Equation how to find out number of shuffles required to return pack to same order
 		1994 The Collected Works of Alex Elmsley — Volume 2 306
Alex Elmsley Stack Transformations how faro shuffles affect a stack
 		1994 The Collected Works of Alex Elmsley — Volume 2 307
Alex Elmsley The Restacking Pack stack whose value distribution is not affected by faro shuffles
Related toVariations 1994 The Collected Works of Alex Elmsley — Volume 2 309
Alex Elmsley Binary Translocations
  • 1) to bring top card to any position with faros
  • 2) to bring card to top with 2^x cards
  • 3) variation of 2)
Related toVariations 1994 The Collected Works of Alex Elmsley — Volume 2 311
Alex Elmsley Penelope's Principle bringing center card to position corresponding with number of cards in cut-off pile
Related toVariations 1994 The Collected Works of Alex Elmsley — Volume 2 313
Alex Elmsley The Obedient Faro shuffling a card to any position up to twenty with two shuffles, for magicians
 		1994 The Collected Works of Alex Elmsley — Volume 2 346
T. Nelson Downs A Real Dovetail Shuffle observation that eight perfect (faro) shuffles restore order
 		1994 More Greater Magic 1084
T. Nelson Downs Four Perfect Riffle Shuffles to Restore Full-Deck Order no perfect faros, but blocks are released (riffle shuffle stacking type)
 		1994 More Greater Magic 1085
Ellison Poland Slough-Off Mathematics how top few cards move during Slough-Off Control
 		1994 Wonderful Routines of Magic — The Second Addendum 5
Claude Rix Subtilités avec Faro notes on faro an memorized deck
 		1995 Claude Rix et ses 52 partenaires 111
The Mathematical Basis of the Perfect Faro Shuffle
  • Mathematical Principles
 		1998 Card College — Volume 3 692
Pit Hartling Elimination - Faro Ordering removing cards so they can be ordered later with faro shuffles
 		2003 Card Fictions 22
Iain Girdwood Unicycle Stack values recycle after one shuffle
  • The 16 Card Unicycle Stack
  • The 30 Card Unicycle Stack
Inspired by 2003 Card Conspiracy — Vol. 2 68
César Fernández Lightning Divination thought card, number corresponding to value is removed from deck and card divined
Related to 2004 Semi-Automatic Card Tricks — Volume 5 189
Juan Tamariz Royal Location divining card in memorized stack after doing out-faros
Inspired by 2004 Mnemonica 142
Juan Tamariz A Special Idea: The Eight Mnemonicas
2004 Mnemonica 151
Jack Avis The Weave and Waterfall Bottom Palm "A Simple Action Palm"
 		2006 Rara Avis 184
Doug Edwards Faro Lap lapping card while cascading cards after faro shuffle
 		2006 Brass Knuckles 30
Denis Behr Faro and Anti-Faro Combination
2007 Handcrafted Card Magic 50
18/35 Principle
Related to 2008 Dexterity Manual 48
Calculating Positions after One Faro memorized deck
 		2012 Lessons in Card Mastery 32
Gary Plants, Richard Vollmer, Roberto Giobbi Seven position of selection in small packet is predicted, anti faro principle
Inspired by
  • "A Four-tunate Choice" (Gary Plants, Genii, Sep. 1997)
 2012 Confidences 177
Persi Diaconis, Ron Graham A Look Inside Perfect Shuffles Describes the mathematics of perfect faro shuffles, how to stack the deck using in and out shuffles
 		2012 Magical Mathematics 92
Persi Diaconis, Ron Graham All the Shuffles Are Related Explains how perfect faro shuffles, reverse faro shuffles, Monge shuffles, milk shuffles and down-under shuffles are related
 		2012 Magical Mathematics 99
Mahdi Gilbert Dueling Pianos Handling for the Piano Card Trick, bringing in a subtlety from Thieves & Sheep
Inspired by
  • "Piano Card Trick" (Uncredited, Stanyon's Magic, Aug. 1902)
Related to 		2015 Semi-Automatic Card Tricks — Volume 9 194
Pepe Lirrojo A.C.A.A.N. Teórico
Inspired by 2016 Panpharos 47
Alex Elmsley Faro Fan
2018 Solomon's Secrets 85
Ryan Murray Properties of the faro shuffle
  • Cycling Order
  • Controlling a Card to any Position
 		2018 Curious Weaving x
Steve Forte The Faro Process general comments and rules regarding faro stacking
 		2020 Gambling Sleight Of Hand — Volume 1 258
Greg Chapman Eight Perfect Out-Faros
2020 Faro Fundamentals 25
Greg Chapman Fifty-two Perfect In-Faros
2020 Faro Fundamentals 25
Greg Chapman Calculating the New Position for Any Card After Each Out-Faro
2020 Faro Fundamentals 27
Greg Chapman Out-Faro Charts
2020 Faro Fundamentals 28
Greg Chapman Six Belts
Related to 2020 Faro Fundamentals 29
Greg Chapman Fixed Floating Key Cards in a Stacked Deck 18-35 principle
    • High Card
    • Hold'em
  • Work in the Cards
  • Other Positions
 		2020 Faro Fundamentals 30
Greg Chapman In-Faro Calculations: Calculating the new position for any card after each in-faro
2020 Faro Fundamentals 37
Greg Chapman 51-Card Faro allowing cuts
 		2020 Faro Fundamentals 38