Mathematical Facts & Curiosities
108 entries in CategoriesCardsSleightsShuffles (non-riffle)Faro ShuffleMathematical Facts & Curiosities
Search within these entries:
Creators Title Comments & References Year Source Page CARC Categories
Unknown The Eighteenth Card using 18-35-faro-principle with honest shuffle, risky 1940 Expert Card Technique 150
Fred Black The Shuffle faro tables see also "A Correction" (Edward Marlo, 1958) 1940 Expert Card Technique 145
Fred Black The Endless Belts see also "A Correction" (Edward Marlo, 1958),
"Ring Diagrams" (Karl Fulves, 1986)
1940 Expert Card Technique 147
Fred Black Chart of Seventeen see also "A Correction" (Edward Marlo, 1958) 1940 Expert Card Technique 147
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
Edward Marlo A Correction commentary on ECT tables, see also new hardcover edition for further commentary inspired by "The Shuffle" (Fred Black, 1940) 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 see also "The Restacking Pack" (Alex Elmsley, 1994) 1958 The Cardiste (Issue 10) 14
Russell "Rusduck" Duck Perma-Stack based on Elmsley's Restacking Pack idea see also "The Restacking Pack" (Alex Elmsley, 1994) 1958 The Cardiste (Issue 10) 15
Alex Elmsley In and Out Definition 1958 The Faro Shuffle 1
Alex Elmsley In and Out Shuffle Definition 1958 Faro Notes 1
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 "The Restacking Pack" (Alex Elmsley, 1994) 1964 Faro Controlled Miracles 18
Unknown 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 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 as "A Faro Tree" (Roy Walton, 1979) 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 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 "The Restacking Pack" (Alex Elmsley, 1994) 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 (Section "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 (Section "Faro Techniques") 2
Karl Fulves Position Determination following a card's position during in and out faros 1969 Faro & Riffle Technique (Section "Faro Techniques") 3
Karl Fulves The Triple Faro Ring 1969 Faro & Riffle Technique (Section "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 (Section "Faro Techniques") 7
Karl Fulves The Fourth-Order Deck discussing transpositions of two cards within the deck 1969 Faro & Riffle Technique (Section "Faro Techniques") 12
Karl Fulves Three Way Transposition note 1969 Faro & Riffle Technique (Section "Faro Techniques") 15
Karl Fulves Fractional Transforms note 1969 Faro & Riffle Technique (Section "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 see also "The General Recycling Problem" (Karl Fulves, 1979),
"Introduction" (Karl Fulves, 1986)
1969 Faro & Riffle Technique (Section "Faro Techniques") 16
Karl Fulves The 3n Deck properties related to the Triple Faro 1969 Faro & Riffle Technique (Section "The Triple Faro") 46
Karl Fulves Recycling The 3n Deck with the Triple Faro 1969 Faro & Riffle Technique (Section "The Triple Faro") 47
Karl Fulves Inverse Shuffles properties of the Triple Faro 1969 Faro & Riffle Technique (Section "The Triple Faro") 47
Karl Fulves Adjacencies problem of bringing two cards at random position together with faro shuffles 1970 Faro & Riffle Technique (Section "First Supplement") 53
Edward Marlo 1835 Prediction card at chosen number is predicted, using 18-35 faro principle, three methods (duplicate card, equivoque, ..) 1975 Hierophant (Section "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 2-52 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 see also "The Theoretical Faro" (Karl Fulves, 1979) 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 see also "The Theoretical Faro" (Karl Fulves, 1979) 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
see also "Other Forms Of The Transposition" (Karl Fulves, 1969),
"(2) Primitive Cycles" (Karl Fulves, 1979)
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 by "Binary Translocations" (Alex Elmsley, 1994),
see also "Beginning Again" (William Zavis, 1970)
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 see also "Binary Translocations" (Alex Elmsley, 1994) 1977 Faro Concepts 42
Murray Bonfeld & Alex Elmsley Principles and Routines applications inspired by "Penelope's Principle" (Alex Elmsley, 1994) 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
see also "Unit Restorations" (Murray Bonfeld, 1977),
"Unit Shuffles" (Murray Bonfeld, 1977)
1979 Faro Possibilities 6
Karl Fulves The Null Faro an idea similar to Alex Elmsley's Restacking concept see also "The Restacking Pack" (Alex Elmsley, 1994) 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 see also "Morray Bonfeld's Faro Program" (Murray Bonfeld, 1979) 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 as "A Faro Tree" (Roy Walton, 1967) 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 see also "The Recycling Problem" (Karl Fulves, 1969),
"Introduction" (Karl Fulves, 1986)
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 see also "Other Forms Of The Transposition" (Karl Fulves, 1969),
"The Principle of Internal Shuffling" (Murray Bonfeld, 1977)
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 see also "The Interrogation Technique" (Karl Fulves, 1970) 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 see also "Faro Shuffle Machines" (Karl Fulves & Steve Shimm, 1979) 1979 Interlocutor (Issue 29) 112
Karl Fulves Fake Shuffles fake faro shuffle and fake false shuffle with gaffed red/blue decks 1981 Octet 38
Steve Beam False Faro diagonal pressure and swivel cut see also "The Trapdoor Issue #4" (Steve Beam, 2011) 1984 The Trapdoor - Volume One (Issue 4) 59
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 see also "The Endless Belts" (Fred Black, 1940),
"Shuffle Diagrams" (Karl Fulves, 1986)
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" see also "Unit Transpo" (Karl Fulves, 1970),
"Shuttle Shuffle" (Karl Fulves, 1973),
varied by "Transpoker III" (Karl Fulves, 1986),
"Transpoker II" (Karl Fulves, 1986)
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 see also "Carbon Copy" (Karl Fulves, 1973) 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 see also "Ring Diagrams" (Karl Fulves, 1986) 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
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 see also "Perma-Stack" (Russell "Rusduck" Duck, 1958),
"Faro Favorites" (Russell "Rusduck" Duck, 1958),
"The Null Faro" (Karl Fulves, 1979),
"Primitive Cycles" (Karl Fulves, 1986),
varied by "On the Re-Stacking Pack" (Edward Marlo, 1964),
"Marlo Re-Stacking Pack" (Edward Marlo, 1969),
"The Permanent Deck Principle" (Woody Aragon, 2011)
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)
see also "Faro as a Control" (Edward Marlo, 1958),
"Oil Always Floats" (Paul Swinford, 1971),
"The Core" (Pit Hartling, 2016),
varied by "Any Card, Any Number - The First System" (Murray Bonfeld, 1977)
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 see also "Principles and Routines" (Murray Bonfeld & Alex Elmsley, 1977) 1994 The Collected Works of Alex Elmsley - Volume 2 313
Alex Elmsley The Obedient Faro shuffling a card to any position up to 20 with 2 shuffles, for magicians 1994 The Collected Works of Alex Elmsley - Volume 2 346
T. Nelson Downs A Real Dovetail Shuffle observation that 8 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
Unknown The Mathematical Basis of the Perfect Faro Shuffle listing of 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
Denis Behr Faro and Anti-Faro Combination 2007 Handcrafted Card Magic 50
Unknown 18/35 Principle see also "The Eighteenth Card" (1940) 2008 Dexterity Manual 48
Unknown Calculating Positions after 1 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 2012 Confidences 177
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) Add a reference,
see also "Thieves and Sheep" (Lillian Bobo, 1952)
2015 Semi-Automatic Card Tricks - Volume 9 194