Mathematical Facts & Curiosities

152 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."
Also published here 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 Work in Progress faro mathematics, defining Out- and In-Faro, then the Binary Translocations:
  • 1) to bring top card to any position with faros
  • 2) to bring card to top with 2^x cards
  • 3) edge-marked deck with 2^x cards, bringing any card to top
Related toAlso published here Sep. 1957
Ibidem (Issue 11)
21
Edward Marlo The Backward Faro out-jogging every other card, with some properties, also for setting up for a "false" shuffle
Related toVariations Dec. 1957
Ibidem (Issue 12)
4
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
Related to 1958
Faro Notes
12
Russell "Rusduck" Duck Faro Favorites Elmsley's Restacking Pack
Related to July 1958
The Cardiste (Issue 10)
14
Russell "Rusduck" Duck Perma-Stack based on Elmsley's Restacking Pack idea
Related to July 1958
The Cardiste (Issue 10)
15
Mel Stover MelMath on mixing/controlled shuffling, especially with faro shuffles
Related to Mar. 1958
Ibidem (Issue 13)
9
Ronald A. Wohl (Ravelli) On Binary Translocations on having a formula for getting a card to the top with faro shuffles with not 2
Related to Mar. 1958
Ibidem (Issue 13)
10
Ronald A. Wohl (Ravelli) Tips from Ravelli: Binary Translocations not necessary to shuffle perfect faros in certain cases
Related to Sep. 1958
Ibidem (Issue 14)
7
Edward Marlo Calculating Removed Card from memorized deck, card removed and eight faros shuffled in total
Related to Dec. 1958
Ibidem (Issue 15)
19
Russell "Rusduck" Duck Rusduck Thinks on the reverse faro
  • suit separation by going through the deck only once
  • reverse faros to pre-set a stack
  • poker deal application
Inspired byRelated to Oct. 1959
Ibidem (Issue 18)
19
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
The Hierophant (Issue 7 Resurrection Issue)
55
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."
Also published here 1975
Charles T. Jordan: Collected Tricks
131
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
Ronald A. Wohl (Ravelli) The Mathematics of the Weave Shuffle "A Study by Ravelli"
  • Introduction
  • 1. The Terminology
  • 2. The Out Weave Equation
  • 3. To Return a Pack to the Original Order with Out Weaves
    • a) An obvious solution
    • b) The Theorem of Fermat
    • c) A Binomial Pattern Approach
    • d) Factor Analysis
  • 4. To Reverse a Pack with Out-Weaves
    • a) By Calculating and By Using the Binomial System
    • b) Factor Analysis
    • c) An Obvious Solution
  • 5. The General In-Weave Equation
  • 6. To Return a Pack to the Original Order with In-Weave
  • 7. To Reverse a Pack with In-Weaves
  • 8. Summary of the Out- and In-Shuffle
  • 9. The Odd-Pack Weave
    • How to Make an Odd Pack out Of An Even Pack
    • Top and Bottom Odd Weaves
  • 10. Card at n-Fold Places
    • Partial n-fold place
    • Cards at 1/n places
  • 11. In and Out Odd-Weaves
  • 12. The Behavior of a Difference in Faro Shuffles
  • 13. The Mathematics of the Double, Triple and Multiple Faro
 		Mar. 1979
Ibidem (Issue 36)
2
Ronald A. Wohl (Ravelli) The Memorized Faro Shuffle Pack calculating the order after faro shuffling a memorized deck, from one to eight shuffles
Related to Mar. 1979
Ibidem (Issue 36)
23
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
Related to 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) edge-marked deck with 2^x cards, bringing any card to top
Related toVariationsAlso published here 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