The document discusses different methods of shuffling cards, including pile shuffling, overhand shuffling, and riffle shuffling. It analyzes the effectiveness of each method at randomizing cards based on mathematical models. Pile shuffling is shown to not actually randomize cards. Overhand shuffling is a true randomization method but requires too many movements to be practical. Riffle shuffling is effective if done more than 1.5 log2(n) times, where n is the number of cards, making it the best practical shuffling method.

1. Everyday I'm Shuffling
Judge Conference – GP Madrid
14/11/2014
by Pedro Gonçalves, L2

2. Table of Contents
● What “shuffling” means
● Pile Shuffling
● Overhand Shuffling
● Riffle Shuffling
● Musings on shuffling
● Bibliography
● Acknowledgements
Judge Conference – GP Madrid
14/11/2014

3. Shuffling
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4. Shuffling
● CR 701.16a
“To shuffle a library or a face-down pile of cards,
randomize the cards within it so that no player knows
their order.”
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5. Shuffling
● CR 701.16a
“To shuffle a library or a face-down pile of cards,
randomize the cards within it so that no player knows
their order.”
● How can we guarantee the randomness of the cards in a
library?
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6. Pile Shuffling
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7. Pile Shuffling
● Cards are simply dealt out into a number of piles, then the
piles are stacked on top of each other.
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8. Pile Shuffling
● Cards are simply dealt out into a number of piles, then the
piles are stacked on top of each other.
● It does NOT randomize the cards, the process is
completely deterministic.
Judge Conference – GP Madrid
14/11/2014

9. Pile Shuffling
● Cards are simply dealt out into a number of piles, then the
piles are stacked on top of each other.
● It does NOT randomize the cards, the process is
completely deterministic.
● May be useful to check the deck's legality (by counting
the cards).
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10. Pile Shuffling
● Cards are simply dealt out into a number of piles, then the
piles are stacked on top of each other.
● It does NOT randomize the cards, the process is
completely deterministic.
● May be useful to check the deck's legality (by counting
the cards).
● Can be used by less honest players to try to cheat – for
example, using the so called “Double Nickel”.
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11. Pile Shuffling – Double Nickel
● You “shuffle” in 5 piles of cards for 2 times, starting with a
deck with all the lands/spells previously stacked.
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12. Pile Shuffling – Double Nickel

13. Pile Shuffling – Double Nickel

14. Pile Shuffling – Double Nickel

15. Pile Shuffling – Double Nickel

16. Pile Shuffling – Double Nickel

17. Pile Shuffling – Double Nickel

18. Pile Shuffling – Double Nickel
● You “shuffle” in 5 piles of cards for 2 times, starting with a
deck with all the lands/spells previously stacked.
● Lands and spells are distributed in such a way to avoid
'mana screw' and 'mana flood', regardless of where the
deck is cut.
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19. Pile Shuffling – Double Nickel
● You “shuffle” in 5 piles of cards for 2 times, starting with a
deck with all the lands/spells previously stacked.
● Lands and spells are distributed in such a way to avoid
'mana screw' and 'mana flood', regardless of where the
deck is cut.
● It looks like there is no discernible pattern, but it is clear
that there was a library manipulation.
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20. Overhand Shuffling
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21. Overhand Shuffling
● A group of cards on the bottom (or top)
of the deck is lifted sideways out of the
deck, and then placed on the top (or
bottom).
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22. Overhand Shuffling
● A group of cards on the bottom (or top)
of the deck is lifted sideways out of the
deck, and then placed on the top (or
bottom).
● Is this really an actual randomization
method?
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23. Overhand Shuffling
● A group of cards on the bottom (or top)
of the deck is lifted sideways out of the
deck, and then placed on the top (or
bottom).
● Is this really an actual randomization
method? YES!
● Jonasson (2006) proved this is an actual randomization
method.
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24. Overhand Shuffling
● A group of cards on the bottom (or top)
of the deck is lifted sideways out of the
deck, and then placed on the top (or
bottom).
● Is this really an actual randomization
method? YES!
● Jonasson (2006) proved this is an actual randomization
method.
● However, the same paper concludes that, in order to
shuffle a deck with n cards, we need approximately n2
log(n) movements.
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25. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
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26. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
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27. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
n 40
n2 log(n)
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28. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
n 40
n2 log(n) 5902.207
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29. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
n 40 60
n2 log(n) 5902.207
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30. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
n 40 60
n2 log(n) 5902.207 14739.640
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31. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
n 40 60 99
n2 log(n) 5902.207 14739.640
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14/11/2014

32. Overhand Shuffling
● However, the same paper concludes
that, in order to shuffle a deck with n
cards, we need approximately n2 log(n)
movements.
● In practice, this is not an effective
shuffling method:
n 40 60 99
n2 log(n) 5902.207 14739.640 45036.767
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33. Riffle Shuffling
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34. Riffle Shuffling
● Half of the deck is held in
each hand with the
thumbs inward, then cards
are released by the
thumbs so that they fall to
the table interleaved.
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35. Riffle Shuffling
● Half of the deck is held in
each hand with the
thumbs inward, then cards
are released by the
thumbs so that they fall to
the table interleaved.
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14/11/2014

36. Riffle Shuffling
● Half of the deck is held in
each hand with the
thumbs inward, then cards
are released by the
thumbs so that they fall to
the table interleaved.
● There's a risk of damage to the cards –
casinos replace their cards often, but
Magic cards (as well as other TCGs'
cards) are less replaceable.
Judge Conference – GP Madrid
14/11/2014

37. Riffle Shuffling
● Half of the deck is held in
each hand with the
thumbs inward, then cards
are released by the
thumbs so that they fall to
the table interleaved.
● There's a risk of damage to the cards –
casinos replace their cards often, but
Magic cards (as well as other TCGs'
cards) are less replaceable.
Judge Conference – GP Madrid
14/11/2014

38. Riffle Shuffling - GSR
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39. Riffle Shuffling - GSR
● The Gilbert-Shannon-Reeds model, as described in
Gilbert (1955) and Reeds (1981), is a good model for how
people usually riffle shuffle. (Diaconis,1988)
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40. Riffle Shuffling - GSR
● The Gilbert-Shannon-Reeds model, as described in
Gilbert (1955) and Reeds (1981), is a good model for how
people usually riffle shuffle. (Diaconis,1988)
● The deck is split into 2 halves, choosing the cut point with
according to a binomial distribution.
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41. Riffle Shuffling - GSR
● The Gilbert-Shannon-Reeds model, as described in
Gilbert (1955) and Reeds (1981), is a good model for how
people usually riffle shuffle. (Diaconis,1988)
● The deck is split into 2 halves, choosing the cut point with
according to a binomial distribution.
● Choose a card from the bottom of one half of the deck,
with a probability that is proportional to the number of
cards in that half – if the 1st half has A cards and the 2nd
half has B cards, the probability of getting a card from the
1st half is A/(A+B).
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42. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
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43. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point':
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44. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point': 1.5 log2(n)
Judge Conference – GP Madrid
14/11/2014

45. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point': 1.5 log2(n)
n 40 60 99
1.5 log2(n)
Judge Conference – GP Madrid
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46. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point': 1.5 log2(n)
n 40 60 99
1.5 log2(n) 7.982
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47. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point': 1.5 log2(n)
n 40 60 99
1.5 log2(n) 7.982 8.860
Judge Conference – GP Madrid
14/11/2014

48. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point': 1.5 log2(n)
n 40 60 99
1.5 log2(n) 7.982 8.860 9.944
Judge Conference – GP Madrid
14/11/2014

49. Riffle Shuffling - GSR
● How can we measure the effectiveness of a riffle shuffle?
● Aldous (1983) arrives at a 'cutoff point': 1.5 log2(n)
n 40 60 99
1.5 log2(n) 7.982 8.860 9.944
● Notice that players don't usually shuffle that much.
(Although they should!)
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50. Riffle Shuffling - GSR
● In the remarkable paper Trailing the Dovetail Shuffle to its
Lair (1992), Bayer and Diaconis improved Aldous's result.
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51. Riffle Shuffling - GSR
● In the remarkable paper Trailing the Dovetail Shuffle to its
Lair (1992), Bayer and Diaconis improved Aldous's result.
● The total variation distance, ||Qm – U||, between the GSR
and the uniform distributions is:
Judge Conference – GP Madrid
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52. Riffle Shuffling - GSR
● In the remarkable paper Trailing the Dovetail Shuffle to its
Lair (1992), Bayer and Diaconis improved Aldous's result.
● The total variation distance, ||Qm – U||, between the GSR
and the uniform distributions is:
with
n 〈 n
∥Qm−U∥=Σr=1
r−1〉∣(2m+n−r
n )
2nm − 1
n!∣
〈 n
r
(−1) j (n+1
r−1〉==0
j
Σj )(r− j )n
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53. Riffle Shuffling – ||Qm – U|| for 40 cards
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54. Riffle Shuffling – ||Qm – U|| for 40 cards
1 2 3 4 5 6 7
1 1 1 0.9909 0.7688 0.444 0.2230
8 9 10 11 12 13 14
0.1153 0.0582 0.0292 0.0146 0.0073 0.0036 0.0018
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55. Riffle Shuffling – ||Qm – U|| for 60 cards
1 2 3 4 5 6 7
1 1 1 1 0.9764 0.7132 0.4062
8 9 10 11 12 13 14
0.2069 0.1052 0.0530 0.0266 0.0133 0.0066 0.0033
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56. Riffle Shuffling – ||Qm – U|| for 99 cards
1 2 3 4 5 6 7
1 1 1 1 1 0.9801 0.7401
8 9 10 11 12 13 14
0.4260 0.2200 0.1115 0.0556 0.0277 0.0138 0.0069
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57. Musings on shuffling – Technical details
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58. Musings on shuffling – Technical details
● Pile shuffling is an isomorphism. From Wikipedia:
“isomorphic objects may be considered the same as long
as one considers only these properties and their
consequences.”
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14/11/2014

59. Musings on shuffling – Technical details
● Pile shuffling is an isomorphism. From Wikipedia:
“isomorphic objects may be considered the same as long
as one considers only these properties and their
consequences.”
● Can we do a random pile shuffle?
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14/11/2014

60. Musings on shuffling – Technical details
● Pile shuffling is an isomorphism. From Wikipedia:
“isomorphic objects may be considered the same as long
as one considers only these properties and their
consequences.”
● Can we do a random pile shuffle?
● Some people can perform a perfect riffle shuffle, which is
equally useless for shuffling purposes!
Judge Conference – GP Madrid
14/11/2014

61. Musings on shuffling – Technical details
● Pile shuffling is an isomorphism. From Wikipedia:
“isomorphic objects may be considered the same as long
as one considers only these properties and their
consequences.”
● Can we do a random pile shuffle?
● Some people can perform a perfect riffle shuffle, which is
equally useless for shuffling purposes!
● Other potential problems with riffle shuffling: bottom/top
cards, clumps due to humidity (!)
Judge Conference – GP Madrid
14/11/2014

62. Musings on shuffling – Technical details
● Pile shuffling is an isomorphism. From Wikipedia:
“isomorphic objects may be considered the same as long
as one considers only these properties and their
consequences.”
● Can we do a random pile shuffle?
● Some people can perform a perfect riffle shuffle, which is
equally useless for shuffling purposes!
● Other potential problems with riffle shuffling: bottom/top
cards, clumps due to humidity (!)
● Actual randomness vs. de facto randomness
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63. Musings on shuffling – General details
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64. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
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65. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
● Mash shuffling can be considered a rough approximation
to a riffle shuffle.
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14/11/2014

66. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
● Mash shuffling can be considered a rough approximation
to a riffle shuffle.
● Less shuffling effects, better sleeves, teaching the players
how to shuffle properly.
Judge Conference – GP Madrid
14/11/2014

67. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
● Mash shuffling can be considered a rough approximation
to a riffle shuffle.
● Less shuffling effects, better sleeves, teaching the players
how to shuffle properly.
● Could Pile shuffle be considered Slow Play?
Judge Conference – GP Madrid
14/11/2014

68. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
● Mash shuffling can be considered a rough approximation
to a riffle shuffle.
● Less shuffling effects, better sleeves, teaching the players
how to shuffle properly.
● Could Pile shuffle be considered Slow Play?
● World Record for memorizing a 52-card deck:
Judge Conference – GP Madrid
14/11/2014

69. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
● Mash shuffling can be considered a rough approximation
to a riffle shuffle.
● Less shuffling effects, better sleeves, teaching the players
how to shuffle properly.
● Could Pile shuffle be considered Slow Play?
● World Record for memorizing a 52-card deck: 21,19s!
Judge Conference – GP Madrid
14/11/2014

70. Musings on shuffling – General details
● Is Riffle shuffle practical enough for tournaments?
● Mash shuffling can be considered a rough approximation
to a riffle shuffle.
● Less shuffling effects, better sleeves, teaching the players
how to shuffle properly.
● Could Pile shuffle be considered Slow Play?
● World Record for memorizing a 52-card deck: 21,19s!
● Should Battle of Wits be legal?
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14/11/2014

71. Bibliography
● Flores, Micheal J. (2009). How to Cheat. Five with Flores.
http://fivewithflores.com/2009/05/how-to-cheat/ Retrieved
05/09/2014
● Jonasson, J. (2006). The Overhand Shuffle Mixes in O(n2
log(n)) Steps. The Annals of Applied Probability, vol.16,
#1, pp. 231-243
● Gilbert, E. (1955) Theory of Shuffling. Technical
memorandum, Bell Laboratories
● Reeds, J. (1981) Unpublished manuscript.
● Diaconis, P.(1988) Group Representations in Probability
and Statistics. IMS, Hayward, California

72. Bibliography
● Aldous, D. (1983). Random walk on finite groups and
rapidly growing Markov Chains. Semináire de Probabilités
XVII. Lecture Notes in Math. 986 pp.243-297. Springer
● Bayer, D. Diaconis, P. (1992) Trailing the Dovetail Shuffle
to its Lair. The Annals of Applied Probability, vol.2, #2, pp.
294-313
● Levin, D. Peres, Y. Wilmer, E (2008). Markov Chains and
Mixing Times. American Mathematical Society
● http://en.wikipedia.org/wiki/Shuffling Retrieved 12/11/2014
Judge Conference – GP Madrid
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73. Acknowledgements
● To the Portuguese Judges, for listenening patiently to an
earlier version of this presentation.
● To Frederico Bastos, Frank Karsten and Luís Gobern, for
providing some feedback from a player's perspective.
● To the organizers of this conference, for the opportunity to
give the presentation.
● To everyone in the audience, for your attention.
Judge Conference – GP Madrid
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THANK YOU!!!