(Metagame Archive) Strength in Numbers, Shuffling, Part 1: Stopping Shady Shuffles

By Olav Rokne

From my experience at large tournaments, the more obsessively I shuffle my deck, the better I do. If I were of a superstitious bent, this correlation could be attributed to some generation of positive karma through good sportsmanship. A more practical, analytical explanation is that the greater the number, variety, and randomness of shuffles performed on a deck, the greater the card motility (in other words, the less clumping you’ll experience).*

Probably the most common shuffling technique used in trading card games such as Vs. System is the riffle shuffle. When performing this shuffle, a player will split his or her deck into two roughly equal portions. Holding one of these portions in each hand, the player will lift up a corner of each and fan them together in such a way that the two piles are brought together by interspersing cards from one pile with cards from the other. To prevent stacking the top or bottom card of the deck, this should be done so that when the two portions are fanned together, the top card of the portion that started out on top is shuffled below the top card of the portion that started out as the lower half of the deck.  

This is the meat and potatoes of shuffling, and its importance cannot be underestimated, because small errors in the size of the two portions and variations in the speed of the fanning together provide the fastest and most truly random method of shuffling available to the average card gamer.  

The riffle shuffle, however, has several key flaws that can be pitfalls to those who rely on it exclusively and can be exploited by cheaters.

The largest problem with this technique is that while absolute card motility between regions of the deck is high in a riffle shuffle, relative motility between neighbors is not. As an example, take a sixty-card deck in which the cards have been numbered sequentially, with the top card as #1 and the bottom card as #60. If you perform a perfect, no-errors, mechanical riffle shuffle in a sixty-card deck, you will end up with an order something like: 31, 1, 32, 2, 33, 3, and so on. In this example, although card #32 has moved within the deck twenty-nine absolute spots, it has only moved one spot away from its previous neighbors. The portions of the deck that have the highest card motility (both relative and absolute) are those closest to the top, to the bottom, and to the cut. Portions in the middle see the least motility.  

Unscrupulous players who are incredibly clever can perform this shuffle perfectly and accomplish what is known as a faro shuffle, which if done a specific number of times in a row, will return the cards to their original order.§

The mathematician and magician Persi Diaconis once calculated that a deck does not even begin to become truly random until five riffle shuffles have been performed. To randomize properly, one should perform a riffle shuffle seven times. If your shuffling technique is not up to snuff, it will likely take upward of twelve such shuffles to produce an acceptably randomized pile of cards. An example of poor shuffling technique in the riffle shuffle would be when a player flips multiple cards together (rather than single cards) while shuffling the two halves of the deck together.  

A popular solution to this lack of relative card motility is what is known as the pile shuffle. In this shuffle, you take your deck and deal out multiple piles (usually six or seven) from the top of the deck onto a table before gathering these piles up and stacking them back together as a deck. Using the earlier model of a numbered 60-card deck, if you deal out six piles, you would end up with piles that have orders of 1, 7, 13, and 19; 2, 8, 14, and 18; and so on.

While this may produce desirable relative card motility (and depending on the order in which you replace the piles, an equally enviable absolute card motility), the pile shuffle also has problems.  

The first is that it is entirely mathematically predictable, and predictability is anathema to a good shuffle. Since there is no way to introduce randomness to an uninterrupted string of pile shuffles, you should never use this method exclusively.
The second problem is that the relative card motility happens in preordained increments. That is to say that cards will always be exactly six slots apart.
A thoroughly clever London-based Vs. player and statistician named Ian Vincent worked out a solution to the second problem. This involves a pile shuffle based on co-primes, wherein the deck is dealt out into seven piles from left to right, and then the two piles on the right are dealt out again from right to left. This process is repeated in alternating directions until the deck is out; then the deck is reassembled, starting with one of the center three stacks and following with the stack three to the right of the previous one.
The advantage of this method is that if it’s performed between a series of riffle shuffles, it gives a wide range of relative card motility values; a card can move zero, two, three, four, five, or seven spots from its original location within the deck with relative ease.
A rarely used shuffling technique is called a wash shuffle. It’s an easy one to learn, and although it can damage card sleeves and requires a large and unoccupied table, it does produce a pleasantly chaotic card distribution.
During a wash, you spill your deck face down over a table and just slide the cards around, pushing them this way and that before trying to reassemble the deck. There’s rarely the time or space to do this in your average tournament, but it’s a good one to try out once in a while.**
If you have failed to shuffle properly, your draws will suffer. Let me illustrate with an example in which someone is playing a simple deck with 30 “A” cards and 30 “B” cards. After sorting the deck, the plot twists are at the bottom of the pile and the characters are on the top. The player riffle shuffles two times and pile shuffles once with a standard six-pile sort.
The first riffle shuffle will likely order the deck into a pattern of A, B, A, B, A, B. The second riffle will then fold those into A, A, B, B, A, A, B, B. These patterns will occur through most of the deck no matter where you cut the deck before riffle shuffling. In the third iteration, you will either cut through the pattern in the sequence between “A” and “B” or between a pair of the same kind of card (As or Bs).
If you end up in the first situation, you’ll have an A, B, A, B, A, B pattern again throughout large portions of the deck. In the second instance, you’ll end up with a pattern of A, A, A, A, B, B, B, B throughout large portions of the deck, which is somewhat more problematic than the first scenario.
In the second eventuality, when you then pile shuffle, you’ll end up with several sequences in the deck that are approximately ordered A, A, A, A, B, A, B, B, B, B, A, B.
Thus, if you see the first four cards of your deck after this shuffling sequence, you can reasonably predict the next set of the sequence of cards—and predictability is anathema to a good shuffle. Unless you have pre-planned this shuffle (at which point it’s cheating) and started out with a very specific order of cards (which is incredibly difficult and utterly immoral), the fact that you know the sequence of plot twists to characters will not be advantageous because several copies of the same card will remain together within the sequence.
With four riffle shuffles and one six-pile shuffle, things actually get worse. There is a statistically significant chance (approximately one in forty††) that the first eight cards of the deck will be all “A” or all “B.”
If we extend this analogy to a Vs. deck, that’s a 1 in 40 chance of a near-automatic loss. In a perfectly random deck, in which every card has an equal chance of being in any particular position, the chances of this happening are fewer than 1 in 300.
That means that over the course of the first day of a Pro Circuit, if everyone shuffles perfectly, there will be about five people who have a complete deck failure once.
In most casinos that don’t use mechanical shuffling, you’ll find that the wash, along with other shuffling techniques, is used. Often the dealer will wash, riffle twice, perform a series of cuts, riffle again, and make a final cut.
The most orderly your deck will ever be is immediately after you have written out your decklist. Immediately after you have done your decklist, you should perform a minimum of seven riffle shuffles interspersed with a co-primes pile shuffle or two.
The best way to foil deck stacking is to give your opponent’s deck a shuffle at every opportunity. Without fail, do one riffle shuffle every time he or she has randomized the deck; cutting your opponent’s deck is not sufficient, as there are techniques to shuffle important cards to the middle of a deck to fool the cut. Often, the same miserable wretches who use false shuffles will use sleazy gamesmanship tactics to avoid their opponent’s cut or shuffle. This can include but is not limited to:
Drawing a hand before giving you a chance to shuffle the deck, then disingenuously making a false claim that you had passed on the opportunity to do so.
Holding the deck in hand as he or she offers you the opportunity to cut, thus making it likelier that you just take the top half as your cut (and avoid a shuffle).
Tapping your deck in a friendly gesture that indicates that he or she “doesn’t need to cut your deck,” and then assuming that you will do the same.
Some decks, such as New School, EMS, GLEE, Fantastic Fun, and Both Guns Blazing, have enough search effects that opponents will get tired of always shuffling the deck and stop bothering.
There are several things to stay on guard for when watching your opponent shuffle. First off, watch your opponent’s hands; if your opponent repeatedly shuffles so that the original top pile is always fanned in last, the top card is staying on top. Secondly, if your opponent uses nothing but simple pile shuffles, stay on guard and make an extra effort to shuffle well.
Oh, and be on the lookout for people wearing Sunderland Football Club jerseys.
Olav Rokne

olavrokne(at)gmail.com
For further reading, I would suggest:
Diaconis, Persi. “Analysis of Top to Random Shuffles.” Combinatorics, Probability

 Computing 1: 135-55.
Diaconis, Persi. “Riffle Shuffles, Cycles and Descents.” Combinatorica 15: 11-29.
Erdnase, S. W., An Expert at the Card Table
* Card motility is a term used to define the ability of a card to travel from one point in a deck to any other point in a deck within a shuffle. This should be looked at in two ways: first, in terms of a card’s ability to travel in relation to its nearest neighbors, and secondly, in terms of absolute position within the sixty cards.


Many of whom, in my humble opinion, cheer for Sunderland Football Club, which is an inferior and unpleasant soccer club that no real Geordie would cheer for. I’m not sure if any Sunderland supporters actually play Vs. System, but if they do, this Newcastle United fan is dropping the gauntlet.


Such as those who would support the Sunderland FC.

§ Thankfully, the only player I know of who is dexterous and clever enough to pull off this feat is not a Sunderland supporter. He is, in fact, a fellow of unmitigated honesty.


I don’t fully understand the mathematics of this, but he explained that it is based on a combination of statistics, co-primes, and the teachings of an ascended, illuminated Shaolin master he studied with in Nepal.


The seven piles represent the seven chakras (or plexuses) of the human body in Cabbalist teaching, the fact that there are three piles that only get dealt to once in each round represents the holy trinity, and the overall nine steps represent the fact that there were nine Police Academy movies.
** Tim Willoughby’s personal favorite shuffle is to throw his semi-shuffled deck at a wall and then pick it up. This neatly randomizes his deck and lets it know who’s boss.


††
Several statisticians calculated this differently depending on assumptions made off the bat. This seemed like a reasonable estimate. From my experience at large tournaments, the more obsessively I shuffle my deck, the better I do. If I were of a superstitious bent, this correlation could be attributed to some generation of positive karma through good sportsmanship. A more practical, analytical explanation is that the greater the number, variety, and randomness of shuffles performed on a deck, the greater the card motility (in other words, the less clumping you’ll experience).*

Probably the most common shuffling technique used in trading card games such as Vs. System is the riffle shuffle. When performing this shuffle, a player will split his or her deck into two roughly equal portions. Holding one of these portions in each hand, the player will lift up a corner of each and fan them together in such a way that the two piles are brought together by interspersing cards from one pile with cards from the other. To prevent stacking the top or bottom card of the deck, this should be done so that when the two portions are fanned together, the top card of the portion that started out on top is shuffled below the top card of the portion that started out as the lower half of the deck.
 

This is the meat and potatoes of shuffling, and its importance cannot be underestimated, because small errors in the size of the two portions and variations in the speed of the fanning together provide the fastest and most truly random method of shuffling available to the average card gamer.

 

The riffle shuffle, however, has several key flaws that can be pitfalls to those who rely on it exclusively and can be exploited by cheaters.

 

The largest problem with this technique is that while absolute card motility between regions of the deck is high in a riffle shuffle, relative motility between neighbors is not. As an example, take a sixty-card deck in which the cards have been numbered sequentially, with the top card as #1 and the bottom card as #60. If you perform a perfect, no-errors, mechanical riffle shuffle in a sixty-card deck, you will end up with an order something like: 31, 1, 32, 2, 33, 3, and so on. In this example, although card #32 has moved within the deck twenty-nine absolute spots, it has only moved one spot away from its previous neighbors. The portions of the deck that have the highest card motility (both relative and absolute) are those closest to the top, to the bottom, and to the cut. Portions in the middle see the least motility.

 

Unscrupulous players who are incredibly clever can perform this shuffle perfectly and accomplish what is known as a faro shuffle, which if done a specific number of times in a row, will return the cards to their original order.§

 

The mathematician and magician Persi Diaconis once calculated that a deck does not even begin to become truly random until five riffle shuffles have been performed. To randomize properly, one should perform a riffle shuffle seven times. If your shuffling technique is not up to snuff, it will likely take upward of twelve such shuffles to produce an acceptably randomized pile of cards. An example of poor shuffling technique in the riffle shuffle would be when a player flips multiple cards together (rather than single cards) while shuffling the two halves of the deck together.

 

A popular solution to this lack of relative card motility is what is known as the pile shuffle. In this shuffle, you take your deck and deal out multiple piles (usually six or seven) from the top of the deck onto a table before gathering these piles up and stacking them back together as a deck. Using the earlier model of a numbered 60-card deck, if you deal out six piles, you would end up with piles that have orders of 1, 7, 13, and 19; 2, 8, 14, and 18; and so on.

 

While this may produce desirable relative card motility (and depending on the order in which you replace the piles, an equally enviable absolute card motility), the pile shuffle also has problems.

 

The first is that it is entirely mathematically predictable, and predictability is anathema to a good shuffle. Since there is no way to introduce randomness to an uninterrupted string of pile shuffles, you should never use this method exclusively.


The second problem is that the relative card motility happens in preordained increments. That is to say that cards will always be exactly six slots apart.


A thoroughly clever London-based Vs. player and statistician named Ian Vincent worked out a solution to the second problem. This involves a pile shuffle based on co-primes, wherein the deck is dealt out into seven piles from left to right, and then the two piles on the right are dealt out again from right to left. This process is repeated in alternating directions until the deck is out; then the deck is reassembled, starting with one of the center three stacks and following with the stack three to the right of the previous one.


The advantage of this method is that if it’s performed between a series of riffle shuffles, it gives a wide range of relative card motility values; a card can move zero, two, three, four, five, or seven spots from its original location within the deck with relative ease.


A rarely used shuffling technique is called a wash shuffle. It’s an easy one to learn, and although it can damage card sleeves and requires a large and unoccupied table, it does produce a pleasantly chaotic card distribution.


During a wash, you spill your deck face down over a table and just slide the cards around, pushing them this way and that before trying to reassemble the deck. There’s rarely the time or space to do this in your average tournament, but it’s a good one to try out once in a while.**


If you have failed to shuffle properly, your draws will suffer. Let me illustrate with an example in which someone is playing a simple deck with 30 “A” cards and 30 “B” cards. After sorting the deck, the plot twists are at the bottom of the pile and the characters are on the top. The player riffle shuffles two times and pile shuffles once with a standard six-pile sort.


The first riffle shuffle will likely order the deck into a pattern of A, B, A, B, A, B. The second riffle will then fold those into A, A, B, B, A, A, B, B. These patterns will occur through most of the deck no matter where you cut the deck before riffle shuffling. In the third iteration, you will either cut through the pattern in the sequence between “A” and “B” or between a pair of the same kind of card (As or Bs).


If you end up in the first situation, you’ll have an A, B, A, B, A, B pattern again throughout large portions of the deck. In the second instance, you’ll end up with a pattern of A, A, A, A, B, B, B, B throughout large portions of the deck, which is somewhat more problematic than the first scenario.


In the second eventuality, when you then pile shuffle, you’ll end up with several sequences in the deck that are approximately ordered A, A, A, A, B, A, B, B, B, B, A, B.


Thus, if you see the first four cards of your deck after this shuffling sequence, you can reasonably predict the next set of the sequence of cards—and predictability is anathema to a good shuffle. Unless you have pre-planned this shuffle (at which point it’s cheating) and started out with a very specific order of cards (which is incredibly difficult and utterly immoral), the fact that you know the sequence of plot twists to characters will not be advantageous because several copies of the same card will remain together within the sequence.


With four riffle shuffles and one six-pile shuffle, things actually get worse. There is a statistically significant chance (approximately one in forty††) that the first eight cards of the deck will be all “A” or all “B.”


If we extend this analogy to a Vs. deck, that’s a 1 in 40 chance of a near-automatic loss. In a perfectly random deck, in which every card has an equal chance of being in any particular position, the chances of this happening are fewer than 1 in 300.


That means that over the course of the first day of a Pro Circuit, if everyone shuffles perfectly, there will be about five people who have a complete deck failure once.


In most casinos that don’t use mechanical shuffling, you’ll find that the wash, along with other shuffling techniques, is used. Often the dealer will wash, riffle twice, perform a series of cuts, riffle again, and make a final cut.


The most orderly your deck will ever be is immediately after you have written out your decklist. Immediately after you have done your decklist, you should perform a minimum of seven riffle shuffles interspersed with a co-primes pile shuffle or two.


The best way to foil deck stacking is to give your opponent’s deck a shuffle at every opportunity. Without fail, do one riffle shuffle every time he or she has randomized the deck; cutting your opponent’s deck is not sufficient, as there are techniques to shuffle important cards to the middle of a deck to fool the cut. Often, the same miserable wretches who use false shuffles will use sleazy gamesmanship tactics to avoid their opponent’s cut or shuffle. This can include but is not limited to:


Drawing a hand before giving you a chance to shuffle the deck, then disingenuously making a false claim that you had passed on the opportunity to do so.


Holding the deck in hand as he or she offers you the opportunity to cut, thus making it likelier that you just take the top half as your cut (and avoid a shuffle).


Tapping your deck in a friendly gesture that indicates that he or she “doesn’t need to cut your deck,” and then assuming that you will do the same.


Some decks, such as New School, EMS, GLEE, Fantastic Fun, and Both Guns Blazing, have enough search effects that opponents will get tired of always shuffling the deck and stop bothering.


There are several things to stay on guard for when watching your opponent shuffle. First off, watch your opponent’s hands; if your opponent repeatedly shuffles so that the original top pile is always fanned in last, the top card is staying on top. Secondly, if your opponent uses nothing but simple pile shuffles, stay on guard and make an extra effort to shuffle well.


Oh, and be on the lookout for people wearing Sunderland Football Club jerseys.


Olav Rokne

olavrokne(at)gmail.com


For further reading, I would suggest:


Diaconis, Persi. “Analysis of Top to Random Shuffles.” Combinatorics, Probability

 Computing 1: 135-55.


Diaconis, Persi. “Riffle Shuffles, Cycles and Descents.” Combinatorica 15: 11-29.


Erdnase, S. W., An Expert at the Card Table


* Card motility is a term used to define the ability of a card to travel from one point in a deck to any other point in a deck within a shuffle. This should be looked at in two ways: first, in terms of a card’s ability to travel in relation to its nearest neighbors, and secondly, in terms of absolute position within the sixty cards.


Many of whom, in my humble opinion, cheer for Sunderland Football Club, which is an inferior and unpleasant soccer club that no real Geordie would cheer for. I’m not sure if any Sunderland supporters actually play Vs. System, but if they do, this Newcastle United fan is dropping the gauntlet.


Such as those who would support the Sunderland FC.

 

§ Thankfully, the only player I know of who is dexterous and clever enough to pull off this feat is not a Sunderland supporter. He is, in fact, a fellow of unmitigated honesty.


I don’t fully understand the mathematics of this, but he explained that it is based on a combination of statistics, co-primes, and the teachings of an ascended, illuminated Shaolin master he studied with in Nepal.


The seven piles represent the seven chakras (or plexuses) of the human body in Cabbalist teaching, the fact that there are three piles that only get dealt to once in each round represents the holy trinity, and the overall nine steps represent the fact that there were nine Police Academy movies.


** Tim Willoughby’s personal favorite shuffle is to throw his semi-shuffled deck at a wall and then pick it up. This neatly randomizes his deck and lets it know who’s boss.


††
Several statisticians calculated this differently depending on assumptions made off the bat. This seemed like a reasonable estimate.
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