An Introduction to Full-Deck Stacks

In magical parlance, a “stack” is a specific, intentional ordering of cards. A “stacked deck” can be partial (some of the cards are arranged in a stack) or full (all of the cards are so arranged). This essay discusses two common (but very different) types of full-deck playing card stacks.

Sequential Stacks

A sequential stack permits one to determine the card following (and, in most cases, that preceding) any given card. Such stacks are relative in nature (i.e., no particular card is designated as being at the top of the deck), and designed to be “cyclic” (indeed, they are sometimes termed “circular” or “rosary” stacks). Consequently, the pack may be given any number of single complete cuts without disturbing the functionality of the stack.

Classic examples include the venerable Si Stebbins (a numeric progression: see analysis, below; download a copy of Si Stebbins’ original book, “Card Tricks and The Way They Are Performed”) and Eight Kings (a rhyming mnemonic progression: Eight kings threatened to save, nine fine ladies for one sick knave. ⇒ 8–K–3–10–2–7–9–5–Q–4–A–6–J) stacks. There are mnemonic sequences other than Eight Kings (cf. Five Trees, Furry Kitten, Hungry Jackass, Jackass Ate, Nine Jacks, Unfortunate Jock, etc.), and numeric progressions other than Si Stebbins (see discussion below), but the concepts are the same. The basic versions of these classic stacks exhibit a rotating suit (and thus alternating colour) sequence that is not very desirable; there are methods for eliminating this, but they add to the complexity of the next-suit calculation. Better (more deceptive, and just as easily used) stacks exist, though.

For sequential stacks, the principal goal is an effective compromise between (1) the ease of determining the next — and previous — cards in the stack, and (2) the degree to which the stack appears to be a random assortment of cards. “New deck order” is an example of an arrangement in which it is trivial to determine adjoining cards, but one that does not look at all random. At the other extreme is a stack like the Breakthrough Card System, in which it is quite difficult (for most people; mathematicians are not so easily deceived) to determine that the cards are ordered, but that requires a considerable number of “processing” steps in order to make the next — and previous — card determinations. More contemporary sequential stacks (including my own well-reviewed DAO Stack) strive instead for an optimal compromise somewhere between these extremes, combining an ease of operation comparable to Stebbins/Kings with an ordering that appears random to all but extremely rigourous examination.

In truth, an accomplished artist is not plagued by audiences demanding to examine the cards. It’s wise to recall that superstar mentalist Chan Canasta built his stellar career on the Eight Kings stack, and world-class magicians Stewart James and Gene Anderson chose the Si Stebbins above all others! That said, unless it’s necessary to exploit the particular ordering of Stebbins/Kings-like systems, there is no good reason for modern entertainers not to employ more deceptive approaches.

Memorized Stacks

A memorized stack (aka “memorized deck”, “memdeck”) is one in which you simply(!) know the position of every card, and — conversely — the name of the card at any location. Clearly, this is suitable for anything requiring a knowledge of preceding and following cards, but it enables a much wider realm of possibilities.

Do not confuse this with the concept of rapidly “memorizing a deck” as performed by memory experts, which simply means learning the order of the cards in a shuffled deck. Such demonstrations do not generally concern themselves with the numerical positions of the individual cards.

There is no “trick” to this; as the name suggests, the stack is actually memorized. There are, however, four alternative approaches to the learning process.

The first, and most straightforward, is with the use of rote memory. Decide on the pack arrangement you want to use (ensure that it appears to be random), and just sit down and memorize it. It’s not as difficult as it sounds, but it’s not trivial either. And some people do find it beyond their capacity.

The second approach is the use of classical mnemonic tools as a “stepping stone”. The well-known mnemonic alphabet (T/D=1, N=2, M=3, etc.) can be used to devise images for each of the 52 positions in the stack. Similarly, images can be created for each of the 52 cards in the deck. Then scenarios can be imagined, pairing the card images with their corresponding stack position images. So when given a card name (or stack position), one can recall the associated images to reconstruct the relationship, and the corresponding position (or name). This won’t be truly useful/effective, of course, until you have learned the relationships so well that you no longer have to think about the images, but can simply (and instantly) recall the association directly. The most widely-used such stacks are currently those by Simon Aronson and Juan Tamariz, extensively described in their respective books, though this solution can be applied equally to any of the many other published stacks … those by Steve Aldrich, Laurie Ireland, Bob Klase, Ed Marlo, William McCaffrey, Herbert Newell, Claude Rix, Rusduck, Mike Skinner, Rufus Steele, and Audley Walsh, to name only some of the better-regarded ones.

It’s also worth noting that Bob Farmer has devised an easily-learned mnemonic system (not requiring knowledge of the mnemonic alphabet) for memorizing arbitrary playing card sequences. Another useful playing-card-specific mnemonic code can be found in lesson seven of David Roth’s venerable Memory Course.

The above two approaches yield a pair of useful benefits: they allow for the most random appearance, and they permit stacks that have been “wired” to perform very specific effects, commonly spelling tricks (which I have never found to be particularly strong performance material), poker deals, and the like. They are challenging to learn, however, and also have a significant drawback: unless you are regularly doing a lot of memorized deck work, it is easy to forget a particular association in the heat of performance.

A third approach is used in Martin Joyal’s Six-Hour Memorized Deck and Chris Matt’s Six Kicks stack. In place of a classical mnemonic system, these each employ a set of “rules” (Joyal uses fourteen, Matt thirteen) as stepping stones to enable learning and remembering the necessary relationships. By way of an example, the rule for the four deuces (2s) in the Joyal stack is “even positions containing the digits 2 and 4: 22-40-42-44”. The equivalent rule for the Matt stack is “positions ending with the digit 2: 12-22-32-42”. One can see that these are not always precise, specific rules (they are more like clues), and some additional memorization is clearly required. Nonetheless, such an approach makes it significantly easier to get to the stage where you can match card names and stack positions. But there is no magic road to the point where you can instantly recall those associations … that will take a similar amount of time in any case.

The fourth approach is an algorithmic one, in which a formula of some kind is used to relate card values and positions. This approach is particularly popular among those who want to do memorized deck work, but not make it a life’s work (particularly mentalists and others who don’t do a lot of card work, but recognize the miracles that can be performed with a memorized deck). Its advantage lies in the fact that a single algorithm relates any card name to its corresponding position (and vice versa). This yields two specific benefits: first, it enables one to perform a significant number of “memorized deck effects” without truly memorizing the stack; second, if the memorized relationship is temporarily forgotten, there’s still a reliable (albeit slower) fallback position.

As with sequential stacks (which are also simple algorithmic methods), a variety of approaches have been proposed. Here are half a dozen representative examples:

1. Michael Weber (Card Kindergarten, 1990) and I (SnapStack, 2021) have both described algorithmic stacks that are almost instantaneous to both learn and utilize, but are not even remotely examinable, rendering them useful in narrow circumstances only.
2. Jack Yates (1978), Boris Wild (1996), and Charles Gauci (2002) have each created modified versions of the Si Stebbins stack. They’re not particularly random in appearance, with regular suit rotation (thus alternating colours as well) and clearly detectable sequences, but the calculations can be made quite quickly.

   The Si Stebbins Family: The so-called “Si Stebbins” stack (popularized by William Henry Coffrin, aka William Vino, aka Si Stebbins, in 1898), itself a variation of the Horatio Galasso stack (1593, in his Giochi di Carte Bellissimi di Regola, et di Memoria)! Some versions are quite weak … the Wild stack, for example, is comprised of 13 four-card groups (each in strict sequence, both numerically and with respect to suit), with all the court cards clustered at the end, and thus unlikely to survive any but the most cursory examination. This is not necessarily a show-stopper (any card arrangement can be hidden by a sufficiently skilled performer), but there are no commensurate benefits: the computations necessary for Wild’s name/position conversions are no simpler than with (for example) QuickStack, which offers a considerably more random presentation. Anyone interested in further details of this “Si Stebbins family” of stacks can explore a spreadsheet that I constructed to illustrate the algorithm. It is interesting to note that Coffrin himself did not use the stack he published for magicians, but rather the original Galasso stack (the ordering of which is somewhat less obvious).

3. The other end of the spectrum is occupied by the Bart Harding Stack (1962), which is very random in appearance (though, having at least four repeating patterns, will not withstand intensive scrutiny). The algorithm is not completely consistent either, having exceptions that must be rote memorized. Finally, it requires considerable calculation, thus takes the longest of all these examples to convert between card names and stack positions.
4. More recently, The Solution stack (2015), proposed by Atlas Brookings & Raven Gairloch, offers a similarly random appearance to the Harding stack. The algorithm is quite complex, however, involving four differently-sized “banks” of cards, half a dozen different rules, and arithmetic with two-digit numbers. Further, more than a sixth of the cards are not addressed by the algorithm, so must be rote memorized.
5. My own QuickStack (2002) is specifically designed as a tetradistic stack (see below for the advantage of this), which by definition is not completely random in appearance, but will still withstand casual examination. Doing the conversions, however, is considerably faster than with the Harding or Brookings/Gairloch systems.
6. My more recent Q Stack (2021) offers yet another approach to an algorithmic solution, incorporating the fastest algorithm I know that yields an examinable stack that is truly random in appearance (unlike many pseudo-random sequences, it exhibits value and suit pairs, longish runs of identically-coloured cards, etc.).

By way of a summary, here is a brief comparison of the tradeoffs associated with the four different memorized deck techniques:

	rote memory	mnemonics	rule-based	algorithmic
ease of learning the associations	very difficult	difficult	moderate	easy
support of random “built-in” effects ¹	good	good	very limited	limited
backup strategy if memory fails ²	none	fairly poor	poor	good

It’s important to understand that “ease of learning the associations” in this comparison refers to exactly that, and not the total time necessary to reach the stage where translations between card positions and values can be performed instantly, without conscious thought. The latter is primarily a function of practice, the additional acquisition time being roughly comparable in all cases. Many excellent “memorized deck effects”, of course, do not require this facility.

Realize as well that there are specific exceptions to these broad, general characteristics. For example:

¹ The above summary comparison refers to random/arbitrary “built-in” effects. As rote memory and classical mnemonics can be used with any sequence of cards, one has considerable freedom to construct arrangements in support of specific card tricks. But this hardly rules out support for effects with other memorization methods, particularly the algorithmic approach. Certain structurally symmetric orderings (such as stay stacks, tetradistic stacks, etc.) enable a considerable variety of extremely strong effects, and algorithms can be designed to produce such orderings (in fact, this is exactly how QuickStack was created). Many — myself included — would argue that this is superior to methodologies that only support a handful of very specific, predetermined effects (especially when those effects were predetermined by someone else). Top card man Alan Ackerman has written, “I personally feel the tetradistic stack is the strongest of all stacks.” And the justly celebrated Juan Tamariz, after having used his (built-in-effect-style) "Mnemonica" stack for many years, eventually came around to the use of a structurally symmetric stack as his primary tool (in fact, his "Exhibition Stack", with its strict suit rotation and highly visible ordering, may be the least examinable of such stacks; here are the first dozen cards: AD QS KH AC 2D KS AH 2C 3D AS 2H 3C). So these issues are not as simplistic as some would have you believe.

² The use of rote memory or classical mnemonics to memorize a sequential stack yields a functional backup strategy (since, if an association is forgotten, one can apply the sequential rule in conjuction with an adjacent card); unfortunately, this largely nullifies the “built-in” effect advantage. Understand also that this chart refers to backup techniques that are inherent to the learning method: obviously, external crib sheets could be constructed for any stack, though their use can be problematic in the heat of performance.

Occasionally one reads disparaging remarks about algorithmic and rule-based solutions, claiming that they are not “real” memorized decks. This is uninformed nonsense (and a common consequence of confusing the organization of a stack with the issue of whether or not it is memorized). A memorized deck is simply that, and that alone … one in which the practitioner knows the positions of all 52 cards; the method initially used to learn the card name/position relationships is irrelevant. With any approach, translations made while one is still learning the stack will be too slow for some effects (though perfectly sufficient for many others). It’s certainly true that in the case of an algorithmic solution, one can simply learn the algorithm and never actually memorize the stack (this, in fact, is one of the benefits of this approach), but then it’s not really a “memorized deck”.

If you want to know the card at position #46 in the Aronson stack, you either just “know” that it’s the Eight of Hearts, or you apply the various mnemonic rules to work it out: four is an “R”; six is a soft “J”, “SH”, “CH”, or “G”; that suggests a “roach”; that reminds you of a hive filled with roaches; the “H” in “hive” indicates a “Heart”; the “V” is an “Eight”. In QuickStack, you either “know” that #46 is the Ace of Diamonds, or you use an algorithm to work it out: forty-six denotes bank four, the seventh (6+1) card, an “Ace”; the natural suit of an Ace is Spades; in the fourth bank, it is three (4-1) greater, a “Diamond”. Neither approach is “better” in any absolute sense; they are just different. The tradeoff is that the algorithmic solution can be learned a lot more quickly (a single, straightforward algorithm vs. a mnemonic alphabet, 104 word images, and 52 word-pair relationships), but constrains the order of the cards, thus limiting the possibility of building in arbitrary effects (but still enabling the incorporation of effect methodologies, an alternative preferred by many). Anyone using either approach regularly will soon “know” all the card positions anyway, though it’s nice to be able to calculate them if you forget!

For the sake of completeness, I also note that both Barrie Richardson and Lewis Jones have published clever algorithmic systems that are extremely easy to learn, but cover only half the pack: either all the even cards (Jones), or all those of one colour (Richardson). These can be quite effective for certain effects.

Which is the “Best” Stack?

I hope that the above exposition will help lay to rest the absurd notion that there is a “best” stack. There is not, despite an abundance of grandiose proclamations (with very little justification one way or another) that the proclaimer’s personal choice of stack is such.

My own view, for the record:

1. No stack is “best” for everyone, though any particular stack may be “best” for your needs. Anyone who claims otherwise is either naïve, or marketing.
2. Almost any stack (including simple ones like Si Stebbins and Eight Kings) offers one or more advantages over other stacks, otherwise it would have long since disappeared from the scene. You might not consider such elements advantageous, but that’s to be expected (see point #1).
3. If the stack that you are using appears sufficiently random (no obvious recurring sequences of values, suits, or colours, say), and you can call out the stack in sequence (for a sequential stack) or randomly (with positions, for a memorized deck), at a cadence of not worse than about one second per card, then that stack is probably sufficient for your needs (though it may lack some feature for a specific trick), and you likely have no reason (other than intellectual curiosity) to explore others. If not, you might.
4. There is no compelling reason (other than marketing) to campaign for others to use the same stack that you do. Quite the opposite, in fact: the fewer people who use your favourite stack, the less widely it’s known, and the more deceptive it will remain.

… Doug Dyment