When people first learn how to play Zendo, they tend to compare it to the well-known game Mastermind. This comparison is understandable—in both games, players attempt to discover something hidden by setting up colored pieces and observing responses in the form of black and white markers. However, there’s an important difference between the two games. Mastermind is a game of deductive logic, while Zendo is a game of inductive logic.
What’s the difference? Well, one way to explain it is that with deductive logic, you start with a collection of rules and facts, and then you “deduce” other facts from them. With inductive logic, you start with a collection of facts, and then you try to “induce” a rule or set of rules that might explain them.
Deductive logic is a relatively well-understood and well-explored branch of mathematics and philosophy. Systems of symbolic logic provide methods of encoding rules and facts, along with methods of deriving other rules and facts from them (that is, “deducing” them). Deduction revolves around the concept of proof; if your premises are true, and if you follow the rules of deductive logic correctly, your conclusions must be true.
Inductive logic, on the other hand, is a poorly-understood branch of mathematical philosophy, shot through with conundrums and uncertainty. The theory of induction has been causing philosophical headaches ever since the eighteenth century philosopher David Hume argued that induction is philosophically unjustified. This battle rages on into our own century, in which there is still no consensus about whether the “induction problem” has been solved, whether it can be solved, or whether it’s even a coherent problem.
By these definitions, Mastermind is a pure deduction game. After a relatively small number of trials, you’ll be able to deduce with certainty what the hidden sequence must be (assuming that the “master” hasn’t mis-marked one of your tests). In fact, it’s even possible to deduce from the outset the most efficient way of performing your tests. It’s been shown that any four-peg, four color Mastermind puzzle can be solved in just five moves.
Zendo, on the other hand, is a true induction game. You are presented with a collection of facts—koans, marked with white and black stones—and you must induce the pattern that ties these facts together. There’s no simple formula for solving a Zendo puzzle; there’s no algorithm for determining the fewest number of moves required. Like all induction problems, it’s impossible to know for certain that a given hypothesis is correct—no matter how many confirming instances you’ve seen, it’s always possible that your next play will uncover an exception that you haven’t seen yet. I believe that this is what gives Zendo its unique fascination.
In actual fact, the process of solving a Zendo rule requires a mix of both inductive and deductive techniques. As you’ll see, it is possible to achieve deductive certainty about some facts during a game. And, even given the mysterious nature of the inductive aspect of Zendo, it is possible to get better at solving Zendo problems, and to keep getting better at them over time. At the time of this writing, I have played hundreds of games of Zendo, and I now routinely solve rules that would have been almost impossible for me to solve a year ago. The purpose of this chapter is to pass along some of the tricks, tips, and techniques that I, along with the rest of the Zendo players at Wunderland, have developed over the past years of intense play.
Positive and Negative Features
When you set out to solve a Zendo rule, what you’re really trying to do is determine what feature is causing koans to have the Buddha-nature. Every Zendo rule imaginable can be stated in the following fashion: “all koans that have feature x will have the Buddha-nature, and all other koans will not”.
What counts as a feature? Basically, any fact that you can name about a koan counts as one of its features. A koan may have the feature “contains exactly three blue pieces”, or “contains a piece pointing at another piece”, or even “contains an odd number of medium pieces touching the table”. These features can be combined into “compound features” like “contains a red piece and a green piece”. (Later on, we’ll talk about the special issues that arise when dealing with compound features.)
For any given koan—even a simple one—you’ll be able to name a long list of features. You’ll come up with some of these features without even having to think about it; for instance, it’s not hard to notice that a koan contains exactly three blue pieces. On the other hand, you’re very unlikely to notice that a koan contains an odd number of grounded medium pieces, unless you have a specific reason to do so.
Koans also have what I call “negative features”. If a koan does not contain exactly three red pieces, that counts as one of its features. For any given koan, there are even more negative features than positive ones; in fact, any given koan has an infinite number of negative features, because there are infinitely many things that are not true about that koan. Negative features are the most difficult features to notice; indeed, you shouldn’t notice them unless there is a specific reason to do so. It may be true that a koan contains no blue pieces that are pointing at yellow pieces, but you must have a specific reason to select this feature to be noticed out of the infinity of alternatives. Of course, negative features are important, and learning to spot them is a crucial aspect of advanced Zendo play.
Features (both positive and negative) can be categorized according to type—there are “color” features, “size” features, “pointing” features, and so on. Some features (like “contains a yellow piece pointing at a blue piece”) fall into multiple categories. The Terminology page provides a useful guide to the most common types of features that may or may not be important during any individual game of Zendo.
Positive and Negative Rules
Due to the existence of negative features, any positive Zendo rule can be restated in negative form. For instance, the rule “a koan has the Buddha-nature if it contains a red piece” can be restated as “a koan does not have the Buddha-nature if it contains no red pieces”. Negatively stated rules can contain positive features, and vice-versa. (“A koan does not have the Buddha-nature if it contains a blue piece pointing at a yellow piece.”) Since negative rules are (in my opinion) easier to think about than negative features, I tend to state rules negatively if it allows me to state the features positively. Of course, rules can refer to a mix of positive and negative features (“a koan has the Buddha-nature if it contains a red piece and if it contains no blue pieces”); such rules will contain negative features no matter how you choose to state them. Ultimately, you must learn to think fluidly in both positive and negative terms, viewing rules and features from many different perspectives.
When you begin a game of Zendo, your first goal should be to determine what kind of rule you’re dealing with—in other words, to figure out what matters. The first step toward solving a rule is to determine which of the basic types of features the rule seems to be about. This is one of the few areas in Zendo in which absolute knowledge is possible. If, for instance, a rule has something to do with color, it’s possible to know with certainty that color matters, because you’ll be able to find a black koan and a white koan that are identical except for the colors of their pieces.
It’s interesting to note that, while it’s possible to know for certain that color matters, it will never be possible to know for certain that color doesn’t matter. No matter how many experiments you do which fail to uncover color differences, it’s always possible that there are some exceptions that you haven’t found yet. Nevertheless, the longer you go without finding them, the more certain you can be that color doesn’t matter—and finding out what doesn’t matter is an essential part of finding out what does.
The Reduction Phase
The first step toward figuring out what matters is to reduce the number of features you have to work with. The more pieces the initial koans contain, the more features you’ll have to sift through to determine what’s important. Therefore, your first turns should be spent reducing the initial two koans to their bare-essentials—in other words, making less complex copies of them until you find the smallest example you can of a white koan and a black one. This will make it easier to pinpoint which features are causing koans to be marked one way or the other.
Let’s take a look at how this process might work. Imagine that the Master’s initial white koan contains four pieces, and the black one contains three. Your first task should be to simplify the most complex of the two koans—in this case, the white one. On your first turn, make a copy of that koan, but leave out one of its pieces. Let’s assume that the status of new koan is black. You can then be sure that the piece you removed has something to do with why the initial koan has the Buddha-nature. Take note of that piece’s color and size, the orientation it’s in, and how it’s related to the other pieces in the koan—these are important clues to guide your experiments later.
On your next turn, make another copy of the initial koan, leaving out a different piece this time. Let’s say that this time the new koan is white. Good! You’ve successfully eliminated an extraneous piece, and with it, a large number of positive features. Keep this process going; on your next turn, make a copy of the new white koan, leaving out yet another piece. If this new koan is also white (let’s say it is), you’ve succeeded in reducing your smallest white koan from four pieces to two. Finally, remove the one piece that you haven’t tried removing yet, leaving only the piece that you know is important. If the new koan is black, then you know that both of those pieces are important in some way. If the koan is still white, then you’ve reduced your smallest white koan down to a single piece—a very powerful result indeed.
The reduction process leads naturally into an exploration of the single-piece koans. These koans are extremely useful, because they’re so simple—they hardly contain any positive features at all. In a four-color game, there are only 24 possible single-piece koans—they’re made up of the twelve unique Icehouse pieces, which, in isolation, can only be placed in one of two positions: upright or flat.
In some games, the reduction process will uncover differences among the single-piece koans—in other words, some of them will be marked white, and some black. This is a powerful discovery; it’s the most sure-fire way of establishing the importance of at least one of the basic features (color, size, and orientation). For instance, if two differently-colored pieces of the same size and orientation are marked differently, you can be certain that color matters. If two differently-sized pieces of the same color and orientation are marked differently, you can be certain that size matters. And if two identical pieces of different orientations are marked differently, you can be sure that orientation matters. You may even determine that all of these features matter. Remember, however, that if you don’t find differences among the single-piece koans, you can’t automatically conclude that these features don’t matter. Color, size, and orientation may turn out to be important even if no differences exist among the single-piece koans.
Even when such differences do exist, you’re not always able to discover them right away. It would be tedious and time-consuming to test all 24 single-piece koans at the beginning of every game; a more sensible approach is to allow new information in the mid-game to suggest single-piece koans that you haven’t tried yet. However, I do find it helpful to try a basic sampling of single-piece koans during the beginning stages of a game—say, one example of each color, size, and orientation.
Worrying About the Other Players
You may ask at this point: what are the other players doing during all of this? Well, if they’re experienced players, they’ll be working right along with you to accomplish these goals. Some of the simplifying plays described here will probably be performed by other players as well.
But what if the other players don’t play in this way? Isn’t it a bit unfair that you’re spending your time doing good experiments, while other players are off doing whatever they want? My advice is: don’t worry! The other players aren’t gaining any special advantage by doing other things while you’re making good plays. If you’re the only player performing intelligent experiments in a game, then it’s likely that you’re the only player who understands why they’re intelligent. You’re also the player who’s most likely to make a play that “blows the lid off” the rule, and then you’re the one who gets to take the next guess and win. (Of course, you’d better have a guessing stone! If you think your next koan might “blow the lid off” the rule, make sure you call mondo…)
The Experimentation Phase
Once you’ve simplified the initial pair of koans, it’s time to begin making experimental copies of these koans. Your primary goal has not changed—you’re still trying to hone your knowledge of what matters.
The most common—and costly—mistake that players make in the early portions of a game is to decide that some feature matters when it actually doesn’t. My friend John Cooper refers to these mistaken conclusions as “superstitions” (perhaps because they arise in Zendo much as superstitions might arise in real life). An early superstition can keep you confused, misdirected, and running in circles for a good portion of a game; by the time you recognize a superstition for what it is, you’re likely to be well behind the other players in the race to solve the rule. At other times, an entire group of players can get caught up in a superstition, at which point it becomes a “religion”. Learn to recognize what you should and should not conclude from each experiment performed, and avoid group religions at all costs.
The best way to avoid developing superstitions as you perform your experiments is to make copies that only change one type of feature at a time. If you change more than one type of feature at once, and if the new koan then has a different status, you’ll have no idea which of your changes caused this result. If you fail to take into account all of the different types of features you’ve changed, you’re opening the door to superstition. Therefore, if you change something having to do with color, don’t change the sizes or positions of any pieces; if you change something having to do with size, don’t change the colors or positions of any pieces. If you reposition pieces within a koan, don’t change the colors or sizes of pieces. Avoid adding new pieces to your copies during this stage of the game, because doing so changes all three of these basic types of features. Try working with what you have for a while, only making copies that change the colors, sizes, and positions of pieces in existing koans.
If a rule has something to do with color, it will always be possible to find two koans which are marked differently, but which are identical in every way except for the colors of their pieces. If you find such a pair koans, you may conclude with certainty that color matters. Until you find such a pair, any belief that color matters counts as superstition, and nothing more. Clearly, the best way to avoid superstition and to accurately determine whether or not color matters is to build koans that are identical to existing koans except for color.
Note that even if color does matter, this doesn’t mean that any koan’s status may be changed just by altering its colors. Consider the rule “a koan has the Buddha-nature if it contains three pieces of the same color”. Now imagine a koan containing two pieces. No matter how you change the colors of the pieces, this koan will never have the Buddha-nature under this rule. However, there are many other koans that can be changed simply by changing the colors of their pieces.
You may find it useful to make a distinction between “specific color rules” and “general color rules”. A specific color rule is one that makes explicit reference to one of the colors, such as “red”. A general color rule is one that makes no reference to any specific color. The rule discussed above is a general color rule: “a koan has the Buddha-nature if it contains three pieces of any one color.” An example of a specific color rule is: “a koan has the Buddha-nature if it contains three red pieces.” It’s possible to determine with certainty that a rule makes specific reference to one or more colors (although it may be difficult to determine which colors it refers to).
You can do this using a method I call “color swapping”. Change all the pieces of one color into another color, and at the same time change all the pieces of the second color into the first color. (If a koan does not contain all four colors, changing all the pieces of one color into an unused color also counts as color swapping.) If this experiment causes a koan’s status to change, you can conclude with certainty that the current rule makes specific reference to at least one of the colors you swapped.
Homogenization is a general technique that can be used to quickly establish whether color, size, or orientation is important. Homogenization refers to the act of changing all of the pieces in a koan to be identical in one specific way. For instance, “color homogenization” means changing all of the pieces in a koan to a single color (leaving their sizes and positions intact, of course). Although homogenization often changes many pieces at once, it changes them in a uniform way, and in fact represents another way to simplify koans, thus reducing the number of features you have to work with. My experience is that homogenization is one of the quickest routes to uncovering color, size, and orientation differences, if they exist.
Homogenization is also one of the best techniques for determining when these features are not important. The more homogenization you perform which does not cause a koan’s change status, the more strongly you may suspect that the type of feature in question is not important. Of course, you can never be absolutely certain that some feature is not important, but it is possible to become certain enough to allow your suspicions to guide the rest of your experimentation.
All of the ideas presented so far in relationship to color may also be applied to the feature of size. However, working with size is a bit trickier than working with color, because changing sizes may cause positional changes as well. For instance, if a koan contains a stack of large pieces, you can’t change the size of the piece on the bottom of the stack without grounding the piece above it. If the status of that koan changes, you can’t be certain whether it was due to the size change or the grounding change. If there was a flat piece pointing at that stack, then that piece may now be pointing at two of the pieces in the tower instead of just one. And so on. There’s really nothing you can do about these issues except to be aware of them. Try to start with changes that cause the least number of unwanted side-effects.
The notion of “pip-count” also complicates matters. The technique of “size-swapping” (changing all of the pieces of one size into another size, and vice-versa) is not as useful as “color-swapping”, because rules that refer to specific sizes can often generate the same kind of behavior as pip-count rules. For instance, consider the rule “a koan has the Buddha-nature if it contains an equal number of small pieces and medium pieces”. This is “specific size” rule, but the behavior it generates makes it look very much like a pip-count rule—at first, it looks like the pip-count of the white koans is always divisible by three. Nevertheless, you can often still use the technique of homogenization (changing all of the pieces in a koan to the same size) to determine that the rule has something to do with size.
There are many rules in which the positions and orientations of pieces do not matter at all. In fact, such rules are so common that they deserve a special name. I refer to them as “population rules”, because they refer only to the “populations” of koans. In other words, it only matters which pieces are in a koan; it doesn’t matter what those pieces are doing in the koan. Homogenizing the positions and orientations of pieces in koans is especially useful for determining whether or not you’re dealing with a population rule. If you make copies of many koans with pieces standing upright in straight lines, and these koans never change status as a result, you may begin to suspect that position and orientation (and, by extension, pointing, touching, and groundedness) simply don’t matter.
The easiest way to figure out whether or not pointing matters is to find a simple koan which contains a flat piece pointing at something (and preferably nothing else pointing at anything), and make a copy of that koan with the piece turned to the side so that it’s no longer pointing at anything. If the koan’s status changes, you can be reasonably certain that pointing is important. However, be on the lookout for rules like “contains two pieces pointing in the same direction”, in which it doesn’t actually matter what the pieces are pointing at, just what direction they’re pointing.
Don’t forget that, by the standard definitions, an upright piece is considered to be pointing at any pieces stacked on top of it. Because of this fact, pointing rules can fool you into thinking that “stacking” is important. A good way to test this is to perform an operation that I call “stack dissection”. Take a stack apart and lay its pieces flat in a line, each pointing at the next. If you really want to be careful, touch each tip to the base of the piece it’s pointing at, so you’re changing as few types of features as possible. If the koan does not change status, slide all of the pieces apart so they aren’t touching each other. If the koan still doesn’t change status, you can begin to point these pieces away from each other to determine if “pointing at” is important here. Of course, there are plenty of ways for this technique to fail—for instance, the rule may have something to do with “an ungrounded piece being pointed at”, or “an upright piece pointing at something”. Such is the endless fascination of Zendo.
Orientation, Grounded, Touching
By now, you should be able to find ways of applying all of the above ideas to the other types of features, such as orientation, groundedness, and touching. Note that weird pieces and ungrounded pieces are always touching other pieces—something to keep in mind when attempting to isolate some of these features from each other.
Once you feel like you have a pretty good handle on what types of features are important in a given game, it’s time to start figuring out how those types of features are important. Really, you’re still asking same old question—what matters?—but this time you’re asking it at a greater level of detail. Ultimately, you’re hoping to narrow your search to the exact feature which causes koans to have (or not to have) the Buddha-nature.
During this phase of your experimentation, you’ll need to begin adding additional pieces to your copies of koans, and perhaps even building koans that are very different from anything currently on the table. In doing so, you’ll often find yourself changing multiple types of features at once—a necessary evil at this stage of the game. Fortunately, it’s somewhat counterbalanced by the fact that you now have more information about what types of features are important, and what types of features can probably be ignored.
Although it’s possible during your experimentation to change only one type of feature at a time, it’s important to note that it’s impossible to change a single feature at a time. For instance, if you change a single upright blue piece into an upright red piece, you’re not just increasing the number of red pieces in the koan; you’re also decreasing the number of blue pieces in the koan. In addition, you’re changing the number of red and blue pieces in the koan in relation to the number of yellow and green pieces. You’re changing the number of upright red and blue pieces, and the number of grounded red and blue pieces. You’re changing the number of red and blue pieces that are pointing at nothing. And of course, you’re also changing the pip-counts of all of these features. And the list goes on. Try as you might, you’ll never manage to only change a single feature at a time. The best you can do, therefore, is simply to be aware of all of the different features that your experiments are changing, so that you don’t draw unjustified conclusions from the results.
Half-Patterns and Whole-Patterns
So far, all of our talk has been about copying individual koans with modifications and observing the results. This is a very active approach to Zendo, which generally focuses on only one koan at a time. Now I’d like to start talking about a different but equally important aspect of play—searching for global patterns that exist across all the koans. This activity represents a kind of holistic complement to the focused experimentation we’ve been discussing so far—a meditative “yang” to the active “yin” of direct experimentation.
At all times throughout a game of Zendo, you should be scanning the koans on the table and asking yourself the following question: what is it that’s true about all of the white koans on the table, and is not true about any of the black ones (or vice-versa)? Repeat this question over and over to yourself like a mantra; you’ll be amazed at how often new ideas come bubbling up out of your sub-conscious. Use all of the information you’ve acquired about what matters to guide your questioning. If you’ve determined, for instance, that pointing matters, ask yourself “what is it that’s true about pointing in all of these white koans that’s not true in any of the black ones?” If you can find an answer to this question, you’ve come up with a guessable theory. Coming up with guessable theories is one of your most important tasks in Zendo—indeed, the whole goal of the game is to come up with a guessable theory that turns out to be correct.
Of course, finding a theory that works for everything on the table can be quite difficult. To help the process along, try concentrating on only half of the question. Ask yourself, “what is it that’s true about all of the white koans on the table?”, or “what is it that’s not true about all of the white koans on the table?” If you find an answer to a question like this, you’ve found what I call a “half-pattern”. It’s a half-pattern because even though all of the white or black koans contain a certain feature, some of the other koans contain this feature as well. Notice that every guessable theory is really just a pair of complementary half-patterns. For this reason, I like to refer to guessable theories as “whole-patterns”.
The first thing you should do when you discover a guessable theory is to try to falsify it. If a theory does turn out to be incorrect, it’s better to have discovered this as a result of your own experimentation. Not only does guessing a theory out loud cost you a stone; it also allows the other students to hear exactly what you’re thinking. If you theory is correct, it will resist all of your attempts to falsify it.
A common mistake that beginning Zendo players make is to test a theory by making plays that confirm it. Because of the nature of induction, no number of confirming plays can prove with certainty that a theory is correct, whereas a single counter-example will absolutely prove that a theory is not correct. That’s why the best way to test a theory is to attempt to falsify it, even though you hope your attempts will fail. Try to think of ways that your current theory might be incorrect; look for things that haven’t been tried yet.
If you do manage to falsify your theory, don’t be too quick to give up on your idea immediately. A single counter-example to a whole-pattern will almost always break only half of it, leaving you with a serviceable half-pattern. Only when you find a counter-example to that half-pattern should you really worry that you’re on the wrong track completely (although even then, the issue is tricky, as you’ll see later).
Half-patterns are not guessable theories; if you were to state one as a guess, the Master would simply point at one of the existing koans which doesn’t follow your pattern. Nevertheless, half-patterns represent important building-blocks that help you discover guessable theories. Whenever you’ve found a half-pattern, you should focus your attention on those koans that are keeping the pattern from being a whole-pattern. Try to look for similarities between these troublesome koans, with the aim of coming up with a deeper theory that incorporates them and explains them.
It might seem as though there’s a contradiction here: when you have a guessable theory, your goal is to try to falsify it (turning it into a half-pattern), and when you have a half-pattern, your goal is to try to transform it into a guessable theory! But of course, these two approaches are actually complimentary; the movement from whole-patterns to half-patterns and back again to whole ones represents an upwardly spiraling process.
Conjecture and Refutation: An Example
Let’s take a look at a concrete example of this process at work. Let’s say that, during the early portions of a game, you’ve determined for certain that color matters, and you suspect that size and position do not matter. Using this knowledge, you scan the table and notice that every white koan on the table contains at least one red piece. You check the black koans, and, sure enough, none of them contain red pieces. You’ve found a whole-pattern—a guessable theory. However, it’s too early in the game to be very confident about this theory, so you begin looking for ways to falsify it. Unsurprisingly, after a few rounds, your theory has been falsified—you and the other players have found a few koans that have red pieces but do not have the Buddha-nature. Notice that your whole-pattern has not been completely destroyed—it’s simply been reduced to a half-pattern. It’s still true that every white koan contains a red piece; the problem is that some of the black ones do as well.
At this point, those black koans that contain red pieces are the most important koans on the table. Study them carefully and ask yourself: “Why are these red-piece koans black, while all of the other koans that contain red pieces are white? What is it that’s true about this handful of black koans that isn’t true about any of the white koans?” Sound familiar? It’s the same pattern-finding question we’ve been asking all along, applied to a sub-group of koans. We’ve found yet another way of reducing the number of features you have to work with and isolating important features.
Suppose you now notice that all of the black koans that contain red pieces also contain blue pieces, while none of the white koans contain blue pieces. You’ve just found a new whole-pattern: “a koan has the Buddha-nature if it contains red pieces but no blue pieces”. At this point you’re starting to worry about the other players, so you decide to go ahead and guess this rule. The Master breaks your guess by creating a white koan that contains a blue piece and two red pieces.
Now what? Well, although you may feel discouraged, it’s not time to give up on these theories yet. Your initial half-pattern has still not been broken—it’s still true that all the white koans contain at least one red piece. And, although it’s a bit hard to spot, your last guess did highlight another half-pattern—there are no black koans that contain red pieces but no blue pieces.
The most important koan on the table is now the counter-example that the Master just set up—the one that contains a blue piece and two red pieces. Keeping in mind everything you’ve learned so far, what experiments suggest themselves? Well, you could try removing the blue piece from the koan. However, you already have good reason to believe that this koan would stay white, since you’ve never seen a black koan that contains red pieces but no blue pieces. This is an example of the kind of confirming play I warned against earlier. It merely supports half-patterns that you already suspect are true, so it’s unlikely to teach you anything new.
You could try adding a red piece to the koan. However, at this point the evidence suggests that adding a red piece to a koan that already has the Buddha-nature will not cause it to change. Therefore, this is also seems like a relatively weak play. At this point in the game, it’s quite possible that you’re not going to get another turn, so you’d better make this turn count.
In my view, the most interesting play at this point would be to add a blue piece to the koan. You’ve never seen a koan that has two red pieces and two blue pieces, and you’re unsure of what its status will be. This is an important point—the best experiments are the ones whose outcomes are hardest to predict, because unpredictable experiments provide you with new information.
So let’s say you add a blue piece to the koan, and it goes black. This suggests yet another half-pattern: no white koans contain an equal number of red and blue pieces. Now you’re really closing in on it. White koans always have red pieces; any koan that has red pieces and no blue pieces is white; and any koan that has an equal number of red and blue pieces is black. All of these together suggest a pretty strong whole-pattern: a koan has the Buddha-nature if it has more red pieces than blue pieces. This pattern works for every koan you’ve seen so far. At this stage of the game, any whole-pattern you find has a good chance of being correct. You guess it immediately, and win.
Think back over this example, and note the way that half-patterns and whole-patterns play off of each other and ratchet themselves upward into more powerful theories. Also note the way that local experimentation and global pattern-seeking are inextricably intertwined. Your local attempts to discover what matters narrow and guide your search for global patterns, while the global patterns you find indicate new local experiments to try. These two approaches grow out of each other like the two halves of a yin-yang.
Side-Effects and Emergent Patterns
In the above example, the most important half-pattern found was “all white koans contain red pieces”. However, it’s important to notice that the actual rule doesn’t explicitly state that koans must contain red pieces to have the Buddha-nature. It only states that a koan must have more red pieces than blue pieces. Of course, in practice this means that any white koan will contain at least one red piece. I like to say that this kind of half-pattern is an “emergent” pattern. It’s not explicitly stated as part of the rule, but it emerges as a kind of side-effect of the rule. You’ll find that most half-patterns are in fact emergent in this sense.
In the above example, it’s easy to see the connection between the half-pattern and the actual rule. However, in other circumstances, the connections aren’t always so easy to spot. Consider: “a koan has the Buddha-nature if it contains an equal number of grounded and ungrounded pieces”. This rule generates an interesting half-pattern: all white koans have an even number of pieces. It’s quite likely that you’ll notice this half-pattern during the course of trying to solve this rule; if you conclude that the rule must therefore explicitly refer to the concepts of “even” and “odd”, you might end up stumped for quite a while. Whenever you’re faced with a strong half-pattern, don’t forget that it could be the emergent result of a very different kind of rule.
So far I’ve limited all of the discussion in this chapter to rules which involve single features. However, it’s perfectly legal for a Master to create a “compound rule”, which involves two or more different features. The most common kinds of compound rules are “and” rules and “or” rules. To follow an “and” rule, a koan must contain all of the features in question; to follow an “or” rule, a koan may contain any one of the features in question (though it may contain more than one of them).
“And” rules are generally not too hard to get a handle on. Consider the rule: “A koan has the Buddha-nature if it contains a red piece and a green piece”. This rule will generate two clear half-patterns: all white koans contain red pieces, and all white koans contain green pieces. Of course, some black koans will also contain red pieces, and some will contain green pieces; however, it’s not too difficult to notice that the white koans are the ones that contain both features. In general, the difficulty of an “and” rule can be thought of as the “sum” of how difficult it is to notice each of its features. Of course, if you combine two difficult features together into an “and” rule, you’ll end up with a very difficult rule. “And” rules tend to be about as difficult as they sound.
“Or” rules, on the other hand, tend to be a bit more difficult than they sound, and our previous discussion of patterns makes it easier to understand why—they don’t generate very noticeable whole-patterns. Consider the rule: “A koan has the Buddha-nature if it contains a red piece or a green piece”. The tricky thing about this rule is that it doesn’t generate an easily noticeable pattern among the white koans. What is it that’s true about all the white koans on the table? It’s not true that all of them contain red pieces, and it’s not true that all of them contain green pieces. The only thing that’s really true about all of them is a restatement of the rule itself: they all have either a red piece or a green piece. This is a difficult pattern to notice; some people might even argue that it’s not technically a pattern (although I do consider it to be one).
It’s important to notice that this rule does create two half-patterns: all of the black koans contain no red pieces, and they also contain no green pieces. In fact, it’s possible to restate this rule negatively as an “and” rule, using negative features: “A koan does not have the Buddha-nature if it contains no red pieces and no green pieces.” This is an important general finding. It turns out that any “or” rule can be restated as an “and” rule (and vice-versa). In other words, there are really no “and” rules and “or” rules; there are only “and/or” rules. However, in practice, it’s easier to think in terms of positive features rather than negative features, so some rules will be stated more naturally as “ands”, and some will be stated more naturally as “ors”.
The difficulty of an “or” rule can be determined by translating it into its corresponding “and” rule and seeing how difficult each of its half-patterns are to find. In the case of our example “and” rule, the fact that every white koan contains a red piece and a green piece would be relatively easy to spot. In the case of our “or” rule, the fact that every black koan contains no red pieces and no green pieces is more difficult to spot. This explains why, all else being equal, “or” rules are generally more difficult to solve than “and” rules. Of course, understanding these issues better equips you to solve both kinds of rules.
There’s one other common type of compound rule that we haven’t discussed yet—”exclusive-or” rules (sometimes known as “xor” rules). In an “xor” rule, a koan must contain exactly one of the features in question. Here’s an example of such a rule: “A koan has the Buddha-nature if it contains either a red piece or a green piece, but not both at the same time.” It’s “exclusive” because only one of the important features is allowed at a time.
“Xor” rules are substantially more difficult than “and/or” rules. Many of the ideas we’ve discussed about pattern-finding go right out the window when it comes to “xor”. The reason is that these rules often don’t generate any clear patterns that are simpler than the rule itself. Consider our sample rule. It won’t be true that all white koans contain red pieces, nor will it be true that all white koans contain green pieces. It won’t be true that all black koans contain no red pieces, nor will it be true that all black koans contain no green pieces. In short, there are no simple half-pattern stepping-stones leading you up to the whole-pattern you’re looking for.
As you may remember, we were able to convert any “and” rule into an “or” rule (and vice-versa). Is there anything that an “xor” rule can be converted into? Yes, there is: an “xor” rule can always be restated as a “both-or-neither” rule. Our example rule can be restated as follows: “A koan does not have the Buddha-nature if it contains a red piece and a green piece, or if it contains neither.” Although this still doesn’t generate any clear half-pattern to sink your teeth into, it’s still good to be aware of the dichotomy. I personally find the idea of “both-or-neither” to be a bit easier to get my head around than “exclusive-or”, but this is probably a matter of individual psychology.
Fortunately, all of the techniques for determining what matters can still be applied to “xor” rules. Knowledge of what matters is even more important when half-patterns are difficult to come by. As you gain more experience, you may even learn to recognize the distinctive flip-flop pattern that an “xor” rule can generate. Think about a couple of the important features you’ve found, and check to see whether all the white (or black) koans always contain exactly one of them, or always contain both or neither of them.
We’ve talked about how to figure out what matters, and we’ve talked about how to spot global patterns. Now let’s talk about how to go about using that information to win a game of Zendo.
First, let’s talk about what you should do when another player calls mondo. The strategy here is pretty simple—answer the mondo according to all the information you’ve gathered so far during the game. If you’ve found a working whole-pattern, answer the mondo according to that. If you’ve found multiple whole-patterns, answer according to the one that you think is most likely to be true. If you haven’t found any whole-patterns, answer according to any applicable half-patterns you’ve found.
If you haven’t even found an applicable half-pattern, you’ll have to fall back on your intuition. Keep in mind what matters, and try to answer based on similar koans you’ve seen already. As a last resort, you can try a technique that I’ve found to be surprisingly effective: go with the odds. Most rules do not generate an equal number of black and white koans; some rules tend to make more black koans than white ones, and some the reverse. Use this information to your advantage when answering a mondo. If there are six black koans on the table and only one white one, the answer to a mondo is more likely to be black than white.
I should point out that this technique becomes less and less useful as the game progresses. When a rule tends to generate many black koans, players automatically begin trying to build white ones, and vice-versa. Therefore, most games tend to balance themselves as play continues. Also, the longer the game goes on, the more trustworthy the patterns you find become, and it’s always better to answer a mondo based on the evidence you’ve discovered during play. However, “playing the odds” is still a powerful technique for answering mondos in the early portions of a game.
As strange as it may sound, there is actually a (relatively rare) case in which it makes sense to answer a mondo in a way that you suspect is incorrect. Imagine that one of the other players sets up a mondo that suddenly sparks a new theory in your mind. It’s a brilliant theory, it works for everything on the table, and it seems quite likely to be true. The problem is, it’s pretty obvious that this is exactly what the other player is thinking, too. Therefore, the best move at this point is to answer the mondo against what the new theory specifies. If the answer to the mondo ends up supporting the theory (which is the most likely outcome), you won’t win a stone—but it won’t matter, because in that case, the other player’s going to win anyway. If, on the other hand, the answer to the mondo surprises you both by falsifying the theory, you’ll win a stone—and you’ll be more likely to get a chance to use it, since the promising theory was just falsified. This technique was discovered by Jacob Davenport, and he refers to it as “defensive mondoing”. It’s an advanced Zendo strategy that doesn’t come into play very often, but it’s very effective when it does.
The first thing to understand about calling mondo is that you aren’t just trying to win an extra stone for yourself—you’re also trying not to give stones to the other players. The best-case scenario is that you answer the mondo correctly and all the other players answer incorrectly; the worst-case scenario is that you answer incorrectly and all the other players answer correctly. Giving extra stones to other players may not seem like such a big deal to you right now, but as you’ll see below, an experienced opponent who owns three or four stones can pose a serious threat.
When should you call mondo? My own personal rule-of-thumb is that you should call mondo only when you’re in the process of testing (trying to falsify) a whole-pattern. Why do I believe this? Well, first of all, I don’t believe that it’s productive to call mondo if you’ve only found half-patterns. Let’s say that you’ve noticed that all white koans contain red pieces (but some black koans contain red pieces as well). It’s not very helpful to call mondo on a koan with red pieces in it, because your half-pattern actually doesn’t tell you whether that koan is going to be white or black. It would be more sensible to set up a koan with no red pieces and call mondo on that, since your half-pattern does indicate that it should be black. However, the problem here is that you aren’t getting any useful work done while you do this mondo. You are making a weak confirming play, which is unlikely to teach you anything new about the rule you’re working on. You’ll probably win a stone, but on such an easy mondo, the other players are likely to win stones as well.
If you haven’t even found a half-pattern, calling mondo is even more counter-productive. You don’t have anything in particular to base your answer on (except, perhaps, the “odds”); meanwhile, you could be giving easy stones to your opponents, who may know more than you do. In general, I don’t recommend calling mondo in the early rounds of a game. Other students may choose to do so; just play the odds and take your free stones.
When you’ve found a whole-pattern, you have something concrete on which to base the answer to a mondo, and that’s one reason why I believe that this is the best time to call one. However, there’s more to my reasoning than this. A whole-pattern is a guessable theory, and when you’re ready to present this theory to the Master, it’s going to cost you a stone to do so. By calling mondo during the testing of your theory, you are essentially earning the stone to be spent on that theory. Therefore, if you are planning to guess at the end of your turn, you should definitely call mondo.
A possible objection runs as follows: if the best experiments are the ones whose outcomes are hardest to predict, isn’t that a bad time to call Mondo? Well, consider this point carefully: if your answer is incorrect, that means that your theory is incorrect. The koan you just set up falsified your theory by providing a counter-example to it—which is exactly what the Master would have done if you’d spent a stone and presented that theory as a guess. You didn’t win a stone, but you eliminated a theory that would have cost you a stone, so your net gain is the same. In fact, it’s actually better to falsify a theory on your own if you can, because guessing an incorrect theory out loud may spark another player to come up with the correct answer and win. This is yet another reason why it’s better to try to falsify theories rather than confirm them. Making very predictable plays will earn you (and everyone else) more stones, but then you’ll just end up paying the Master to falsify your theories for you. In the meantime, you’ll be tipping your hand by constantly stating your theories out loud for all to hear.
It may seem to you that, during the times when you absolutely need a stone, it makes sense to do an “easy” mondo that virtually guarantees you a stone. The above reasoning should be enough to show why this is never a good move. If you have no guessing stones and are desperate to take a guess, the best move is still to perform an intelligent mondo that attempts to falsify your theory. If your theory is correct, you will win the stone that you need. If you don’t win the stone, that means that your theory was incorrect, so you no longer have a desperate need for that stone. If you choose to do an “easy” mondo, you’ll be providing free stones to everyone, and you’ll be forced to state your theory out loud (since you’re making no attempt to falsify it with your play). If your theory is correct, this won’t matter, but if it’s not correct, this is the worst possible outcome of a mondo.
Since the outcome of a good experiment is, by definition, more difficult to predict, other players are less likely to earn stones when you call mondo on a good experiment. Furthermore, it may actually make some sense to add extra complexity to your mondos to further confuse your opponents. The most obvious way to add complexity is to add extraneous pieces to your koan. When you add more pieces, you’re creating more features, and therefore, more confusion. However, be aware that this strategy can backfire—you may end up confusing yourself, or invalidating your own test. Remember that your primary goal during a mondo is to perform a good test of your whole-pattern. My own experience is that it’s usually wiser to perform clean, simple experiments than to try to confuse your opponents with subterfuge and misdirection.
One final point. If you manage to collect a healthy number of stones (say, five or more), it may be wise to refrain from calling mondo, even before guessing. If you have that many stones, you probably don’t need any more, and you may just end up giving free stones to the players who need them.
Guessing the Rule
Now that we’ve talked about how to win stones, let’s talk about how to spend them. Perhaps the most obvious question is: how long should you wait before you guess a whole-pattern? I think it depends on how long the game has been going on. The longer a game has gone on, the more likely it is that a whole-pattern you see will turn out to be correct. If you find a whole-pattern in the later stages of a game, it’s almost always wise to guess it as soon as you possibly can. If you find a whole-pattern in the beginning stages of a game, it’s almost always wise to attempt to falsify it yourself, at least for a turn or two. One of the main reasons to hold off on guessing is that you don’t want to give away valuable information about what you’re thinking to the rest of the players. On the other hand, the longer you wait, the more you’re risking having another player steal your guess out from under you. It takes experience to know how long you ought to wait before you jump in with a guess.
One of the most important considerations you should make when deciding whether or not to guess is how many stones you currently have. I’ve talked around this issue quite a bit; now let’s take a look at why I believe it’s important, and why you should try not to give other players too many free stones.
The most obvious case in which multiple stones are useful is when you’ve found multiple whole-patterns, any of which could be the correct answer. Having a healthy stash of stones allows you to guess all of your theories at the end of a single turn, greatly increasing your chances of winning. In such a case, simply guess the theory that seems most likely; if the Master breaks it, you can then spend another stone to guess another theory, and so on. There’s an obvious rule-of-thumb here—it’s dangerous to start guessing if you have more theories than stones. If you don’t even have enough stones to cover all of your current theories, you may end up spending your entire stash, only to find that you’ve simply narrowed down the possibilities for the next player. At that point, you may never get another turn. Unless you’re already pretty sure you’re not going to get another turn anyway, it’s probably best to refrain from guessing until you’ve gathered enough stones or falsified enough theories so that you can pay for all of your guesses.
However, even if you only have a single guessable theory, it’s nice to have a healthy number of stones in reserve. Here we come to one of the most important concepts in advanced Zendo play. If you take a guess, and the Master falsifies it, don’t just pass your turn on to the next player. You’re still allowed to take another guess, and now is the time to bring everything we’ve discussed in this chapter to bear. The Master’s counter-example has probably left you with at least one half-pattern intact. Focus on that koan, and try to integrate it into another whole-pattern. If you find one, guess it immediately; if it’s falsified, repeat this process yet again. Use everything you’ve learned about what matters, and the interplay between whole and half-patterns, to ratchet yourself upward to the winning guess.
John Cooper has dubbed this technique the “guess-barrage”, and it is one of the most difficult and advanced concepts of Zendo strategy. It may sound like an impossible feat, and, of course, sometimes it is impossible. However, you may be surprised at how often it is possible, especially with the aid of another concept: the “show-me” guess.
“Show-me” guesses are guesses that, while they work for everything on the table, are so inelegant and full of exceptions that they’re unlikely to actually be correct. They’re often cobbled together when attempting to integrate a single counter-example into a broken theory, which explains why they’re often made up of “ands”, “ors”, and other exceptions. When you take a “show-me” guess, you aren’t expecting to win with that guess. You’re expecting the Master to “show you” another counter-example that may give you enough information to come up with a real guess. You may end up taking more than one “show-me” guess in a row, but it’s all (hopefully) leading up to a theory that has a real chance of being the correct answer.
This explains why it’s so important to have a stockpile of stones when you begin guessing. In general, it’s dangerous to take a “show-me” guess with your last stone, because you’ll be unable to reap the benefits of the new information provided. (On the other hand, it is often wise to spend your last stone on a real guess, because holding off may cost you the game. This is a tricky case—if your guess is close but not quite correct, you may be handing an easy win to the following player.)
A nice thing about the “show-me” technique is that it opens your mind to patterns that you might habitually reject as “too ugly” or “too unlikely”. In fact, any whole-pattern, no matter how “ugly” or “unlikely” it sounds, can be an important stepping-stone on your way to the correct answer. The act of trying to sum up what you see in words, however convoluted, will often lead you to insight in unexpected ways. And, of course, there’s another reason why “show-me” guesses are a good idea—they sometimes turn out to be correct! Most elegant rules can be stated in inelegant and convoluted ways, and you may find that your “show-me” guess turns out to be the correct answer disguise. Or it may actually be that the rule you’re dealing with is ugly and convoluted! If you’re not willing to take “show-me” guesses, you’ll never be able to solve all of the bizarre rules that Masters are bound to throw at you.
The “Show-me” Guess in Action
In order to bring all of this to a focus, here’s an example of how I once used a “show-me” guess to crack a tough rule.
The game had been going on for quite a while, and the other two players and I were pretty stumped. I’d determined that the rule had something to do with orientation and pip-count—specifically, that it had something to do with the relationship between the pip-count of the upright pieces and flat pieces. In fact, I’d noticed a more specific pattern than this, but it seemed so arbitrary that I wasn’t even willing to put it into words for quite some time. The pattern was this: in every white koan, the pip-count of flat pieces was two and upright pieces six, or vice-versa… or the pip-count of flat pieces was three and upright pieces four, or vice-versa. It seemed clear to me that there must be some elegant rule that could explain this pattern, but I couldn’t see it. (Had I written down the pairs “2-6” and “3-4”, and asked myself what they had in common, I might have figured it out immediately, as perhaps many readers already have. Unfortunately, I wasn’t so resourceful at the time.)
I began to suspect that there must be a white koan in which the pip-count of flat pieces was one, and the pip-count of upright pieces was… something else. If I knew what this “something else” was, I’d have a third pair of numbers, and maybe then I could figure out what the real pattern was. I spent a fair amount of time trying to find this koan, and trying to analyze how I might find it, with no luck. Finally, it occurred to me how silly I was being—why not force the Master to show me this koan? I had a pile of guessing stones in front of me, and a theory that refused to be falsified—what was I waiting for? Of course, I was waiting for enlightenment—for an elegant rule to occur to me which tied it all together. I’d been forgetting that any pattern that can be stated unambiguously is a guessable theory, and guessable theories are made to be guessed.
Fortunately, I wised up and took my guess: “a koan has the Buddha-nature if the flat pip-count is two and the upright pip-count is six, or vice-versa, or if the flat pip-count is three and the upright pip-count is four, or vice-versa.” The Master set out a white koan that had a single small flat piece and four large upright pieces. We now had three pairs of numbers: (1, 12) (2, 6) (3, 4).
We achieved enlightenment.
Note the way that my guess forced the Master to “show me” the counter-example that I couldn’t find myself. Also note that, even if I somehow managed to miss the obvious answer that this new koan provided, I still could have incorporated the Master’s counter-example into an even more convoluted “show-me” guess—and it would have been correct! It doesn’t matter whether you see the elegant underlying rule or not—if you can correctly and completely describe which koans will have the Buddha-nature and which will not, you have solved the rule.
The Sanzen Exercise
In spite of all this, you may feel that the “guess-barrage” is really only possible in rare and specific circumstances. While I can’t guarantee that it’s possible in any given case, I can propose the following exercise. Find a willing Master to come up with a rule for you—preferably an easy one to start off with. Have the Master set out the two initial koans. Begin guessing, and have the Master set out a counter-example after each guess, just as in normal Zendo. Keep guessing until you solve the rule. Since the Zen word “sanzen” refers to the private interview session between Master and student, I like to refer to this as the “sanzen exercise”.
You may be surprised at how often you’re able to solve a rule simply by taking guesses. Indeed, I suspect that any rule can be solved in this fashion. The implication is that, given a large enough pile of stones, and enough time, you should be able to win a game of Zendo in a single turn. In practice, of course, your time and stones are limited, and that’s part of the fun and challenge of competitive Zendo.
This sanzen exercise can not only increase your skill at finding whole-patterns and performing guess-barrages, but it can help you gain confidence in the very idea of a guess-barrage. Keep this in mind the next time you find yourself playing Zendo with a large pile of stones in front of you.
The Psychology of Zendo
As logical as the process of solving a Zendo rule might sound, I continue to be fascinated by how psychological the actual gameplay can be. Aspects of group dynamics arise that you might not expect simply by reading the rules-of-play. I’ve already mentioned the concept of “religions”—superstitions that come to be shared by an entire group of players. This is a very real phenomenon, and religions can be surprisingly difficult to break away from. The psychological sense that “the group must be correct” can be as compelling (and as damaging) as it often is in real life.
Although players rarely discuss their thoughts about what the current rule might be, a surprising amount of non-verbal communication occurs during the course of a normal game. Players may learn something about what you’re thinking simply by observing the experiments you chose to perform. Furthermore, they can get a feel for how well you’re doing by observing how you respond to mondos. Indeed, this is an interesting psychological aspect of mondo that we haven’t discussed—calling mondo gives you a chance to see, to some extent, what the other students are thinking.
Believe it or not, there are even elements of bluff in Zendo. If you suddenly come up with an extremely good theory during someone else’s turn, conceal your excitement at all costs! It is a fascinating fact of human psychology that people are much more likely to solve a problem if they know that someone else has already solved it. By signaling that you’ve made a major discovery, you are greatly increasing the chances that someone else is going to steal the win away. Learn to cultivate your “Zendo face”—an inscrutable mask worthy of a Zen Master. Only gloat after you toss in your stone and win.
Zen and the Art of Zendo
Many people have commented upon the ironic fact that the game of Zendo, which is so influenced by the flavor and terminology of Zen, is ultimately based on such a completely western activity—solving a logic problem through the use of the scientific method. My response to this is “Yes! How ironic! How paradoxical! How Zen!”
More seriously, I feel that the theme of Zen is appropriate in ways that might not be immediately apparent. I’ve found that the game has more to do with intuition than the analytical tone of this chapter might lead you to believe. Zendo is at least as much a game of intuition as it is of logic. I believe that this surprising fact is directly related to a well-known feature of the inductive method: it provides no algorithm for generating hypothesis. A theory (or “whole-pattern”, in our terminology) cannot simply be “read off” the data in any systematic way—it requires a creative leap of insight on the part of the player. (Of course, once you have a theory, the process of testing it is a thoroughly logical one.) While I don’t believe that this kind of insight is beyond the reach of analysis—indeed, this entire chapter is an attempt to analyze it—it’s clear that the process resists easy systemization, and that’s what gives inductive problem-solving its unique fascination. Perhaps this “eureka!” experience, this intuitive bursting forth of enlightenment, is nothing like the “satori” of Zen, but it certainly lies at the core of the game of Zendo.
In my view, many of the concepts of Zen can be applied to any human activity, whether it’s archery, flower arrangement, or finding patterns within groups of small colored pyramids. Learning to play Zendo can be much like learning to perform a martial art. The game can be approached on many different levels, from the simple enjoyment of solving a puzzle, to the fascination of group dynamics and competitive play, to the intricacies and mysteries of the discovery process itself. The more you play, the more you may come to appreciate the beauty of the process, above and beyond all issues of winning and losing. At its highest levels, playing Zendo can even become an exploration of your own mind. How do you go about solving problems, and how are you affected by other people’s ideas during this process? What are your weaknesses and strengths? What are the mistakes you habitually make? What things tend to infuriate you, or enlighten you? Ultimately, as you learn more and more about playing Zendo, you may also be learning more and more about yourself.
That’s a bit of the yin and the yang of it.