Zendo Rulebook

Table of Contents

Table of Contents

• What You Need
• Setup
• Koans
• Choosing a Rule
  • Examples
• Initial Koans
• Turn Order for Students
  • 1. Build a Koan
  • 2. Say "Master" or "Mondo"
  • 3. Guess the Rule (optional)
• Winning
• Rules for Beginners
• Terminology
  • How To Use These Definitions
  • The Structure of a Rule
  • Some Problem Cases
  • Teaching Terminology
• Details and Clarifications
  • Selecting a Master
  • Creating Rules
  • Building Koans
  • Breaking Down Old Koans
  • Marking Difficult Koans
  • Asking About Koans
  • Guesses and Previously-Existing Koans
  • Katsu
  • Master's Mistakes
• Tips for the Master
  • Creating a Rule
  • Building the Initial Koans
  • Selecting the First Student
  • Marking Koans
  • Answering Questions about Koans
  • Understanding a Student's Guess
  • Responding to a Student's Guess
  • Helpful Masters and Tight-Lipped Masters
  • Giving Hints, and Giving Up
• Tips for the Student
  • Positive and Negative Features
  • Positive and Negative Rules
  • What Matters?
  • The Reduction Phase
  • Single-Piece Koans
  • Worrying About the Other Players
  • The Experimentation Phase
  • Color
  • Homogenization
  • Size
  • Population Rules
  • Pointing
  • Orientation, Grounded, Touching
  • Further Experimentation
  • Half-Patterns and Whole-Patterns
  • Falsifying Theories
  • Conjecture and Refutation: An Example
  • Side-Effects and Emergent Patterns
  • Compound Rules
  • Exclusive-Or Rules
  • Answering Mondos
  • Calling Mondo
  • Guessing the Rule
  • The "Show-me" Guess in Action
  • The Sanzen Exercise
  • The Psychology of Zendo
  • Zen and the Art of Zendo
• Glossary
• Acknowledgments

What You Need

• At least four stashes of Looney Pyramids.
• A fair number of black and white stones.
• A fair number of stones of a third color.
• Or pick up the commercial release, which includes everything needed to play.

Setup

Choose someone to be the Master. The other players are the Students. Give each Student a black and a white stone, to serve as "answering stones". The remaining black and white stones are "marking stones", and the stones of the third color are "guessing stones". Put all of the marking stones and guessing stones in front of the Master, and put all of the pyramids into a global stash within reach of all the Students.

The Master must choose a rule, create two initial koans, and pick someone to go first.

Koans

Over the course of the game, players will create different arrangements of one or more pyramids on the table. Each arrangement is referred to as a "koan", pronounced "KO-ahn". Koans can be set up in any fashion, as long as they don't touch other objects or other koans.

Choosing a Rule

When you are selected to be the Master, your first task is to devise a secret rule that will be used during this game of Zendo. For your first several games, you may want to use one of the rules listed under Rules for Beginners below. These are good rules for new players. See Terminology below for any terms on this page that are not clear. When the players are more experienced, you can invent your own rule.

According to your rule, some koans will "have the Buddha-nature", and others will not. For the Students, the object of the game is to try to figure out what your secret rule is. As the Master, your job is to act as facilitator; you are not actually a player, and you are not in competition with any of the players.

Examples

• A Simple Rule: A koan has the Buddha-nature if and only if it contains one or more green pieces.
• A Very Difficult Rule: A koan has the Buddha-nature if and only if it contains an odd number of pieces pointing at other pieces.
• A "Negative" Rule: A koan does not have the Buddha-nature if it contains exactly three pieces touching the table; otherwise it does.

Initial Koans

As the Master, start the game off by building two koans in the middle of the playing field. One should have the Buddha-nature according to your rule; place a white stone next to it. The other should not; place a black stone next to it. You will be marking all of the koans in this way throughout the game. Starting koans need not be complicated, even with experienced players.

Turn Order for Students

1. Build a Koan

• Create a new koan using one or more pyramids from the global stash.

2. Say "Master" or "Mondo"

• Master: The Master will immediately mark the new koan with a black or white stone.
• Mondo: All Students must guess if the new koan has the Buddha-nature or not. Pick up your own answering stones and hide your answer (black or white) in one fist. Hold that fist out over the playing field, and wait for all of the other Students to do the same. When everyone is ready, reveal your guess. The Master will mark the koan with the correct answer, and will award a guessing stone to each player who answered the Mondo correctly.

3. Guess the Rule (optional)

• Make a Guess: If you have any guessing stones, you may choose to spend one or more of them to try to guess the Master's rule. Give a guessing stone to the Master and then state your guess as clearly as you can.
• Clarify the Guess: If the Master does not fully understand your guess, or if it is ambiguous in some way, the Master will ask clarifying questions until the uncertainty has been resolved. Your guess is not considered to be official until both you and the Master agree that it is official. At any time before that, you may retract your guess and take back your stone, or you may change your guess. If any koan on the table contradicts your guess, the Master should point this out, and you may take back your stone or change your guess. It is the Master's responsibility to make certain that a guess is unambiguous and is not contradicted by an existing koan; all Students are encouraged to participate in this process.
• Master Disproves Guess: After you and the Master agree upon an official guess, the Master will disprove it, if possible. The Master builds a koan which has the Buddha-nature but which your guess says does not, or builds a koan which does not have the Buddha-nature but which your guess says does.
• Repeat: Once the Master has built a counter-example and marked it appropriately, you may spend another guessing stone, if you have one, to take another guess. You may spend as many of your guessing stones as you wish during this portion of your turn. When you are finished, the action passes to the Student on your left.

Winning

If the Master is unable to disprove your official guess, you've achieved enlightenment—you've discovered the Master's secret rule and have won the game!

Rules for Beginners

The following rules are suitable for beginning students.

A koan has the Buddha-nature if and only if…

• all its pieces are the same color.
• all its pieces are the same size.
• all its pieces are flat.
• it contains at least one red piece.
• it contains at least one small piece.
• it contains at least one piece of each of the four colors.
• it contains no green pieces.
• it contains no large pieces.
• it contains at least one medium yellow piece.
• it contains exactly two pieces.
• it contains two or more upright pieces.
• it contains a piece pointing at another piece.
• it contains an ungrounded piece.
• it contains at least one green piece and at least one blue piece.
• it contains at least two pieces that are touching each other.

Terminology

The following terms and definitions can make it easier for Masters and students to conceptualize and discuss Zendo rules. These definitions are merely conventions, and are not part of the official rules of Zendo.

Color—The standard Zendo colors are red, yellow, green, and blue, though other pyramid colors can be added or substituted. A koan "contains a color" if it contains any pieces of that color. A rule will occasionally refer to the "number of colors" contained within a koan, or within a subset of its pieces.

Size—There are three sizes of pieces: small, medium, and large. A koan "contains a size" if it contains any pieces of that size. As with color, a rule will occasionally refer to the "number of sizes" found within a koan, or within a subset of its pieces. Unlike color, sizes are related to each other in specific ways, such as "larger", "smaller", "largest", and "smallest". A koan always contains a "largest size", and all of the pieces of that size are referred to as "the largest pieces in the koan"; similarly, a koan always contains a smallest size, and all of those pieces are referred to as the smallest pieces in the koan. If a koan contains exactly one size, those pieces are both the largest and the smallest pieces in the koan.

Pip Count—Each piece is marked with "pips" to indicate its size—one pip for a small piece, two for a medium piece, and three for a large piece. This number represents a piece's "value" or "pip-count". A group of pieces has a pip-count equal to the values of all its pieces added together. A rule may refer to the pip-count of an entire koan, or any subset of pieces within a koan. For instance, a rule might refer to the pip-count of all of a koan's red pieces, or even the pip-count of all red pieces that are touching the table and pointing at blue pieces. Note that when a koan contains no red pieces, the pip-count of red in that koan is zero. When employing the concepts of "odd" and "even", don't forget that zero is an even number!

Orientation (Upright/Flat/Weird)—Each piece within a koan exists in one of three "orientations": "upright", "flat", or "weird". A piece is upright when it's pointing straight upward, with its base parallel to the table, either on it or above it. A piece is flat when its lowest triangular side is parallel to the table, either on it or above it. A piece is "weird" if it's neither upright nor flat. Note that these three orientations are mutually exclusive—a piece cannot be both flat and weird at the same time. Also note that a piece's orientation has nothing to do with whether or not that piece is touching the playing surface—a piece which is lying nested inside another piece or lying across the tips of upright pieces is still flat. Finally, note that a group of pieces has the same "orientation" as long as they're all upright, all flat, or all weird, even if they're not all pointing in the same direction.

Grounded and Ungrounded—A piece is "grounded" if any part of it is touching the playing surface; otherwise it's "ungrounded". Note that this attribute is entirely independent of a piece's "orientation".

Direction—This term is used slightly differently than "orientation"; it refers to the direction that pieces are pointing relative to each other. Although absolute direction is not allowed, multiple pieces may be said to be pointing in the "same direction", or "different directions", or perhaps even "opposite directions".

Pointing—Every piece has an imaginary "pointing ray" that shoots directly out of its tip and extends outward into space. The pointing ray will pass unhindered through any other piece; however, if the ray hits the table, the ray will bend to skim along the table's surface. A piece is "pointing at" all of the pieces its pointing ray touches. A piece may be referred to as "the first piece being pointed at", or "the second piece being pointed at", and so on. A piece may "point through" one or more pieces at some other piece.

Touching—If two pieces are making physical contact in any way, they're touching. Unlike pointing, if one piece is touching another piece, the other piece is always touching it back.

Tower/Stack—A tower or stack is any connected group of one or more pieces in which the tip of each piece is fully touching the underside of the tip of the piece above it. A tower may be upright, flat, or weird. Pieces are still said to be "above" other pieces in a tower, even if the tower is lying on its side. The pieces within a tower do not form sub-towers—a three-piece tower does not contain two two-piece towers within it. Single pieces are also referred to as towers. It follows that every piece in a koan belongs to exactly one tower.

Height—The physically highest point of a piece marks the vertical position of that piece within the koan. A piece may be higher than, lower than, or at the same height as any other piece in the koan. There will always be at least one highest piece and at least one lowest piece in a koan. It's possible for the same pieces to be both the highest and the lowest in a koan.

Exactly/At Least—Avoid using indefinite statements like "contains a red piece", or "contains two upright pieces", because it's not clear whether you mean exactly that number, or at least that number. Always use definite statements like "contains exactly one red piece", or "contains at least two upright pieces".

Majority/Plurality—When there are more red pieces in a koan than any other color, the koan contains a "plurality" of red pieces. When there are more red pieces in a koan than all the other colors put together, the koan contains a "majority" of red pieces. Many people use the word "majority" when they mean "plurality". If players use either of these terms while you are the Master, ask for clarification.

How To Use These Definitions

The purpose of this list is twofold: to provide a way for players to conceptualize rules, and to provide a common language to aid the communication process between Masters and students. For the student, formulating and communicating a guess is one of the most difficult aspects of the game. For the Master, understanding and probing that guess is just as difficult. Standard terminology makes this process easier for both parties.

Some Masters worry that the adoption of this standard terminology will restrict their options when creating new rules, but it's important to understand that these definitions are not restrictive in any way. If you want to make a rule in which upright pieces do not point at pieces which are above them, you don't need to redefine pointing. Simply specify in your rule that only non-upright pointing pieces matter. If you want to make a rule in which pieces can't "point through" other pieces, simply specify in the rule that only the first piece being pointed at matters. If you want to use a totally new concept that isn't included in this list at all, feel free to do so. For instance, you're perfectly free to create a rule that involves the concept of "pointing with the corner of a piece", even though that style of pointing is not defined at all on this page. The definitions given here only define the most common concepts, which are used over and over again by many different rules.

Since these standard definitions are merely conventions, and since there are concepts like "pointing with a corner" that have no standard definitions, as the Master you must always make sure that you understand a student's terms. When a student guesses a rule, you are obliged to interpret that guess based on the student's terminology, not your own. Therefore, unless you already know that you and the student share a common language, you should always ask the student to clarify terms, even basic ones like "pointing". The students are free to ask you about your own terminology as well; you are not obliged to answer, but you are free to discuss standards of terminology that might make communication easier.

The Structure of a Rule

The easiest way to state any rule is to use one of the two following forms:

• A koan has the Buddha-nature if (and only if) X.
• A koan does not have the Buddha-nature if (and only if) X.

By convention, the phrase "and only if" is usually omitted. If a rule explicitly states that certain koans have the Buddha-nature, you may assume that the rest of the koans do not; if a rule explicitly states that certain koans do not have the Buddha-nature, you may assume that the rest of the koans do.

Some Problem Cases

Even if all of the players have agreed upon standard definitions and terminology, pitfalls and problems can arise. Consider the following:

• "A koan has the Buddha-nature if all of its red pieces are upright." What if a koan contains no red pieces?
• "A koan has the Buddha-nature if all of its pieces are pointing in the same direction." What if a koan only contains one piece?
• "A koan has the Buddha-nature if its largest piece is green." What if a koan contains more than one largest piece?

There are three basic approaches to these conundrums. I call the first the "natural language" approach. According to this approach, these statements all imply that the referent must exist in order for the koan to follow the rule. For instance, the rule "a koan has the Buddha-nature if all of its red pieces are upright" implies that there must be at least one red piece, so a koan with no red pieces at all would be marked black. The rule "a koan has the Buddha-nature if all of its pieces are pointing in the same direction" implies that there must be at least two pieces, so a koan with only one piece would be marked black. The rule "a koan has the Buddha-nature if its largest piece is green" implies that there must be exactly one largest piece, so a koan with two largest pieces would be marked black.

I call the second approach the "western logic" approach. According to this approach, these rules should be translated into unambiguous logical equivalents. For instance, the rule "a koan has the Buddha-nature if all of its red pieces are upright" can be translated into the rule "a koan does not have the Buddha-nature if it contains a non-upright red piece", so a koan with no red pieces at all would be marked white. The rule "a koan has the Buddha-nature if all of its pieces are pointing in the same direction" can be translated into "a koan does not have the Buddha-nature if it contains two pieces pointing in different directions", so a koan with only one piece would be marked white. The rule "a koan has the Buddha-nature if its largest piece is green" can be translated into "a koan has the Buddha-nature if it contains a piece such that there is no larger piece in the koan, it is the only such piece, and it is green", so a koan with two largest pieces would be marked black. (This is actually the same result that we saw in the natural language approach.)

I call the third approach the "mu" approach. It makes use of the Zen concept of "mu", which means something like "unask the question". According to this approach, if you point at a koan that contains no red pieces and ask "are all of these red pieces upright?", the answer is "mu"—unask the question! In other words, you've made an assumption that isn't valid in this situation. The "mu" approach recommends that, rather than translating phrases like "all the red pieces" or "the largest piece" into "natural language" terms or "western logical" terms, simply state your rule in the unambiguous "translated" form in the first place. If you what you mean is "a koan has the Buddha-nature if it contains one or more red pieces and they're all upright", then say that! If what you mean is "a koan has the Buddha-nature if it contains no red pieces, or if it contains red pieces and they're all upright", then say that. This is the approach that I recommend—try to be as explicit and unambiguous as possible at all times. When other players use phrases like "all the red pieces" or "the largest piece", don't hesitate to ask for clarification.

Teaching Terminology

When you teach new players to play Zendo, do not immediately introduce them to the information contained in this chapter. That would only serve to overwhelm and confuse them. Simply begin playing, selecting rules that only utilize the simplest features like color and size. After you play a few games, you can begin talking about some of the basic terms listed above, since the players will then be prepared to digest them. In fact, after a few games, new players will begin asking about things like "pointing" and "stacking" without being prompted. The best time to discuss terminology is when this discussion arises naturally in the course of play.

Details and Clarifications

Selecting a Master

There are no official rules about selecting a Master. If one person has many new rules to try out, that person may be selected as the Master for the entire evening. If everyone has rules to try out, simply take turns being the Master, or specify that the winner of each game becomes the next Master. If you have one experienced player in a group of new players, that player ought to be the Master, at least for the first few games.

Creating Rules

When you are the Master, you may use any rule that you can imagine, though you should always try to select a rule that is not too difficult for the current group of players. When in doubt, use an easier rule. Beginning Masters vastly underestimate the difficulty of most rules, and players do not enjoy games where the rule is too difficult. Many rules that sound interesting can actually result in frustrating games.

There are some official restrictions, which are all consequences of a basic relational property of koans: a koan is not allowed to refer to anything outside of itself, in space or in time. This has several implications.

• No Outside References: You may not make a rule that specifies whether or not a piece is pointing at one of the players, because players are things that exist outside of koans. You may not make a rule that specifies whether or not a piece is pointing in an absolute direction (say, toward one side of the room), because absolute directions are also things that exist outside of koans. Here's a good rule-of-thumb: rotating any individual koan, or even moving it into another room, should not cause its status to change.
• No References to Other Koans: You may not make a rule in which the status of a koan is affected by the contents of other koans on the table. For instance, the rule "a koan has the Buddha-nature if it contains the same number of pieces as any other koan on the table" is illegal, because koans cannot refer to each other in this fashion. Think of each koan as a tiny microcosm—a small, isolated universe that cannot refer to anything but itself.
• No References to Time: Koans are isolated in time as well as space. You may not make a rule that refers to pieces which used to be in a koan, because a koan's past state is something that exists outside of that koan in time. You may not make a rule that has something to do with the order in which pieces were added to a koan. Here is another good rule-of-thumb: as the Master, you should be able to leave the room while a Student is setting up a koan, and still be able to mark it properly when you return. Note that all pieces of a given size and color are considered to be identical and interchangeable.
• Referring to the Table: The playing surface itself is considered to be part of a koan—it is legal, for instance, to make a rule that specifies whether or not certain pieces are touching the table. However, you may not make a rule that refers to some pattern or design on the playing surface, nor may you make a rule that refers to the edges of the surface. Consider the playing surface to be a flat, featureless plane that extends outward in all directions, "whose center is everywhere and whose circumference is nowhere", as the philosophers might say.
• No References to Stones: The black and white marking stones are not actually contained within koans, so you may not make a rule that refers to them.

Building Koans

When you build a koan, you may arrange its pieces in any fashion, as long as they do not touch another koan's pieces or any other foreign objects, including marking stones. It is legal to lean a koan's pieces against each other or to balance them precariously on top of each other. The Master may move any existing koan in order to help distinguish it from another one or to clear space for more of them. The Master should make the moved koan as similar as possible to the way it was before. The Master may disallow a koan from being built if there is not enough space for it.

Breaking Down Old Koans

If, when building a new koan, the pieces you'd like to use are not available, tell everyone which pieces you're looking for. The Master must decide which koan or koans to break down, if any, taking into account the input of all of the Students as much as possible. If all of the Students agree to allow a certain koan to be broken down, the Master should always do so. If, when building a counter-example koan, the Master wants to use pieces that are not available, the Master will tell everyone which pieces are needed, and will decide which koan or koans to break down, taking into account the input of all of the Students as much as possible.

Marking Difficult Koans

The Master's rule must provide an answer for any koan that a Student could possibly build. Sometimes the Master has difficulty deciding how a certain koan ought to be marked, because of some physical ambiguity, such as "is that red piece just barely pointing at that blue piece, or is it just missing it?" In such cases, the Master must make a silent judgment call, and then mark the koan appropriately. The Master must not indicate that a judgment call has been made.

Asking About Koans

Players may always ask the Master clarifying questions about the physical features of existing koans, such as "Master, is that small green piece pointing at the medium red piece?" or "Master, which pieces are touching that yellow piece?" These questions are free and may be asked at any time. The Master must always answer them, even if they have no bearing on the actual rule. Students may even ask about a koan before they are done building it, such as "Master, is this new koan just like this old one, except that the red piece is now blue?" In all matters of uncertainty, the Master's judgments are final.

Guesses and Previously-Existing Koans

If none of the koans on the table can disprove a particular guess, but a previously existing koan that has since been broken down would disprove the guess, the guess still stands and the guessing stone is not returned. Only koans actually in play are used to determine whether a guess is valid. The Master can build the previous koan again as the counter-example, or may build something entirely new.

Katsu

As a Student, you are never allowed to touch a marking stone, or a koan that has a marking stone next to it. If you ever accidentally knock over or disturb a koan's pieces, someone should say "katsu!" in order to indicate that the board has been disturbed. The Master must then restore the table to its previous state. There is no penalty for katsu.

Master's Mistakes

It sometimes happens that a Master makes a mistake which compromises the fairness of the game for the students. When such an error is discovered, any player may immediately demand that the game be terminated. If all players agree to continue the game, the Master should correct the mistake in the appropriate manner.

• Mismarked Koan: The Master might mark a koan incorrectly and fail to fix it before a player has taken another action. If this happens, the Master should fix the mistake as soon as it's noticed.
• Misunderstood Guess: The Master might not completely understand a Student's guess and make a koan that does not disprove it. If this happens, the new koan must remain on the table and the Master must make another koan after the ambiguity is resolved. As Master, you should understand a Student's guess well enough to play another game of Zendo with it as the secret rule.
• Disproving Koan on the Table: The Master might miss the fact that one of the koans on the table disproves the Student's guess and create another koan to disprove it. In this case, the guess stands, the new koan remains, and the Student does not get the guessing stone back. All Students are encouraged to help the Master confirm that a Student's guess works with all the koans on the table.

Tips for the Master

When you are selected to be the Master for a game of Zendo, you take on a very different role than you would if you were playing as one of the students. You will not be trying to guess the secret rule; instead, you will create the secret rule that the others are trying to guess. You will spend much of your time during the game carefully marking the koans that the players build, and setting up counter-examples to their guesses. This may not sound like much fun, but in fact, almost all of the Zendo players I know enjoy being the Master. The challenge of coming up with an interesting and clever new rule that's neither too easy nor too hard, and the fascination of watching a group of students try to solve it, can itself be viewed as a kind of game. In fact, I know a number of Zendo players who find being the Master more enjoyable than being a student (though I personally find them equally enjoyable).

When you're the Master, it's important for you to remember that you're not really a player; you're a facilitator. You're not in competition with the students; your main objective is simply to provide an enjoyable playing experience for everyone. There are a surprising number of pitfalls along the path to this goal; the aim of this chapter is to provide you with some tips on avoiding these pitfalls and becoming a true Zendo Master.

Creating a Rule

One of the most important choices that you must make when you are the Master occurs before the game even starts—the choice of what rule to use. The most common mistake that beginning Masters make is to create rules that are too difficult for the current group of students. Remember, your goal as Master is not to stump the students with a really tough rule—it's to provide them with an enjoyable experience. Most people don't enjoy trying to solve rules that are too difficult, and when you make a bad choice, they'll let you know about it.

Even experienced players will have a hard time estimating the difficulty level of a rule they've never tried before. The best rule-of-thumb is to remember that rules are usually more difficult than they sound. If you are trying to decide between a few different versions of a rule, go with the one that seems the simplest. It's much better to choose a rule that's too easy than one that's too difficult. An easy rule will still be fun, and the game will probably be over quickly, at which point you're ready to simply start another. In contrast, a rule that's too difficult will generate a long and frustrating game—a punishing experience for both Master and student. My advice is to start with rules that seem stupidly simple, and work your way up slowly until you find a level that everyone's comfortable with.

Remember that what counts as "too difficult" depends not only on the experience level of your current group of players, but also on what they're currently in the mood for. Don't be afraid to discuss rule difficulty with them before you choose a rule. Ask them what kind of rule they feel like playing; if they're in the mood for a tough one, they'll tell you. On the other hand, they may not want to know what difficulty you're choosing. They'll tell you that, too.

The second-most common mistake that new Masters make is to choose a rule that's ambiguous or ill specified. You should try to think about all of the possibilities before you actually start playing the game. If you make a rule like "a koan has the Buddha-nature if it contains a large piece stacked on top of a medium piece", you should decide beforehand whether or not a large piece lying on its side with a medium piece inside it counts as a stack; it's dangerous to make this kind of decision "on the fly", in the middle of a game. You may sometimes be tempted to change a rule in the middle of a game, perhaps in the attempt to salvage a rule that's turning out to be too difficult. This is not a good idea. Not only does your new rule have to match everything on the table, but it also has to match everything that's ever been on the table at any point during the game. Unless you have a phenomenal memory, it will be difficult to guarantee that your new rule matches everything that's happened in the game so far. My advice is: understand your own rule completely before the game even starts, and never alter it once the game's in motion.

Of course, many rules do require you to make judgement calls during the course of the game, but those aren't judgement calls about the rule itself. These judgement calls are about individual koans, in which pieces are just barely pointing at other pieces, or just barely touching them, and so on.

Building the Initial Koans

After you come up with a rule, your next task is to build two initial koans—one that has the Buddha-nature according to your rule, and one that does not. How you choose to build these koans is up to you, and is largely a matter of personal style.

When you first play as Master, you may have the impulse to build large, complex initial koans, so that you will not "give too much away" before the game even starts. As you gain more experience playing Zendo, you will see that this is not necessary. It is impossible to give away any rule with only two koans, no matter how you choose to build them. Remember that your rule seems obvious to you, because you already know what it is. The students will need many more than two examples of koans in order to see the patterns that seem obvious to you. Building large initial koans only serves to make the beginning of the game tedious, as the students reduce the sizes of those initial koans to manageable levels. My own preference is to build initial koans that contain between one and four pieces.

Selecting the First Student

After you set up the initial koans, you must select a student to go first. It really doesn't matter who you select; there's no first-player advantage in Zendo. By the time it matters whose turn it is, you'll be deep into the mid-game.

Marking Koans

As the Master, your most important responsibility during the main portion of the game is simply to mark koans correctly. A single mis-marked koan will probably ruin the game, unless you catch it immediately. It's nearly impossible for students to mentally backtrack and undo the damage caused by faulty information. The best thing you can do to avoid this is to be careful. Don't move too fast; think about every new koan the students make, and be sure you're marking it correctly. Use the "down time" while students are thinking to scan the table for possible mistakes. If you do find that you've make a mistake (and you will make a mistake eventually, if you play as Master often enough), be honest. Let all of the players know which koan you've mis-marked, and let them decide how they want to handle it. They may choose to keep playing with the corrected koan. In our group, we usually prefer simply to scrap the game and start a new one.

A different kind of mistake that the beginning Master often makes after a student builds a koan is to accidentally grab the marking stone before the student calls "master" or "mondo". This basically forces the student to call "master" (since everyone has now seen the answer). Therefore, train yourself to listen for "master" or "mondo" before you ever reach for the marking stones.

Yet another issue that arises during the marking of koans is that it's possible to "give away" certain facts about your rule by the way you study new koans. For instance, if your rule has something to do with "touching", the students may be able to glean this fact simply by watching you study the new koans they create. Therefore, I suggest that you study each new koan as if all of its features mattered, no matter what your rule is. If there's a piece on the far side of a koan, and you can't tell whether it's touching some other piece or not, stand up and take a look, regardless of whether your rule has anything to do with touching. If you make this your standard practice, people will not be able to glean anything in particular about your rule from your behavior. Of course, if this kind of behavior becomes too elaborate, it begins to seem like misdirection, which should not be your goal as Master. Your goal should not be to consciously misdirect the students; it should simply be to allow them to figure out the rule for themselves, without any overt clues from you. As with many of the issues in this chapter, how you approach this delicate issue is largely a matter of personal style.

Answering Questions about Koans

When you're deciding how to mark a koan, you may have to make a silent judgement call about whether one piece should be considered to be touching another piece, or pointing at another piece, and so on. Be careful not to indicate to the students that you're agonizing over a tough judgement call; they will most certainly be able to glean important facts about your rule if you do. The students should be responsible for noticing borderline cases, and asking about them if they feel that they may be important.

Because students are allowed to ask you about borderline cases, yet another issue arises: if you answer such a question without even looking at the koan, that tends to imply that your rule does have something to do with that feature (since you've clearly already checked). However, if you look carefully before answering, that tends to imply that your rule doesn't have anything to do with that feature, since otherwise you would have already checked it in order to properly mark the koan. As always, use your own judgement about how to handle these kinds of issues. I personally always study a koan before answering questions about it, even when I already know the answers to the questions.

Finally, remember that you are only obligated to answer questions about the physical features of a koan; you are not obligated to answer questions about why a koan does or does not have the Buddha-nature.

Understanding a Student's Guess

The first thing that you must do when a student takes a guess is to understand the guess completely. You should not even begin to set up your counter-example until you are certain that you understand exactly what the student's guess is. You should understand the student's guess so well that you'd be able to Master a game of Zendo using that rule. Do not hesitate to ask the student clarifying questions if there's anything you don't understand. Ask the student to define any terms that haven't already been agreed upon as standard terminology. Look for and point out any ambiguity in the wording of the guess, and ask the student to clarify.

Many people fear that such open discussion will unfairly give away too much information to the student who's guessing; however, this fear is groundless. The students will not be able to glean any clues from your questions, because you must clarify ambiguities even when they have no bearing on the actual rule. Just as you must always answer a student's clarifying questions about the facts of a koan, you must always ask students about any ambiguities in their guesses, because it's simply not possible to provide a counter-example to a guess that you don't fully understand. In fact, it's in everyone's best interest to fully understand a guess; the other students are perfectly free to ask clarifying questions along with the Master.

Let's take a look at an example. Say that a student offers the following guess: "a koan has the Buddha-nature if the largest piece in it is red". This is an ambiguous guess, because it assumes that koans always have a single largest piece. It fails to specify how a koan with two or more largest pieces should be marked. In Zen parlance, the answer to such a guess is neither "yes" nor "no"; it's "mu"—"unask the question". In such a situation, your only recourse as Master is to point out the ambiguity and ask the student: do you mean, "a koan has the Buddha-nature if all of the largest pieces are red", or "a koan has the Buddha-nature if any of the largest pieces are red"? Some people are bothered by the fact that you, as the Master, seem to be "helping" the student formulate a guess. However, it's important to understand that you haven't given the student any extra clues about your own rule; all you've done is pointed out possibilities that were already inherent in the student's original guess. Your question does not imply that one of the two possibilities is correct; they could very well both be wrong. There's no reason not to allow the student to make this judgement call after being questioned. Indeed, the student is allowed to reconsider the guess and take the stone back at that point. A student has not officially taken a guess until an unambiguous rule has been stated and you have indicated that it's incorrect by providing a counter-example.

Responding to a Student's Guess

After the student has settled upon an unambiguous guess, your task is to provide a counter-example which shows why that guess is incorrect (assuming that it is). This is probably the most complicated task that you must undertake as Master; you must have a clear conception of both your own rule and the rule that the student has proposed, and you must figure out a specific way in which they are different.

Start by imagining different koans, and thinking about how the student's guess would mark them; then think about how your own rule would mark them, and find one that doesn't match. A common exchange tends to occur when Masters set up counter-examples: Master: "According to your guess, this koan has the Buddha-nature, correct?" Student: "Correct." Master (marking the new koan with a black stone): "Well, in fact, it doesn't have the Buddha-nature."

Many new Masters forget the crucial fact that you can disprove a guess in two ways: by building a koan that has the Buddha-nature but the student's guess says does not, or by building a koan that does not have the Buddha-nature but the student's guess says does. For instance, if a student guesses that "a koan has the Buddha-nature if it contains at least one red piece", you can disprove this guess in two different ways: you can build a white koan that contains no red pieces, or you can build a black koan that contains red pieces. Either one of these possibilities will show the student that the guess is incorrect.

Of course, there are times when only one of these methods will be possible, and this fact often confuses new Masters. For instance, let's say that your rule is "a koan has the Buddha-nature if it contains more red pieces than blue pieces", and a student guesses "a koan has the Buddha-nature if it contains red pieces". You cannot disprove the student's guess by building a white koan that contains no red pieces, because, under your rule, there are no such white koans. In a way, the student's guess is half-correct. However, you can disprove it by building a black koan that contains one red piece and one blue piece. This koan does not have the Buddha-nature according to your rule, because it does not contain more red pieces than blue pieces, but it would be white according to the student's guess, since it does contain a red piece.

There are a few common mistakes that you might make when setting up a counter-example. For instance, you may set up your counter-example and say "according to your rule, this has the Buddha-nature, correct?", and the student may say "no, my rule says that it doesn't have the Buddha-nature, and here's why…" Usually this indicates that some kind of miscommunication has occurred between you and the student; at other times, it's simply an error on your part. (It's conceivable the student could actually be trying to cheat by changing the guess in midstream, but I have never seen this happen, and I don't believe an entire group of players could be fooled by it. However, this is yet another reason why it's important to make certain that you—and the entire group of students—understands exactly what the guess is before you set up your counter-example. If you ever suspect a student of cheating in this way, you always have the prerogative as Master to take the student's stone and let your counter-example stand.)

A similar mistake occurs when you set up a koan that's (say) white according the student's guess, and then you realize that it's actually white according to your rule, too. Oops! While mistakes like these will probably not ruin a game, they do provide an unfair advantage to the student who's guessing, because they provide an extra and potentially important piece of information that the student was not supposed to have. If that student has another stone, this new information may prompt him or her to take another guess and win the game. Misunderstandings and mistakes are impossible to eliminate completely, but expect grumbling from the other students when they occur.

There are many different ways to word any rule, and sometimes it's difficult to sort out whether or not a student's guess is identical to your rule. Zendo has been designed so that you will never accidentally declare a player's guess to be incorrect when it is in fact correct (because the only way to declare a guess incorrect is to provide a counter-example). However, it is possible to accidentally declare a guess correct, when in fact it's actually incorrect. When this happens, someone usually notices after the fact that the two rules are actually different, and a counter-example could have been constructed. Obviously, it's impossible to recover from such a mistake, since you've already told everyone your rule. You may take a small consolation in the fact that, since the two rules were so similar, the game was probably about to end anyway. Nevertheless, you should be on your guard against this common mistake, as it does more or less invalidate the game, even when it's obvious who "would have won".

Helpful Masters and Tight-Lipped Masters

One of the things to keep in mind when you set up a counter-example is that you have a fair amount of control over how much information your new koan provides. This is an area in which your responsibilities as Master are very open to personal interpretation and style. On one extreme, you may choose to set up helpful counter-examples that lead the students away from error and toward the correct answer. On the other extreme, you may take the tight-lipped approach, building counter-examples which give away as little as possible, and perhaps even mislead the students, or reinforce "superstitions" that they've developed.

For instance, let's say that a student guesses "a koan has the Buddha-nature if it contains a red piece pointing at a green piece", when in fact your rule has nothing to do with red pieces, green pieces, or pointing. You could choose to build a very helpful counter-example, by (say) setting up a white koan consisting of a single yellow piece, which will strongly indicate to the student that the rule has nothing to do with any of those things. Alternatively, you can take the tight-lipped approach, and build a complex black koan containing many pieces, including a red piece pointing at a green piece. This doesn't tell the students much at all, and may leave many of their "superstitions" about color and pointing intact.

The important point here is that you are under no obligation to be helpful or tight-lipped as a Master—this is a matter of personal style. The only official requirement is that you set up a new koan that definitively disproves the student's guess. However, I do strongly recommend that you be consistent. Over the course of a single game, choose a style and stick with it. Being helpful gives a slight advantage to the student who's guessing, because that student has the first chance to guess again using the new information. This is perfectly fair, as long as you're consistently helpful after all guesses. However, if you're helpful sometimes, and tight-lipped at other times, you'll be providing an unfair advantage to the students who are lucky enough to get the helpful counter-examples. Of course, even when you've chosen to be tight-lipped, a particularly incisive guess may force you to build a very helpful counter-example. In that case, the guessing student deserves the resulting advantage.

Giving Hints, and Giving Up

There is one situation in which it may be acceptable to switch from "tight-lipped" to "helpful" in the middle of a game—the situation in which it's starting to look like the rule you've chosen is too difficult, and the students are clearly ready for some hints. The least intrusive way to give the students hints is to build helpful counter-examples. You'll notice that, as students continue to struggle with a rule that's too difficult, they'll gradually begin working together, until the game becomes a kind of group effort, with people giving each other suggestions and telling each other their theories. In such a situation, people are unlikely to be bothered if you "unfairly" switch into "helpful" mode.

In fact, at some point, the students may, individually or collectively, begin asking for hints. This is a delicate situation, as not all students may desire hints. As always, you must use your best judgement to assess the situation, and decide whether or not to offer a hint, and what kind of hint to offer.

At some point, students will simply not want to continue playing, and at that point they'll have to decide when it's time to give up and have you tell them your rule. Once again, be sensitive to how all the players feel; one or more players may prefer to continue to try to figure it out for themselves, even if the others choose to quit. Do the best you can to make all players feel good about the outcome.

One final note: when players give up, you might be tempted not to tell the students what the rule was, perhaps planning to save it for another occasion. In my opinion, this is an unacceptable practice. For the students, giving up is the least satisfying way to end a game of Zendo, and it almost always indicates that you've made a mistake—you've chosen a rule that's too difficult. Struggling through a too-difficult rule can be a punishing experience for the students; don't make it worse by refusing to provide them with the closure of knowing the rule they worked so hard on! Tell them what your rule was. Consider that your penance.

Tips for the Student

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 chapter 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.

What Matters?

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.

Single-Piece Koans

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.

Color

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

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.

Size

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.

Population Rules

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.

Pointing

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.

Further Experimentation

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".

Falsifying Theories

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.

Compound Rules

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.

Exclusive-Or 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.

Answering Mondos

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.

Calling Mondo

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.

Glossary

The following is a list of common Zen terms, along with some rather whimsical interpretations of them in the context of the game of Zendo.

Koan—In Zen, the term "koan" (pronounced KO-ahn) refers to a short enigmatic riddle or text which is given to a student to meditate upon. Koans often seem paradoxical or nonsensical; their aim is to "break the mind of logic", thus preparing the student's mind to make the leap of insight that lies at the core of Zen. The most well-known Zen koan is "What is the sound of one hand clapping?" In the game of Zendo, each group of Icehouse pieces poses the riddle "Does this configuration have the Buddha-nature?", and is therefore also a kind of koan. Although the answers to Zendo koans are always logical, they often seem capricious and nonsensical to the students, exhausting their logical capabilities and leading, perhaps, to a moment of insight.

Mondo—A question-and-answer session led by a Zen Master, often in a group setting. The Master may ask koan-like questions in rapid succession, in an attempt to throw the students off-balance and lead them to a moment of insight.

Katsu—A Zen word which has no exact meaning. Masters often shout "katsu!" to startle students into attention, or even into enlightenment. It is sometimes translated as "wake up!" I'm not sure if the similar-sounding term "kotsu" is etymologically related; it refers to the stick that Masters use to hit, poke, or prod students who are falling asleep during meditation. In the game of Zendo, the Master shouts "katsu!" when a student knocks over pieces, or otherwise disturbs the board in some fashion.

Sanzen—An interaction between a Master and a student, usually relating to the student's progress on a koan. In Zendo, the term "sanzen" may be used to refer to the portion of a player's turn when he or she spends stones to guess the Master's rule.

Jakugo—A phrase or expression that summarizes a student's understanding of a koan. In Zendo, this term could refer to the actual guess that a player makes when paying a stone to guess the rule.

Satori—A state of mind which is often associated with "enlightenment". It involves a recognition and understanding of Buddha-nature. This is, of course, the goal of the game of Zendo.

Wato—A response to a koan, which may be a word, a phrase, or even a gesture. In Zendo, the "response" to a koan is a black or white stone (or perhaps, the act of revealing such a stone from a closed fist). Therefore, the black and white stones may be thought of as "wato stones".

Zafu—The small cushion that the Zen student sits on during meditation. This doesn't really have much to do with the game of Zendo, but isn't "zafu" a cool word?

Zendo—A meditation hall. More generally, any place where people go to practice Zen (or play Zendo). For the enlightened, the entire world is a Zendo.

Acknowledgments

• Game Design—Kory Heath
• Game Development—John Cooper, Kristin Matherly, Jacob Davenport
• Playtesting—Peter Hammond, Andy Looney, Kristin Looney, Alison Frane, Gina Mai Denn, David Bender, Michelle Lepovic, Eric Zuckerman, Beth Zuckerman, Mike Sugarbaker, Ryan McGuire, Dan Isaac