The Winning Rule in Zendo and the Zen Nature of the Buddha Nature

Hello Starship Captains!


Playing Zendo with my family gave me a new appreciation for the skill needed as a Student and a Master alike to formulate a rule correctly. I noticed some quirks about Zendo that I like to share with you.

This post will discuss the meaning of the Zendo Winning Rule.

By playing Zendo, I noticed the ability of students to come up with rules that a Master couldn't even remember. In other words, the student's rules would be more difficult than any rule a Master would come up with, or dare to come up with. This led to considerations of simple rules that are possibly equivalent to difficult rules. And thinking about this problem, I stumbled upon a hidden aspect of the Zendo Winning Rule.

The Zendo Winning Rule

The Zendo Winning Rule governs when a Student wins. It can be stated in several ways.

Zendo Winning Rule. A Student wins if at their turn they guess the Buddha Nature and 

1. the Student's guessed rule is logically equivalent to the Buddha Nature (or Master's Rule).
2. the Master can not disprove the Student's guessed rule.

Note that if 1. is the case, then 2. is the case. But not vice versa. I prefer formulation 2. over 1. for reasons I will show in this post.

Summary of Game Play

If a Student has a guessing stone at the end of their turn, they can guess the rule. Once the Student and Master agree on the formulation of the rule, it is clear that formulation 1 of the Winning Rule will never be a problem to decide, given that it is possible to compare rules formally. So, as your skills in Zendo and logic improve, it is possible to always decide whether two rules are logically equivalent or not. If they are, the Student wins. There doesn't seem a problem with this Winning Rule, but there is.

If the rules are not logically equivalent, the Master must disprove the Student's (guessed) Rule and to this end he has two options:

1. Make a koan that follows the Student's Rule but does not have the Buddha Nature. The Student would expect it to be marked white, but in fact the Master will mark it black.

2. Make a koan that does not follow the Student's Rule, but has the Buddha Nature. The Student would expect it to be marked black, but in fact the Master will mark it white.

If the Master can not disprove the Student's guessed rule, it must be equal . . . and the Student wins. (This looks like a hands on way to prove that rules are equivalent, so there is no need to be skilled in formal logic.)

However, I found certain sets of rules that are NOT logically equivalent, yet can not (under any circumstance) be disproved by the Master. So, it would seem that Winning Rule 1, although sufficient, is not necessary to win! If the Master fails to disprove the Student's guess, then by Winning Rule 2, the Student wins nevertheless.

At this point an example could come in handy. See the next section.


We know some of the attributes of pyramids, used in formulations of the Buddha Nature: color, size, groundedness, orientation, etc.

Now, for any attribute A, consider the rule schema S(A): "All pyramids in the koan have attribute A."

Example. Consider the attribute A = red, Then S(A) gives the rule: "All pyramids in the koan are red." It is easy to decide of any koan whether or not it follows this rule.

Example. Now consider the attribute A = ungrounded. Then S(A) gives the rule: "All pyramids in the koan are ungrounded." We run into a problem finding a koan that follows this rule . . . 

It is quickly realized that in any koan (played on Earth), there is at least one pyramid touching the table, and hence is grounded. At least within the confines of the game rules (pyramids are not allowed to touch anything else but pyramids and the table), there is NO koan with at least one pyramid that ever will follow the Buddha Nature. The Master would have a hard time setting up the initial koans to begin with! It is impossible for any koan to follow the just statedThe only koan that follows this rule is the empty koan (consisting of no pyramids at all). A Student guessing this odd rule will be able to win with a guess of the form "the koan contains no pyramids". The Master would not be able to make a counter example, and the Student wins.

Example. Consider the attribute A = weird. Then S(A) gives the rule: "All pyramids in the koan are weird."

In this case it IS possible to make a non-empty koan following this rule, but it requires a minimum number of pyramids. The question is how many exactly. It's larger than 4, because 4 pyramids can not be configured to even form a closed loop (see Fig. 1). In this case, an interesting question arises.

Suppose the minimum required pyramids is N. Then the rule is equivalent to "the koan has at least N pyramids and all koans are ungrounded." Now the problem is this. How would the Master know the correct value of N? This could be a difficult--even unsolved--geometrical question. So, if the minimum is in reality 5, but the Master can only make koans of at least 6 pyramids, the Student will win with the rule "The koan contains at least 6 pyramids and all pyramids are ungrounded." The possibility of the Student winning because of lack of koan building skills on the Master's side, is therefore real.

We are now faced with the fact that the rules the Students can come up with in order to win, are not necessarily logically equivalent to the Buddha Nature, NOR do they even have to be correct from a theoretical standpoint!

Too Few Pyramids

A rule schema like S(A) also poses another problem. With a given set of pyramids, there is sometimes only a finite number (although a very VERY large number) of koans that can be built that follow the Buddha Nature. Consider the Buddha Nature: "All pyramids in the koan are red." In a standard Zendo set, there are exactly 15 red pyramids. Therefore, the rule is equivalent to "The koan exists of no more than 15 pyramids, which are all red." Obviously, this rule is NOT logically equivalent to the Buddha Nature. A student guessing this rule will win, because the Master simply doesn't have more than 15 pyramids, which he would need to disprove the rule.


We see that the first winning rule isn't sufficient to allow the game to end, because there are rules the Master can think of for which he can not (is not, under any circumstance, able to) disprove any Student's guess which is not logically equivalent to the Buddha Nature. And in that case, the Student should win . . . if only for the fact the rule is too difficult, even impossible, to guess.

Therefore, the Zendo Winning Rule: "The Student wins if the Master can not disprove the Student's guessed rule" is the preferred and reasonable rule which will allow Students to win, although sometimes with rules that are NOT equivalent to the Buddha Nature, NOR always correct from a theoretical standpoint!

Isn't that the true Zen nature of the Buddha Nature?

This, I'd like to add, will only add to anybody's fascination for Zendo.


Can you find a koan which contains exactly and only 5 weird pyramids? The best I could come up with was 7 before I wrote this post, but today I reduced it to 6. (See Fig. 2). It is clear you can find koans that have only weird pyramids of any number bigger than 6, by increasing the circle. (See Fig. 3).

See also my next post on solving some of these problems, and perhaps making Zendo more interesting, if only for advanced Zendo players.

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Interesting; I'd interpret these examples differently.

E.g. "All pyramids are ungrounded" and "The koan contains no pyramids" ARE logically equivalent to me, given the basic playing conditions (that pyramids sit on the table or on other pyramids). If we want to consider the possibilities that pyramids are floating, then either (1) we are in some unusual physical situation where that is actually possible or (2) it seems a dubious impractical choice for a rule, like making a koan about some color of pyramids which we don't have on hand, or making a koan about very large numbers. I.e. to me, there seem certain logical givens (pyramids rest on things in gravity, the koans involve the colors we have on hand, the number of pyramids on hand suffices to construct a "reasonable" number of positive and negative examples, etc).

The practical limits of how many pyramids one has on hand seems just that - a merely practical limit, not a logical issue, just as (e.g.) a boardgame might have a perfectly logical coherent rule that you need to roll a die one million times, but that would (for practical reasons) not be physically do-able in reality.


> > >
"The Student wins if the Master can not disprove the Student's guessed rule" is the preferred and reasonable rule which will allow Students to win, although sometimes with rules that are NOT equivalent to the Buddha Nature, NOR always correct from a theoretical standpoint!
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But it seems trivially easy to make such a rule if you consider the physical limits of our pyramids on hand to be meaningful and to mean that the master cannot physically/practically disprove the proposed rule.

E.g. a student has figured out that the rule seems to clearly be "The koan has red pyramids". So they propose "The koan has between 1 and 100 red pyramids". The master's rule and the student's proposed rule agree about all koans with no red pyramids, so to disprove the proposal, the master would need to make a koan with 101 (or more) red pyramids, which the master cannot physically do if they only have 15 red pyramids.

Unless I'm misunderstanding something, I'm not seeing the value added by this alternative formulation of winning, as opposed to the rules as written: the student wins by simply saying "The koan has red pyramids". :)

Hi Russ,

My post is basically an observation that might turn out to be useful for advanced rules (see also my follow-up post (No Pyramid Extension), or when Students formulate a rule clumsily, but in such a way that the Master can not disprove it. I showed in several ways that practical limits may influence with what guess a Student wins and that the first formulation of the Winning Rule is not satisfying in all cases.

In the case "All pyramids in the koan are red", this rule tends to be guessed that way (and I agree that logically it includes the empty koan). This post was not about finding rules that utilize this principle, but to indicate that the Zendo Winning Rule preferably follows the second formulation to prevent a deadlock.

In the Extended Version of Zendo, the rules "All pyramids in the koan are ungrounded" and "the empty koan" are not equivalent. That's the value of this kind of consideration.

There are multiple ways to do exactly 2 all weird or exactly 3 all weird. If you combine one of each, you get exactly 5 all weird.

Hi Jeff, 

perhaps you've proved exactly my point, as I forgot about leaning pyramids! If it were in a real game of Zendo, I'd be at a loss as Master. But since I believed you, I found such a koan within 1 minute!

Even though I found some koans with all pyramids weird, I couldn't find such koans with 2 equal pyramids. The examples I found, are hardly stable, but that is OK, I guess.

So, there are only certain combinations of sizes of pyramids that allow for koans with all weird pyramids, These combinations can be listed in a set S, a subset of {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}, and I believe S is a subset of {(1,2), (1,3), (2,3)}. If you can find other koans, please show them.

Then a Student's guess might be:

"The koan exists of at least two pyramids, which would be in sizes according to S, and all of them are weird."

Again, this rule is logically not equivalent to the Buddha Nature "all pyramids in the koan are weird", but can not be disproved by the Master.

I just wondered if the term "simple" for rules is arbitrary? Considering the aspects of koans that are mostly overlooked, as you get better in Zendo, it becomes less clear what is "simple", because you see all kind of exceptions or restrictions that might apply to such a "simple" rule.

Great response!


It's possible to do 2 pyramids, same size, both weird. After reading your post I was able to construct all three examples, but technical problems prevented me from uploading photographic evidence. Smalls are the hardest to make, but when I finally succeeded the koan was pretty stable. The pyramids in these koans are tip to tip, but it's not actually necessary to perfectly align the tips. Still, I suspect it will be easier with the blunter-tipped Arcade pyramids, although some of my problems with the smalls had to do with center of gravity and not tip alignment.

And this type of discussion, is why Zendo will never be on my Starship Captain's list-it's too confusing of a game to me.

FWIW I have never had these kinds of problems arise in play, and have successfully enjoyably played it with many diverse experienced and new players alike.

We don't consider exact physical position to be relevant, because that seems to introduce various annoying practical problems (well, they are "Features" for some people and "Bugs" for other people) like ambiguity and difficulty of moving koans on the table.

We always play either:

(1) a koan is simply an unordered set of pieces (physical position is irrelevant)


(2) a koan is simply an ordered stack of pieces (there is an unambiguous ordering to the pieces).

So if you're leery of Zendo because of weird theoretical discussions like this thread, I recommend trying it one of these two ways, which seems to eliminate much of the potential "weirdness" and confusion. :)

As Russ said, these things don't come up in practice. Personally, I find this whole discussion confusing, too, and Zendo is one of my favorite games.

That's why my own contribution to this discussion has been limited to what you can do with "weird" pieces. If you don't know what "weird" is, it's pretty easy to understand. Imagine you dump all your pyramids on the table in a heap. Some of the pieces will be standing up and some will be lying down, but many of them will be leaning or hanging or otherwise not standing up or lying down. You might say to yourself, "those pieces are in kind of a weird orientation." That's why Zendo says any piece that's not standing up or lying down is called weird.

Don't let this thread put you off Zendo. The issue raised in this thread can't actually come up, and Zendo is a great game.

I started this thread because I am fascinated with Zendo and like to share some quirks that could come up in advanced play. If anything seems confusing, please play Zendo the way you like. I hoped to add to the appreciation for Zendo.
After all, this is the forum to share insights, together with questions on any Looney Pyramid game. Take the good as you see fit.

Formulation 1 of the Winning Rule could be repaired by changing "logically equivalent" to "equivalent for all constructable koans". Depending on how we define the logical space of the game, there may be logically possible koans that are not actually constructable.

Yes, I agree. "Equivalent" would then mean: "uses the same color of marking stone". Note that your formulation "equivalent for all constructible koans" is now equivalent to formulation 2. This is seen as follows.

If the Student's Guess G and the Buddha Nature B correspond for all constructible koans k, i.e., G(k) = B(k) := the color of the marking stone, then the Master can not disprove G, since he can only make constructible koans.

If the Master can not disprove G, then for any koan k with G(k) <> B(k), koan k is not constructible (otherwise koan k would be a constructible counterexample), and therefore, G and B coincide for all constructible koans.

Thank you P.D.M.



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