The I Ching, random numbers, and why you are doing it wrong

One would think that humanity would not have a need for good random number generators until computers and simulations were invented since, for most practical purposes, tossing a coin or throwing a die should suffice us all. So you can imagine my surprise when I saw in this four to five thousand years old Chinese divination book called I Ching a RNG algorithm that reminds modern Linear Congruential Generators! But why the need for such a complex procedure to render random numbers?

         Artemisia Stems

The I Ching divination process requires to randomly select two trigrams via a rather convoluted process using either stems of Artemisia or Yarrow. And although I acquired this ancestral book a long, long, time ago, truth is that when reading it as an oracle I always used the simplified version for lazy busy people consisting in simply tossing three coins and checking the combination of heads and tails.

I always thought that the traditional form was just a magical way to do the same thing that we can do by tossing three coins, but today, for no particular reason that having too much free time in my hands, I gave a deeper mathematical look to this traditional form and it turns out that it renders a complete different random result that tossing three coins!

Well, a mathematical curiosity you might think, but does it matter? It might! Millions of people seek advice using the simplified coin version to render the I Ching Yin Yang oracles. In this post I will show how the three coins method yields an equal proportion on Old Yin and Old Yang oracles signs whereas the traditional method yields three times more Old Yang signs than Old Yin!

This means that The I Ching, in its traditional form to draw oracles, promotes Yang behaviour over Yin, that is, it promotes among its users action, imagination, creativity, strength whereas, nowadays, with the simplified three coin version, the active and passive answers are even out.

I am not a sinologist nor a psychologist so I cannot really tell what version would have a better influence among practitioners lives, but I know though that the traditional form promotes Yang among those seeking advice which, at first glance, seems like a positive thing to do and, since this book is used by millions of people, maybe experts in the field should advice to practitioners not to use three coins anymore when using the I Ching. For those interested in having a traditionally sound oracle in terms of probability, I will show a few simple ways to achieve just that at the end of this post.

This book has impressed mathematicians like Leibniz, psychologists like Jung, poets like Jorge Luis Borges and all kind of intellectuals all over the world for centuries. And regardless you believe or not whether it has magical properties, what is certain is that it has deep psychological sapiential ones. This is not only the oldest book in human history, but a beautiful one. So, before we plunge into the mathematical details of the traditional algorithm to draw oracles, let’s share this poem from Borges about the I Ching to break the ice.

For a Version of I Ching Para una versión del I King
The future is as immutable
As rigid yesterday. There is nothing
That is no more than a single, silent letter
In the eternal and inscrutable
Writing whose book is time. He who walks away
From home has already come back.
Our life Is a future and well-traveled track.
Nothing dismisses us. Nothing leaves us.
Do not give up. The prison is dark,
Its fabric is made of incessant iron,
But in some corner of your cell
You might discover a mistake, a cleft.
The path is fatal as an arrow
But God is in the rifts, waiting.

El porvenir es tan irrevocable
Como el rígido ayer. No hay una cosa
Que no sea una letra silenciosa
De la eterna escritura indescrifrable
Cuyo libro es el tiempo. Quien se aleja
De su casa ya ha vuelto. Nuestra vida
Es la senda futura y recorrida.
Nada nos dice adiós. Nada nos deja.
No te rindas. La ergástula es oscura,
La firme trama es de incesante hierro,
Pero en algún recodo de tu encierro
Puede haber un descuido, una hendidura,
El camino es fatal como la flecha
Pero en las grietas está Dios, que acecha.

Artemisia / Yarrow vs Three Coins

We will now compare the results from the traditional Artemisia / Yarrow method to obtain the two I Ching oracle trigrams as explain by Richard Wilhelm in his The I Ching: The book of changes with those from the simplified coin version described in the same book.

The Three Coins

For the coin version we toss three coins six times and we build an hexagram from bottom up with the following correspondence and probabilities:

  • Three heads:  Old Yang       (9)    (Pr = 2/16)
  • Two head:       Young Yang  (8)    (Pr = 6/16)
  • Two tails:        Young Yin      (7)    (Pr = 6/16)
  • Three tails:     Old Yin           (6)    (Pr = 2/16)

Every hexagram has a meaning and, if it has Old Yang or Old Yin lines, its meaning is extended by the I Ching with further advises. In this case, the Old Yin / Yang extension is done evenly with the same probability (2/16) when using three coins.

The Artemisia / Yarrow

This ancient method consists in dividing three times and into two groups a set of 50 stems of Artemisia or Yarrow. At each division a calculation is done, a number of stems are left out and the process is repeated. For a complete human language explanation of the process I highly recommend you Mr. Richard Wilhelm translation of the I Ching, but next you can see this process in the more precise and concise R language.

IChing = function(N=6){
set = NULL
for(i in 1:N){
sign = 0
hand = 0
for(j in 1:3){
left = sample(1:(48-hand),1)
right = 49-hand-left
pinky = 1 ; right = right – 1
anular = ifelse(left%%4==0,4,left%%4);
middle = ifelse(right%%4==0,4,right%%4);
fingers = pinky + anular + middle
hand = hand + fingers
sign = sign + ifelse(fingers==9 || fingers==8,2,3);
set = c(set,sign)

view raw


hosted with ❤ by GitHub

The previous R function returns six Yin Yang elements coded from 6 to 9 following the traditional method to obtain I Ching hexagrams as explained by Richard Wilhelm.

Seems clear to me that the intention in this procedure is to randomly choose at each step a number from 1 to 4 by applying a modular arithmetic process at each division, but the algorithm does not exactly calculates that since, when expanding all possible solutions, there is a tiny bias towards 4(3) results. So if we ignore this tiny bias the probability tree the algorithm generates for each Yin Yang code sign follows:

probability treeThe final probabilities in this tree imply the following Yin Yang  hexagram lines probability distribution

ichingSo if we now compare the Artemisia / Yarrow Yin Yang probabilities with those rendered by tossing three coins we have:

  •                                             Old Yin    Young Yang    Young Yin   Old Yang
  • Artemisia / Yarrow:           1/16          5/16                   7/16              3/16
  • Three Coins:                          2/16          6/16                   6/16              2/16

So both methods return equal probabilities for each hexagram since we have a 50/50 chance to have a Yin or a Yang sign, but we don’t have the same probabilities for their interpretations.

The traditional method clearly favors Yang interpretations of the hexagrams which makes the coin procedure a bad alternative for an accurate reading of the I Ching as intended by they ancient Chinese who developed the method and, if we consider how this book is being use all over the world, it might not be a bad idea to stop using the three coins procedure in favor of the traditional one since the later promotes more positive and active interpretations of the hexagrams.


Now, I wondered a number of questions about why this procedure is the way it is, so unless some sinologist in the room says otherwise I am going to make some “common sense” assumptions.

Why the use of stems instead wood sticks?

The use of stems instead straight wood sticks might be due to avoid getting a “feeling” of the sticks’ width and getting always a desired result (similar to what might happen with a deck of cards). the gnarled stems with its different widths help to prevent such bias and increase the randomness of the process.

Why 50 stems of Artemisia / Yarrow?

why 50 stems when the first step is to discard one? Well it could be magical reasons but it could also be an algorithmic left over reason since, at each of the three divisions, one stem is discarded. Another reason could be that it also increases the randomness of the process; by discarding a different stem each time it becomes more difficult to get a “feeling” of the stems and unconsciously obtain desired results.

Why Those uneven probabilities?

I would say it has to do with numerology, the proportions for Yin and Yang signs yield by the algorithm are represented by the first four prime numbers, that is, 1, 3, 5 and 7. Ancient Chinese might have noticed that 1+7 = 5+3 and associate this mathematical property with the Yin and Yang equilibrium in the universe. Once this is set the probabilities for the Old Yang and Old Yin are also set and they decided for a higher degree of action among those using the I Ching.

Why 3 COINS as alternative?

Obviously the Artemisia / Yarrow process is overly complicated, but I don’t know why the alternative chosen to simplify the process were the three coins version since they clearly change the I Ching oracles. Did those developing the alternative knew what they were doing? Was is just a way for lay people to use the book without much hassle and they did not care? Any sinologist comment is more than welcome!

Alternatives to Artemisia / Yarrow than render traditional Hexagrams

Anyhow, now I will explain a couple of alternatives that return proper traditional sign probability hexagrams, let’s begin with my favorite one

The one coin + one 8 sided die


This is my favorite alternative both, for its simplicity and its beauty.

For this alternative we simply need to first toss a coin; if we have tails (0) that means Yin, if we have heads (1) that means Yang.

Then we throw the 8 sided die to determine if the Yin or Yang sign is old or young. If we do so we have the following 16 possible outcomes:

0,1 – Old Yin
0,2 – Yin
0,3 – Yin
0,4 – Yin
0,5 – Yin
0,6 – Yin
0,7 – Yin
0,8 – Yin
1,1 – Yang
1,2 – Yang
1,3 – Yang
1,4 – Yang
1,5 – Yang
1,6 – Old Yang
1,7 – Old Yang
1,8 – Old Yang

Which means that when having a Yin result with the coin, a result of 1 with the die (we could choose other number I just thought to use small numbers for the Old Yin and big numbers for the Old Yang) will make it Old Yin, and when having a Yang result with the coin , a result of 6,7 or 8 will make it Old Yang.

Now we simply have to throw the die six times, build the hexagram from bottom up and that’s it, enjoy your traditional I Ching oracle hexagram!

The one 16 sided die


Another simple way -perhaps the simplest- to build I Ching hexagrams would be to use a 16 side die and then associate its side numbers as follow.

  • 1          : Old Yin
  • 2 – 6     : Yang
  • 7 – 13   : Yin
  • 14 – 16 : Old Yang

Another perhaps more aesthetically pleasant distribution of numbers would be to place all the Yin possibilities (1 + 7) at the top of the bi-pyramid and all the Yang (5 + 3) a the bottom.

Looking for dices I found an amazing web page where we can buy dices, coins and many other beautiful designs… Just for you to know.

Anyhow, happy traditional choice!


I would like to thank all the kind and insightful comments on this post, I do truly appreciate it. A few of you also shared alternative methods to the Artemisa/Yarrow method that also maintain the right probabilities, thank you very much for sharing.

I thought about updating this section with your kind contributions and give you due credit. However, I find more efficient just to recommend here to the visitors to check the comment section below for these alternatives -which are many and very creative!

Thank you all!

81 thoughts on “The I Ching, random numbers, and why you are doing it wrong

      • Thank you guys. I really like this book and when I found out about the Yin Yang probability imbalance I got all excited. 🙂

        • Dear Fran, thank you! I’m too tired today to understand the beauty of your article (I sense its beauty). My significant other is open-minded but does not like it when I I Ching. If I could do it on my PC, it would work better. Is there, anywhere, a programme to be trusted for this? Kindest wishes, Etienne

          • Etienne, dude, no. If you want throwing the I Ching to have any meaning and connection to you and your reality at that moment, you have to do it. Physically pick up yarrow stalks or pull a marble out of a bag or whatever. (distinct groupings of marbles that are exactly the same shape but different colours: 1, 3, 5, 7. 1, the single, is a broken moving line, and 3 (all the same colour) gives a solid moving line. Same probabilities, same interaction except somewhat faster, so just take your time.) Point of the process is to connect you with the randomness of the Universe. The same unseen forces that are affecting you are affecting the item (a bunch of stalks, a marble) you pick. If you don’t actually do that what do you think is actually happening? How is a randomly generated number controlled by a timer chip in a computer triggered by a programmer’s algorithm the same event? It’s not. The whole point of divination is for you to do the work.

  1. Wow fascinating exhaustive article, bookmarked for future reference. I have used the I Ching several times in the past and even have a (don’t laugh) I Ching app on my iphone!

  2. A worth-reading post. I never thought of the probability associated with each “yao” or “yiao.” (!

    I am not sure what the Artemisia method is. However, I’ve derived a different probability distribution for the yarrow method described here – The probability of getting 4 or 8 on the second and third “changes” is not 1/2. I might be wrong. I need to double check my reasoning later.

    • The Yarrow method uses Achillea millefolium, Richard Wilhelm translation mentions Achillea and Artemisia, so basically it is the same traditional method using different stems. Maye I should update the post with both kind of stems, thanks for mentioning.

      The link at eheart basically describes the same method found in Richard Wilhelm translation, they also realized that the probability of getting a changing yin line (Old Yin in Whilhem translation) is smaller but they don’t go further.

      About you having different probabilities, in the post I mentionted.

      Seems clear to me that the intention in this procedure is to randomly choose at each step a number from 1 to 4 by applying a modular arithmetic process at each division, but the algorithm does not exactly calculates that since, when expanding all possible solutions, there is a tiny bias towards 4(3) results. So if we ignore this tiny bias the probability tree the algorithm generates for each Yin Yang code sign follows

      So it is not exactly 1/2, maybe this is why you have different results? but if you think about the procedure I would say it is quite obvious that 1/2 was the intention, in fact, if the number of stems grows to infinity the bias disappears and the probability converges to 1/2.

      I did not calculate it but perhaps there is a larger number of stems that would remove the tiny bias without going to infinity? In any case ancient Chinese probably considered practical issues and thought that 50 stems was a good enough approximation.

      Thanks for your comment JH 🙂

  3. After the first “change,” there will be 40 or 44 sticks left. Removing one stick from one of the piles, there are, e.g., 39 left altogether. The possible remainder combinations are (1,2), (2,1), (3, 4) and (4,3). I made a mistake thin inking the third possibility as (3, 0), temporarily forgetting that four sticks are set aside when the remainder is zero in this case.

    Anyway, yes, if we write out all possible ways of dividing the sticks, the probabilities are not exactly 3/4-1/4 and 1/2-1/2.

    Thanks for your reply.

  4. The link to Leibniz takes one to Jung. Minor error. Enjoyed your post.

    “This book has impressed mathematicians like Leibniz, psychologist like Jung, poets like Jorge Luis Borges and all kind of intellectuals all over the world for centuries. And regardless you believe or not whether it has magical properties, what is certain is that it has deep psychological sapiential ones. This is not only the oldest book in human history, but a beautiful one.”

  5. I use a simple method to maintain the probabilities.

    I have 16 marbles 1 Red (Old Yin) 3 Yellow (Old Yang) 5 Green (Yang) and 7 Blue (Yin)

    Pick them one at a time and return to dish and pick the next etc etc. for the 6 lines.

    You can of course vary the colours provided you keep to the ratios.

    • that is the method I use. I pick the marbles from a bag, meditating with my eyes closed. Until I have pulled out the marble, hehe. I got this from REDISCOVERING I CHING 1995
      by Greg Whincup, which I bought then. He goes back to the original source material and does a translation that also considers all the subsequent scholarship. He removes centuries of obfuscating embellishments and interpretations. Clarity!

  6. Hi,
    I run website based on 3 coin method algoritm. It’s been 9 years and I have collected over 3.000.000 questions and answers in form of 678967. Your article surprised me and I’d like to change my algoritm to yarrow to see how the results will change … thanks for inspiration!!!

  7. Another reference i came up recently since this is part of some studies i do as well, by physicist-biologist S. Petoukhov, which relates the DNA bases and biological symbolic alphabet to the trigrams and hexagrams of the I Ching, and the dualities of Yin/Yang. Quite interesting, and not only for random generation!

    Sergei V. Petoukhov (1999) (

    Click to access pethoukhov.pdf

    Specificaly i quote:

    […]The modern science does not know why the alphabet of genetic language has four
    letters (it could have any other number of the letters in principle)…

    […]The starting point of the original research, stated in the article, was the following. The
    author paid attention to the fact that these four nitrogenous bases represent specific
    poly-nuclear constructions with the special biochemical properties. The set of these four
    constructions is not completely heterogeneous, and it has on itself a substantial system
    of attributes for uniting and distinguishing of its letters (or, more precisely, the set has
    pairs of “attribute – anti-attribute”).

  8. Thanks for the interesting article. A couple of things I would like to add:

    1) We don’t know for sure what exact yarrow stalk method was used by divination officials in late Shang and Early Zhou times. It is quite possible it was much different then the 12th century Zhu Xi explanation which is official today (please see more about that here:

    2) It is very likely coin method has been in use much longer and has been used much more frequently than the yarrow stalk method.

    3) But the most important question is: does it really matter? I think it does not really matter too much because even though we call it a random selection of the lines/hexagrams the basic idea is that cast hexagram IS NOT RANDOM because it must give meaningful answer to the question asked. Otherwise we would accept basic premise that any given hexagram and any given moving line would equally well “answer” the question. Which I do not believe is true.

    4) Probability in favour of more yang lines would not imply inherent logic of Yi Jing oracle is towards action. it just means that one would (on average, when used in the context of large numbers) get more yang moving lines which in turn may mean anything.

    5) If you toss a coin 5 million times on average you will get 2.5 million heads and 2.5 million tails however if you flip the coin only one time this coin will act as if no coin was ever flipped before and no coin will be flipped after. In my opinion traditional statistical probabilities make sense only in the context of large numbers. As we know we do not cast Yi Jing hexagrams 5 million times for one question therefore all those probabilities have very relative meaning.


    • Thank you very much for your points, they are really interesting! Let me comment them briefly here:

      1) Even if we don’t know the original method at least we know that the scholars in the Zhou dynasty decided to hold the yarrow stalk method as official.

      2) According to the author of the link you posted “Although yarrow divination is mentioned in the Zuozhuan, a chronicle of the period 722–468 BCE, the actual procedure is not described in that text.” also “Tradition holds that Wang Xu, a Daoist recluse, invented the coin tossing method in the 4th century BCE.”

      So we have a chronicle vs a tradition holds, you can make your source choice. What I can believe and I would agree with you is that the average person would not want to use the yarrow stalk for being overly complicated and that they would use the simplest tossing coin procedure.

      3) I will disagree on this point with you. It is random, if it wasn’t you would always get the same anagram for the same question which is not the case, or opposite answers for opposite questions which is not the case either. What is not random is our interpretation.

      4) Well, again, the Zhou scholars decided that the yarrow stalk procedure was a sound logic for the Oracle, I am not a scholar myself so I can’t really tell otherwise. Also, if the practitioner associates Yang with action and Yin with caution, which I would say is a reasonable association, then yes, having a larger proportion of moving Yang means something in my opinion.

      5) The inference process that goes from the general to the particular is called deduction, and this is precisely the inference process Probability Theory is good for. Tossing a coin is one of this particular cases.

      Thanks again Zhouoracle!

  9. We use two coins:
    We throw first and then the other coin out system:

    HH Yin (old)
    HT Yin
    TH Yang
    TT Yang (old)

    1/4 + 1/4 + 1/4 + 1/4 = 1
    Equally and fairly.

    • i think the whole point of the discussion here is whether these probabilities, faithfuly represent the sample space. Based on the analysis by Fran on the yarrow-stalk method vs 3-coins method (or 2-coins method), this would not be so. In other words, the sample space may indeed have more old yin or old yang items than new yin / new yang. This is the whole point, and in this sense the yarrow-stalk method would indeed faithfuly sample and represent the event space, than the other methods.

      One possible justification for that would be that since things change (after all this IS the book of changes), it would mean that under any circumstamces changing lines should appear more frequently than non-changing lines and so on. So I Ching places a (justified) emphasis on changing lines, more than non-changing lines (this coupled with the fact that we are talking about trigrams/hexagrams and so on)

      • Sorry, minor correction, the rationale of having more new yin / new yang than old yin / old yang, is related to being a faithful reperesentation of the underlying event space. A possible justification for this distribution has to do with the whole theory behind the I Ching and how changes happen and what they represent.

        • Hi, Nikosmos!

          Why not 1/4 + 1/4 + 1/4 1/4
          instead of 3/16 + 5/16 + 5/16 + 3/16.

          Tell me, why not!

          • It may very well be 1/4 + 1/4 + 1/4 + 1/4. The I Ching has a theory behind it which relates trigrams, hexagrams and the like into a representation. Associated to this theory, which we do not know very well (inspite some “new-agers” claiming otherwise), is the method to sample the event space in order to generate hexagrams for a given situation.

            In this sense and given Fran’s analysis in this post, it is plausible the yarrow-stalk method (given the theory of the I Ching), indeed faithfuly represents and samples the event space. One may choose to sample the event space differently, it does not guarantee faithful sampling though (unbiased sampling if you like), unless one can justify it based on the (theory of the) I Ching itself. And since this theory is not exactly known, one can say that based on the I Ching and the documented yarrow-stalk method, this most probably matches the (largely unknown) theory of the I Ching itself on the event space and how to sample it correctly.

            Also note that ancient Chinese were very good number-theorists among other things (e.g the chinese remainder theorem), and probably knew enough methods of sampling. Using a method for a specific purpose, should have a meaning (assuming the yarrow-stalk method was indeed used as documented) This was my comment.

          • Additionaly have in mind that the I Ching, like other (sacred) books of the east, has in its writing and content, a codified multiplexed context. This in simple terms means that the same text can represent multiple (note, simultaneously compatible and valid) associations which unfold based on external data and stimuli. This extra data (or decipher code if you like, what i refered to as the theory behind the I Ching in the previous comment) are usualy provided on initiation, which further unlocks the whole meaning that has been codified (if this is the case of course) into the text.

    • So, one thing the article glosses over is that, when taking a reading, you look at your old lines as changing and they flip to the other line. If you have moving lines, the hexagram you pick is supposed to represent the current situation. The moving lines each have an interpretation associated with them that reflects the things changing in the situation and how to get to the solution. This leads to an end-state hexagram that represents where things are going.

      The yarrow stalk method gives you the same number of ying (passive, feminine, receptive) & yang (active, masculine, creative) lines. This means that the chance of finding a *starting* hexagram to describe the situation is equal. It gives you more old ying lines than old yang lines – lines that flip – so the *ending* situation and the steps to get there favor a more active approach to problems.

  10. Hi Fran nikosms,
    thank you for your comments – all of them make sense and you might be right on all points, I do not claim monopoly on the truth when it comes to Yi Jing or anything else for that matter 🙂 However since I enjoy civilized and intelligent discussions I will add a few more lines:

    1) Even though Yi Jing (as a text and way of thinking) pre-dates yin-yang theory for many years I still believe yin yang paradigm is inseparable from the Yi Jing logic. In this context one can see randomness as yin and deliberate/intentional choice as yang. They form a pair as any other yin and yang pair (such as for example fate/destiny vs free will). One of the axioms of yin-yang theory is that those two (yin and yang) always come in pairs, they are inseparable, one may be dominant but the other one is always there. For that reason I believe that any method we choose (coin method, yarrow stalk method or any other method) must have an element of randomness just to “immobilize” our ego/mind (yang) and unleash what I would call subconscious faculties (yin) which basically lead us to the answers however I would have REALLY hard time accepting that answers we get are totally random because that would imply that yin exists without yang which is not possible.

    2) Statement “It is random, if it wasn’t you would always get the same anagram for the same question which is not the case” and “it is plausible the yarrow-stalk method (given the theory of the I Ching), indeed faithfuly represents and samples the event space” totally make sense and may very well be true however they sound an awful lot like a children of the scientific paradigm where repeatability (of an experiment) is the only valid proof of objective truth and existence of the phenomenon being investigated. As already stated Yi Jing is the book of CHANGES, expectations that we confirm validity of the given answer by drawing the same hexagram two (or more) times in a row comes from a different (scientific) “cognitive space”. Yes, “sample of events space” as nikosms put it does play a role, I would agree, however I must repeat my original statement – I do not believe one or the other sample of event space would critically affect efficacy of any given method. Why? because it is not all about randomness as I elaborated in the previous point.

    3) Concept that one method may “more faithfully represent sample space” then any other given method leads us again to the same dilemma: are those methods themselves random (yin) or they are carefully and deliberately selected by someone (yang)? Again I think they are both at the same time but I think yang is dominant – they are not totally random methods. For example it is well know that ancient Chinese knew about magic squares (3×3 and others) and probability of 1/16, 3/16, 5/16, 7/16 in Yi Jing numerology can be seen as 1/7, 3/7, 5/7, 7/7 somehow “make more sense” to me than coin method probabilities and it is quite possible that what Nikosms said is true however one thing I know about Chinese people is that they are and they have always been pragmatic: if coin method was inferior to yarrowstalk method they would not use it for so long which leads me to believe it is not all about the method.

    And in conclusion I want to say this: although Yi Jing to most science-minded persons sounds like a pile of hocus-pocus nonsense Yi Jing is as scientific as it can get: ever since oracle bone times people asked important questions, oracle provided them with the answers and with the passage of times those answers were validated by the actual events. In other words everyone can test it: if you have an important question you are not sure about, ask the question, get the answer, wait, and see if the answer was correct or not. If answers you receive are nothing but random nonsense then don’t waste your time on Yi Jing. If however it turns out answers were highly insightful and useful and validated by the upcoming events then pay closer attention to it.


  11. Bug, or oddity, in the R code there: when you start off with 48 in the left hand & 1 in the right you still end up moving 4 from the right hand to the middle finger. Your calculation does, however, preserve the 5,5,5,9 pattern for the total between your fingers as you increment the no. straws in the left-hand division. Still, no-one divides the straws so unevenly, do they? This engenders the reflection that the uniform distribution given by sample(1:(48-hand),1) is unrealistic: owing to the patterns 5,5,5,9 at the first division, & 4,4,8,8 at subsequent divisions, the probability model you’ve shown, rather than exhibiting a “tiny bias” compared to the true model, might be seen as a reasonable approximation to a range of plausible models for how real people divide real bunches of straws.

  12. Or to put it another way, if the intent is, as you suggest, to obtain probabilities of 1/16, 3/16, 5/16, & 7/16, the yarrow-stalk method will be fairly robust to how different people divide different bunches of stalks; considering the exact probabilities when every possible division is assumed equally likely is something of a red herring.

  13. Thanks for this explanation of the probabilities. I have been playing with the I Ching for thirty years and this was the m out complete and cogent discussion on the topic I have seen.

    One method not mentioned here was recommended to me because though it is two coin based, it claims to reproduce the same probabilities as the yarrow stalk method. (I will leave it to the more statistically minded to confirm that claim.) Don’t be thrown off by the 2-coins aspect, as it also requires twelve tosses, two per line.

    The method is described here, among other places:

    Thank you again.

    • I believe you are correct Andy; they way you do it is a clever way to travel with two coins the probability tree that you can find in this post.

      A bit convoluted for my taste since you’re forced to do some calculations but, hey, it renders the correct yarrow stalk probabilities all right. Thank you very much for sharing!

  14. Your randomness theory fits well with ‘The Bicameral Mind’ by Julian Jaynes.

    It’s a while since I read it, but Jaynes suggests various forms of Divination (including the iChing) were taken seriously for thousands of years in all spheres of life in nearly every culture from Ancient Greece to China.

    The common thread seem to involve some random pattern that allows a ‘Eureka’ moment of creative interpretation that relates to whatever your particular problem is.

    The iChings answers are often vague or metaphorical, but your brain instantly applies them to your own situation. Similarly if you opened a Bible or Koran (or any significantly weighty text) while thinking of a current problem and pointed at a random stanza, your (bicameral) brain switches into gear to interpret it to your situation.

  15. I agree, the I Ching was clearly based on the yarrow stalks method. If you draw up a table with all probabilities for the changes of one old sign (three lines) to another, Earth will change to Heaven with 1/512, whereas Heaven will change to Earth with 27/512. This table will show much of the internal workings of the I Ching.

  16. Hello , you guys are really incredibly intelligent people and i admire your power of concentration and excogitation (i found this word in google trad). So, all your cogitations are powerful, but i ask myself: what about the last vers of the Borges poem ? i think no matter the method, coins, stems or app, and mathematical analysis ,someone up there (or inside us?) is watching in ambush (wich is a better translation for “acechar”, i think) and giving the accurate answer…
    Thanks, and congratulations for your work !

    • Hi Gus, yeah, I believe that somebody else hinted that if there is a higher power the method is not relevant for an individual since everything will be according to plan.

      However, even God has to follow Its own mathematical laws since, well, in a way they’re divine and, if this is the case, it does matter the method indeed, it truly does.

      About the translation into ‘watching in ambush’ you are literally right, but when I did the translation I took into account the context, that is why I wanted to remove the feeling of ‘trap’ that the word ambush and similar translations had and decided to leave the more open ‘waiting’ and yet, in my opinion, more precise in this case.

      However English is not my mother tongue so any word meaning “Waiting attentively in a non threatening way but rather compassionate” is more than welcome… I just thought that by placing ‘God’ and ‘waiting’ together people would understand the full meaning of ‘waiting’.

      Anyway, thanks for you comments Gus!

  17. After thinking about it a bit, it might be on purpose that the probabilities are skewed in the way they are, using the yarrow stalk method, but I suspect that they weren’t, given the mirrored nature of the descriptions of the different results and their associated values.

    For my own personal enjoyment and satisfaction, I figured out how to achieve the “proper” normal distribution using the yarrow stalks, instead of 1/16 vs 3/16ths issue as described above: When casting the first piece of a given line (where you end up with 5 or 9 stalks), instead of combining all of the stalks between your fingers, set aside the stalk between your ring and pinky fingers, and see if the two bundles left between your fingers have an equal number of stalks. If they do (2 & 2, or 4 & 4), they represent the number 9 and the value of 2. If they don’t match (1 & 3, 3 & 1), they represent the number 8 and the value of 3. Everything else works exactly the same.

    Thanks for writing this wonderful article!

    • There it is at last! Thanks a lot! I’ve geen looking for this solution for a while. I’ve read that the current “classical” yarrow stalks method which gives unequal probabilities for old yin lines might as well differ from that of the first 1000 years of existence of the Yarrow divination. The current version was reinvented by Zhuxi of the Song Dynasty and it’s possible (from the analysis of Pre-Qin divination results) that the original method produced 6,7,8,9 probabilities very similar to the current coin method. But I like the process of stalks sorting and here I’ve found at last how I can use and get coins probabilites.

  18. Thank you for such a full and thought-provoking post—and I see it’s more than four years since you posted it! A few points:
    (1) In describing the coins, you give “two heads” (8) as Young Yang and “two tails” (7) as Young Yin, when the odd numbers are usually taken as Yang and the even ones as Yin. It makes no difference to the argument, of course.
    (2) In a similar way, in the trigrams, though yang is the more masculine principle, the three daughters have two yang lines and one yin, whereas the three sons have two yin lines and one yang. More yang doesn’t always mean more action — there active daughters (Li) and still sons (Ken), as well as active sons (K’an and Chen) and more passive daughters (Tui and Sun).
    (3) There are I Ching experts who agree with you on the distortion caused by the coins. Wu Wei in his 2005 book “The I Ching: A Book of Answers” writes: “Many people today us a coin method, which is quick and easy. However, because the coins have only three heads and three tails collectively, they lack the mathematical potential to display the unfolding Universe. One of the greatest injuries inflicted on the use of the I Ching was the introduction of the coin method for obtaining answers. It has caused many people to stop using the I Ching because the answers were not quite right all the time. Only yarrow stalks should be used.” (p. 55)
    (4) Some apps at least incorporate yarrow-stalk probabilities or methods of using virtual yarrow stalks (e.g. Yi Jing), though these most definitely lack the ritual and mind-calming element of the stalks themselves, beautifully conveyed by Hermann Hesse in “The Glass Bead Game.”

  19. I am not sure why the word “descuido” translated as mistake in the poem is there?
    The original poem says”
    “LUZ”, which means LIGHT, and changes the meaning of this beautiful sentence in the poem to a much more enlightened meaning.

  20. I think this would be a better translation of the gorgeous poem:

    For a Version of I Ching

    The future is as immutable
    As the rigid yesterday. There is nothing
    That is no more than a single, silent letter
    In the eternal and inscrutable
    Writing of the of book of time.
    He who walks away
    From home has already come back.
    Our life is a future and well-traveled road.
    Nothing abandons us. Nothing leaves us.
    Do not give up. The prison is dark,
    Its walls made of unrelenting iron,
    But in some corner of your cell
    You might discover a light, a tiny opening.
    The path is fatal as an arrow
    But in the rifts, God is awaiting.

    And this is the original version in Spanish:
    Para Una version del I King

    El porvenir es tan irrevocable
    Como el rígido ayer. No hay una cosa
    Que no sea una letra silenciosa
    De la eterna escritura indescrifrable
    Cuyo libro es el tiempo. Quien se aleja
    De su casa ya ha vuelto. Nuestra vida
    Es la senda futura y recorrida.
    Nada nos dice adiós. Nada nos deja.
    No te rindas. La ergástula es oscura,
    La firme trama es de incesante hierro,
    Pero en algún recodo de tu encierro
    Puede haber una luz, una hendidura,
    El camino es fatal como la flecha
    Pero en las grietas está Dios, que acecha.

    Jorge Luis Borges

  21. Thank youforexplaining how the ratios are calculated.I understand now how the prime numbers came about. I knowof another way to make it easier without getting stalks. A simple deck of playing cards can give the same results, and offer up 2 small decks to use. Just remove 4 of one color of court cards, 10 of one color pips and 6 of the other color pip. That nets you a ratio of 2:32, 6:32, 10: 32 and 14: 32. Split them evenly and you cut the deck to 16 cards. To avoid getting the feel of them, and to keep them safe from friction wear, use the card sleeves sold in most hobby and game shops. I have a deck split into 3 small ones, but I had to tweek the definitions a bit to make up for the missing red court cards. I figure this is useful to add since most people have cards and it makes it feel more like a tarot reading for those interested in that without access to the expensive decks. I hope this helps, and thank you once again for the explanation.

  22. Some years ago I devised a simpler 4 coin method which does not use binary patterns yet generates the same probablilites as the yarrow stalk method. I wrote an article discussing the general analysis of the yarrow method and outlining two alternate 4-coin methods, of which I recommend the second (as being much simpler). The article was hosted on, which has since been taken down, but I recently found a new home for on Here’s a link, if you’re interested:

    Click to access 4coinmethod.pdf

    • Your second 4-coins method is a remarkably smart and straightforward method. Congrats and thanks for sharing!

      Maybe you should cut & paste it from your paper. I think nobody addressed it here because they are not going to the trouble of going to the link and reading the paper.

      • Thank you, MMello for your kind response. As you have suggested, I will post the second coin method here:

        Step 1: Toss 1 coin. Heads means strong, tails means weak. Leave the coin where you tossed it.
        Step 2: Toss 3 more coins next to the first one. If you see exactly 3 heads among all 4 coins, then the line is changing; otherwise unchanging.

        If anyone would like to see how I arrived at this, it is described in the article I linked to in my first post (2 posts up from this one). That article also contains a bit of discussion on why I think that human interaction with the yarrow sticks is unlikely to influence the probabilities in practice. (A word of caution: I do not personally use either of the divination methods — I am just a mathematician who has taken an interest in the question.)

  23. Have you considered that systematically tossing coins to exlpore purely mathematical outcomes might yield very different results than tossing them with a puer, authentic intention behind the toss? For example, I have observed that the various ying and yang energies within myself can influence what you call “randomness” outcomes. I have found that in the same way that intention can influence “coincidences” in everyday life, it can also influence coin tosses. With right concentration one can toss multiple heads or tails that consistently defy mathematical probability. If you deny this aspect of the I Ching, you deny its essence. It is not pure algorithm any more than it is pure intention. It is both. This is the essence of the Way.

    • That is remarkable, I recommend you to present this ability to a group of reputable physicists and neuroscientists and publish a peered review paper together… It would be a landmark in the history of human kind.

  24. Very interesting, thank you. I’m just beginning with I Ching, and was looking for a really simple method when I ran into your article. I just realized that two four-sided dice (2d4) may be used in another way, hopefully easier to remember.

    Roll both dice and see what is the highest number.
    1 is Old Yin (1/16) –– (1,1)
    2 is Old Yang (3/16) –– (1,2)(2,2)(2,1)
    3 is Yang (5/16) –– (1,3)(2,3)(3,3)(3,2)(3,1)
    4 is Yin (7/16) –– (1,4)(2,4)(3,4)(4,4)(4,3)(4,2)(4,1)

    It’s so simple I’m afraid it’d be wrong.

  25. By the way, you can get a relevant “non-random” reading, in sync with dream material, from the I Ching by setting in motion any three objects with two sides, e.g. three packets of tissues. Has anyone experimented with this?

  26. I use 16 home made cards with each possibility shown and replace the card in the deck each time after its been draw. But I wouldn’t worry about using coins. Its not suppose to be random, the higher self is directing the result

  27. Hey excellent article I have studied the I Ching for some time and I have seen this same point raised in other places, but one of the comments raises another interesting point; there are non- numerical methods for creating a hexagram, and non-divinatory purposes for using it as well. The method of generating a hexagram is one aspect of the work but I would not call it an essential one. I find the system fascinating because it attempts to create a universal language of events and processes to deepen the vision of the practitioner and encourage associative thinking, while also emphasizing the universality of cyclical change. I hope everyone reading this is able to realize that benefit.
    Also, I have used several apps to generate hexagrams, some better than others, some accounting for accurate probability, others not so much. One program (can’t recall the name) actually creates virtual yarrow stalks and physically shuffles them, which I thought was a cool solution. Thanks for the great article, may you cross all your rivers in peace 🙂

  28. This is AMAZING, thank you! I had no idea.

    If I’m reading it correctly, quick note: “Which means that when having a Yin result, a result of 1 with the die will make it Old Yin, and when having a Yang result, a result of 1,2 or 3 will make it Old Yang.” — The chart above this quote shows using 6, 7, 8 as the old yang numbers. I assume it doesn’t matter which you choose, as long as you’re consistent?

  29. Trying to manually emulate Yarrow stick method with the deck of 50 cards I developed following card algorithm:

    1) Take a deck of cards (52 cards).
    2) Draw 1st card (or shift the deck and see the card). If the suit is hearts, then Result = 2, else Result = 1.
    2) Draw 2nd card. If it is red, add 1 to Result.
    2) Draw 3nd card. If it is red, add 1 to Result.

    If Result = 1 — Old Yang (9, –ø–)
    If Result = 2 — Young In (8, — –)
    If Result = 3 — Young Yang (7, —–)
    If Result = 4 — Old In (6, –x–)

    You can build a probability tree for this algorithm and see that probabilities are the same as for Yarrow stick method. More precise card method with ordered stack of cards can be proposed.

    What do you think about that?

  30. There is a simple way that you can use 16 cards to simultaneously produce hexagrams using the yarrow or coin probabilities. The cards would be marked as follows:

    Cards 1-6: Yin for both yarrow and coin divination methods.
    Card 7: Yin for yarrow method and yang for coin method.
    Cards 8-12: Yang for both divination methods.
    Cards 13-14:Old yang for both divination methods.
    Card 15: Old yang for yarrow method and old yin for coin method.
    Card 16: Old yin for both divination methods.

    You would shuffle the cards six times and each time select a card at random, note the yin/yang symbols on the card, return the card to the deck and apply the noted symbols to form two separate hexagrams starting from the base and working up; one stack would be the hexagram for the yarrow system and the other stack would be the hexagram for the coin system. You could then compare the hexagrams defined by the yarrow and coin methods.

    If you examine the above noted distribution of yin/yang symbols for the cards you will see that the yarrow method differs from the coin method by only 2 cards out of 16. This indicates a probability difference between the two divination systems of only 1/8 or 12.5%. So, the yarrow and coin methods should generally produce very similar results and that is probably why the two methods have worked for effective divination for hundreds of years.

  31. If you want to compare the divination results for coin and yarrow probabilities in a simple way, just consider the divination results achieved by using a “Rule of 16” method. That is, define your six-line hexagram by selecting six random numbers from a pool of numbers 1-16.

    With this method, you can see that in order to satisfy both the coin and yarrow probabilities, numbers 1-6 of the pool are yin, numbers 7-11 are yang, numbers 12-13 are moving yang and number 14 is a moving yin. That leaves you with only numbers 15 and 16 which differ in yin/yang identity based on whether the yarrow or coin probability is required. If the yarrow probability is required, then number 15 is yin and number 16 is moving yang. If the coin probability is required, then number 15 is yang and number 16 is moving yin.

    That’s it. For each draw from the 1-16 number pool, there is a 12.5% chance (i.e., 2 out of 16) that the number drawn will be a 15 or 16 and therefore not the same for both yarrow and coin probabilities. In other words, for each draw, there is an 87.5% chance that the coin and yarrow methods will produce exactly the same result. And this is why the coin and yarrow divination methods have been used for hundreds of years without any complaints. They often produce either the same results or very similar results.

    Of course, there is definitely a difference between coin and yarrow divination probabilities, but that difference is only 12.5% for each draw. This may seem somewhat complex when written down, but the result is easily seen by a simple chart of Rule of 16 possibilities as noted above. I have this chart, but there is no way to attach the associated image to this comment. That is unfortunate, because the chart allows pretty much instant comprehension of the probability difference between the coin and yarrow methods of divination.

  32. Fran: great stuff! Surprised no one previously mentioned Greg Wincup’s 1995 REDISCOVERING I CHING. The 1, 3, 5, 7 marble method is what Greg suggests as a simpler and faster way to get the same probabilities. Plus his translation is an historical milestone.
    Having know about the I Ching since about 1981-2 and done the coin method a couple of times and then set it aside as too confusing and etc. I got Greg’s Rediscovering when it came out and have used the I Ching since. And have observed the bias in the results. They sort of favour my personality, on one hand, but sometimes wish to hear more assertive action suggested. I take it all ‘with a grain of salt’. Sean.

  33. Interesting… I am always open to new ways of doing things, but to be honest I am not so sure that how you generate the numbers really matters as much as how well connected you are to your higher self. The I-Ching seems to be able to give similar answers using different hexagrams and combinations of changing lines. So as long as you have a strong connection to your “Te”, then regardless of the method you use, you will nevertheless get an appropriate answer to your question.

  34. If the Oracle is smart enough to answer a question by drawing a quasi-random number, maybe She is smart enough to know which method you are using, and making the appropriate adjustment. What about using Alexa to choose a random number between 1 and 64, etc.?

    • Well, you’d have to factor in additional logic. “1 to 64” doesn’t factor in if one (or more) line is changing or not. Hence the weird and wobbly probabilities in the original method. But of course plenty of computer programs replicate this probability quite easy.

  35. A few decades ago a way to toss two coins that matched the yarrow stalk divination process was developed.

    The two-coin method of divination is as follows. For each of the six lines of a hexagram, a large coin and a small coin are tossed twice upon a hard, flat surface. The six lines are determined from the bottom up. The coins land either “heads” or “tails” and are written down, noting the large coin first. Each toss may be written down, H for “heads” and T for “tails.” For example, if the first toss is Heads-Heads, and the second toss is also Heads-Heads, then the line is changing yang (see below). Although there are four basic possibilities for each line, there are sixteen different permutations of the two coins tossed twice: eight yang lines and eight yin lines
    The four doubled-pairs, HH, HH; HT, HT; TH, TH; and TT, TT, are the changing lines; The first three of which are yang changing to yin, and the last one is yin changing to yang.

    HH, HH – changing yang, written as a solid line with and x in the middle
    HH, HT – yang, written as a solid line
    HH, TH – yang, written as a solid line
    HH, TT – yang, written as a solid line

    HT, HH – yang, written as a solid line
    HT, HT, – changing yang, written as a solid line with and x in the middle
    HT, TH – yin, written as a broken line
    HT, TT – yin, written as a broken line

    TH, HH – yang, written as a solid line
    TH, HT – yin, written as a broken line
    TH, TH – changing yang, written as a solid line with and x in the middle
    TH, TT – yin, written as a broken line

    TT, HH – yin, written as a broken line
    TT, HT – yin, written as a broken line
    TT, TH – yin, written as a broken line
    TT, TT – changing yin, written as a broken line with an O in the middle

    • The existence of the two coin method is a little annoying as it took me a couple of hours to come up with it myself, only to find it had already been proposed.

      Good news is I can still propose what i consider an improvement: a two coin method that half the time only requires to throw the two coins once for every line.

      In fact, the version I had come up with has the following outcomes (big coin always reported first)

      -H,T: always yang
      -T, H: always yin

      -H, H, then throw again:
      –if at least of of the two coins is H: changing yang
      — if both coins are T, yang

      -H, H, then throw again:
      — if both coins are T, changing yin
      — if at least one coin is H, yin

      It has the same probabilities of the two coin method, but with 25% less coin throws needed. That is, an average of 9 thows per hexagram instead of 12.

  36. Great article . Thank you. I realise that I’m late to the party though I wonder if someone could please answer the question Sonjathegrey posted in 2019. She was querying the 8 die, one coin method – does a die toss of 1,2 or 3 result in old yang as per your narrative? Or is it a toss of 6,7,8 that results in old yang as per the chart? I would be really grateful for clarification as I’m not sure which to use. Many thanks

    • In case Fran doesn’t see your question right away, I’ll jump in. I’m the mathematician who posted the simple 4-coin method somewhere up in this thread. In answer to your question, if your concern is to get the intended probabilities, it doesn’t matter whether you use 1,2,3 or 6,7,8. Because you only interpret rolling a 1 as old yin when the coin toss was tails, it is independent of how you interpret the other cases where the coin toss come up heads. So if your concern is to get the probabilities right, the only thing you need to do is to choose ANY 3 of the die numbers to represent old yang. So you could use 1,2,3 or 6,7,8 or even something random like 2,3,7, and you would get the same probabilities. I think (guessing here) that Fran just used 6,7,8 in his chart for the visual purpose of moving the old yin and old yang cases far apart in the chart for clarity.

      Fran would you please weigh in on this and correct anything I have wrong? I don’t want to put words in your mouth, I’m just trying to get a quick answer out to Amanda’s question.

      • Thank you,Stuart, your response is much appreciated. Any relating hexagram will be different if you were to use 1,2,3 rather than 6,7,8 given the moving lines are different . I’m new to IChing so am now wondering what the point of this method is if you can have two different answers to the same question . Any thoughts? Thanks.

  37. The number 50 can be argued in two ways. First it has to do with the leaving 1 behind in the first round, that meant 49 is needed, one has to use 50.

    However, the original words are said missing something i.e. “The numbers in the heaven and earth is 50 (and 5), only 49 is used”. The argument is 55 not 50 should be the phrases not 50. Why 55? 1+2+…+10 = 55, the human numbers reflected by the heaven endowment — our fingers!

  38. My “Hamming” algorithm to build hexagrams from a bit stream:
    ‘seven’ = 1
    ‘eight’ = 01
    ‘six’ = 001
    ‘nine’ = 000
    Example: 00101110011 = = 687767 = ䷷ = Hex. 56 (The Wanderer).

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