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?
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
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 contains Old Yang or Old Yin lines which 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 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.
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:
So 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 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 few 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
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 four coins
Okay, let’s say we don’t want to buy a die and we only have coins. Then we can still use coins to render properly traditional probability sound hexagrams. For this alternative we need to toss four coins all at once and read from left to right being heads (1) and tails (0), we have 16 possible outcomes:
|0000 – Old Yin
0001 – Yin
0010 – Yin
0011 – Yin
0100 – Yin
0101 – Yin
0110 – Yin
0111 – Yin
|1000 – Yang
1001 – Yang
1010 – Yang
1011 – Yang
1100 – Yang
1101 – Old Yang
1110 – Old Yang
1111 – Old Yang
As an alternative, instead reading from left to right we can also have 4 different coins and assign each a position from 1 to 4. We could also toss four times one coin for each sign but that would make a grand total of 24 tosses per hexagram which, if you ask me, is a bit of a mood killer, but the choices are out there if we need them.
The one 16 sided die
One simple way to do so 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.
The two four sided die
If you happen to have two four sided dies then the procedure is similar to the 4 coins alternative; for the two dice we simply throw the two 4 sided dice and we read the outcomes from left to right. And just like in the case with the coins we could also have two different dice and assign each a position so we don’t have to worry about the order. These are the 16 possible outcomes:
|1,1 – Old Yin
1,2 – Yin
1,3 – Yin
1,4 – Yin
2,1 – Yin
2,2 – Yin
2,3 – Yin
2,4 – Yin
|3,1 – Yang
3,2 – Yang
3,3 – Yang
3,4 – Yang
4,1 – Yang
4,2 – Old Yang
4,3 – Old Yang
4,4 – Old Yang
Anyhow, happy traditional choice!
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- Tao Te Ching 17: The Art of Leadership (beyondthedream.co.uk)
- Tai Chi Chuan and Ancient Chinese Philosophy (interactchina.wordpress.com)
- Fortune telling, the hidden side of doing business in China you may have ignored (teabreakchina.wordpress.com)
- Heated Debate Over 5000-year-old Chinese Inscriptions (medindia.net)