I Ching Probabilities

Nature does not hurry, yet everything is accomplished. - Lao Tze

Life moves in circles; sometimes big ones. Over thirty years ago I sat down with a copy of the Richard Whilhelm translation of the Chinese Book of Changes, the I Ching, and I did myself some maths. Unique among the sacred books of the world, the I Ching is profoundly mathematical. It consists of a set of 64 hexagrams generated from a binary system of broken and unbroken lines, yin and yang. It has been a source of endless fascination for many people including Leibniz who, it is said, used it as a source of inspiration in his pioneering mathematical studies. It can be used as a book of divination, and is commonly consulted using a set of three Chinese coins. Thus:

More traditionally, it is consulted using a complex ritual manipulating a set of fifty yarrow stalks, thus:

The coin method is a relatively modern innovation. The yarrow stalk ritual can take over half an hour. The coins streamline this to a few minutes. It satisfies the modern need for instant gratification. But is it the same thing? That was the question with which I was concerned. I did endless hours of investigation into this question back then. I was not primarily interested in the Book of Changes as a vehicle of divination or fortune-telling. It was a mathematical conundrum.

Consequently, in more recent times, I was very fortunate to teach the I Ching to undergraduates as a module in a unit on sacred texts in a Religious Studies course. This kept me up to speed with the great Chinese classic, or at least it meant I had to revisit it every second semester and explain its history and complexities to young minds. It has therefore never been very far away from me and I remain quite fascinated by it. It is certainly one of the oldest sacred texts in the world, one of the most mysterious and yet one of the most enduring. 

Just recently - a few days ago - I found some of my notes on the I Ching from over 30 years ago. Some of them are now incomprehensible, as you might expect, but amidst my ramblings there was at least one  pearl: my mathematical calculations of the probabilities of coins versus yarrow stalks. Readers really need to have some familiarity with how the Book of Changes is consulted in order to appreciate this, but I think I should set out my figures here and share them with others, just for the sake of it. It is not earth-shattering by any means, but there are, I think, many ramifications for those who dabble with the Ching. 

The coins, you see, have exactly even probabilities. You toss three coins in the air and there are even chances of them falling in any one of several combinations: three heads, three tails, two head/one tail, two tails/one head. Three heads is no more likely than three tails, and so on. Consulting the I Ching, three heads yields a solid (unbroken) line and three tails a broken line. One is yang and one is yin. This is how the oracle is consulted. You toss the coins six times to give you six lines = one of the 64 hexagrams. Then you look up the result in the book. Here are the sixty-four combinations of six lines.

The stalks, however, are another story. The fifty yarrow stalks are divided according to a quite complex ritual pattern that is, by all accounts, very ancient. What are the probabilities involved? I worked it out. The possibilities are not even, as with the coins. Not at all. On the contrary, the probabilities are as follows:

Unbroken line (Unchanging yang) = 5/16ths. 

Broken line (unchanging yin) = 7/16ths

Unbroken line (yang changing) = 3/16ths

Broken line (yin changing) = 1/16th

To put this into the terminology of the coins:

Two heads/one tail = 5/16ths. 

Two tails/one head = 7/16ths

Three heads = 3/16ths

Three tails = 1/16th

That is, using the stalks, there is only a one in sixteen chance of three tails but a three in sixteen chance of three heads. As I say, the coins give exactly even probabilities. Not so the stalks. With the coins you have even chances of throwing three heads or three tails. But the stalks yield a very different set of probabilities. This means that the two methods are not interchangeable. If you use the coins to consult the oracle you will get quite different results than if you use the stalks. If you use the stalks then the extreme yin line (three tails) - or "changing yin" as it is called - is the least likely result of all. The most likely result is the unchanging yin line - seven times out of every sixteen. 

I hope this is clear. Probably not. Or not unless you are familiar with the Book of Changes. But it is worth putting in writing and posting all the same. 

What might it mean in practice? Try this:

Take sixteen pieces of paper, tiles, whatever.

On five of them write "Unchanging yang"

On seven of them write "Unchanging yin"

On three of them write "Changing yang"

On one of them write "Changing Yin"

Now put them all in a hat and draw out six of them one at a time. 

That will give you a hexagram of the same probabilities as the yarrow stalks. The modern coin method is a bastardisation and will not give you accurate results. Better still, learn the yarrow stalk ritual. It is, after all, a meditation.

I'm sure this has all been noted before. The Book of Changes is some 3000+ years old and has been studied in meticulous detail. Modern people, though, are terribly lazy. Thus they use the coin method. A simple study of probabilities reveals that the coin method is defective.

The maths, by the way, is the same maths as the so-called "Knight's tour" in chess, the chess board having sixty-four squares.

Using the Yarrow Stalks

Yours,

Harper McAlpine Black