The purpose of this paper is to provide a mathematical analysis of the oracles used with the Chinese I Ching (pronounced EE CHING). The I Ching is the ancient Chinese Book of Changes. In this document I briefly describe the I Ching and the hexagrams it uses. Then I proceed with a probability analysis of the oracles used to consult the I Ching.

I have not attempted to describe the Book of Changes in detail since that is beyond the scope of my ability. For translations and discussions of the I Ching itself, I refer you to the many fine texts available. Some of the ones I use are:

The I Ching : Or, Book of Changes
C.F. Baynes (Translator from the German)
and R. Wilhelm (Translator)
Princeton Univ Press
The I Ching
James Legge (Translator)
I Ching, the Book of Change
John Blofeld (Translator and Editor)

My purpose here is to provide a small amount of mathematical analysis which I have not seen performed elsewhere. As I studied the I Ching, I found that the procedures used to consult the book have some interesting mathematics associated with them. I became interested in knowing more about how the oracles operate, so I set off to understand better how they work.

This introduction contains a very brief description of the I Ching to allow you to make sense of the mathematical discussion which follows. Once you have a basic understanding of the book, you should be able to understand my presentation on the probabilities of the oracles. If you find that reading this treatise has inspired an interest in the I Ching, I encourage you to read one or more of the above references.

At the end of the mathematical analysis I address some more esoteric issues regarding the use of the I Ching as an oracle and its relevance to mathematics. Those discussions are purely my speculation and may be taken at face value.

If you are already familiar with the I Ching and the hexagrams, you may skip to the next section where the analysis of the hexagrams and the oracles begins.

The I Ching

The I Ching is a very ancient text written in the distant history of China. Its actual origin is not completely known. Some say that the hexagrams resulted from another oracle which involved heating a tortoise shell until it cracked, and then interpreting the cracks to learn the future. The lines of the hexagrams represented the cracks in the shell. The hexagrams evolved into a book of divination where the future was represented by images made up of six horizontal lines. Each group of six lines is called a "hexagram".

Each hexagram represents a situation in the natural world which applies to the lives of individuals and states. Each hexagram has associated with it an "Image", sometimes called a "Symbol". The image of the hexagram is a description of the situation which it represents. For example, the image for hexagram four is (From the R. Wilhelm translation):

"A spring wells up at the foot of the mountain:
The image of YOUTH.
Thus the superior man fosters his character
By thoroughness in all that he does."

This hexagram represents youthful folly. Youthful folly is a life situation which we have all experienced, and this hexagram represents that situation.

The I Ching claims to have captured in its hexagrams the essence of all natural states in the universe.

At some point in the history of the Book of Changes someone realized that the telling of the future was insignificant unless the seeker was also told what to do about it. Thus, the "Judgments" were added to each hexagram to provide guidance to the seeker. At that point the book changed from a book of divination into a book of wisdom. A book of divination tells the seeker what will happen, whereas a book of wisdom tells the seeker what may happen given the wisdom and behavior of the individual.

The current form of the I Ching is officially attributed to King Wn, who lived around 1150 B.C, and his son, the Duke of Chou. King Wn wrote the judgments which are associated with each hexagram. The judgments advise the seeker of information what to do about the situation represented by the hexagram.

The Duke of Chou took over where his father left off and added judgments to each line of the hexagram. In actual use, each line of the hexagram can have the attribute of being either stationary or moving. A stationary line is static and adds no additional wisdom to the judgment. However a moving line represents change in the current situation and carries with it additional wisdom concerning what to do. When the I Ching is consulted, the contributions of the Duke of Chou are applied to those lines of the hexagram which are in motion.

The Hexagrams

Each hexagram consists of six "lines", which are chosen at random using a procedure called an "oracle". The oracle uses a randomising procedure to obtain each line of the hexagram. Disregarding for the moment the motion or stasis of the lines, each line is either a "yielding line" or a "firm line". These are represented graphically by a broken line and a solid line, respectively:

broken, or yielding line
solid, or firm line
The broken line is called a "yin" line since it represents the yin principle. The yin principle has the following characteristics:

The solid line is called a yang line since it represents the yang principle. The yang principle has the following characteristics:

The yin/yang principle represents all the bipolar reciprocals in the universe. The yin and the yang principles are complementary characteristics;they are not opposites, and they are not judgmental. Hence, "good" and "bad" are not yin/yang concepts. Rather, yin and yang are complementary concepts which exist together with equal importance. So yin and yang are represented by female and male, dark and light, yielding and firm. There is no yang without yin; there is no yin without yang. There is no warm without cool. There is no hard without soft. There is no up without down.

There are many discourses on the yin/yang principle, and they are beyond the scope of this paper. You may recognize the yin/yang symbol.

Each hexagram is constructed of six lines which are either yin lines or yang lines. For example, hexagram 1 consists of all yang lines.

This hexagram is called the "Creative". Since all of the lines carry the yang principle this is the most creative of all hexagrams.

Hexagram 2 is constructed of six yin lines.

This hexagram is called the "Receptive". Since all of the lines carry the yin principle this is the most receptive of all hexagrams.

The other 62 hexagrams consist of an admixture of both yin and yang lines.

This section has presented only a brief overview of the construction of a hexagram. The next chapter goes into more detail about hexagrams and their enumeration. It also addresses the movement of individual lines.

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