The solution of the King Wen sequence?

Richard S Cook’s Classical Chinese Combinatorics – a review article


Richard S Cook. 'Classical Chinese Combinatorics: Derivation of the Book of Changes Hexagram Sequence.' University of California, Berkeley: STEDT Monograph Series, Vol. 5, 2006 (Lulu), paperback, xviii + 642 pages. $64. ISBN: 0-944613-44-6.



I am immensely grateful to József Drasny of the Yi-globe website for reviewing Richard Cook's much-ignored massive work that claims to have solved the reasoning behind the King Wen sequence. József has compressed his understanding of it into small compass as an aid to any others who wish to make a study of the book, and given some critical notes on it. So far as I know, this is the first review of Cook's book since it was published in 2006.

I feel it is important that such a complex work making the bold claim to have solved the King Wen sequence should receive public scrutiny, but it has been difficult to find anyone even remotely able to disentangle clarity from this ambitiously obscure work. I hope now that aficionados of the Yijing world can at least get a qualified glimpse at what Dr Cook has been proposing all this time.


King Wen, the mathematician

It is unusual to write a review 13 years after the first publication. At that time, however, I could not get a free copy to read, and spending $100 on a book of unknown contents would have been too risky for me (it has since come down to $64). Since then, I have not found a single critical evaluation of the book though the title was often included in the catalogues of libraries and the references of scholarly articles. Thus, about a year ago, still having an interest in the subject (and getting a free copy from a friend), I began to read Richard Cook's impressive work. This review article summarises my observations.

There is a well-known question in connection with the hexagrams of the Book of Changes or Yijing: Were the 64 hexagrams of the traditionally received sequence (henceforth referred to as the Wen Wang or WW sequence) intentionally arranged? Plus, the follow-up question: In the case of conscious design, what was the basic idea of the plan? In connection with these questions, the author firmly declares:

This study resolves the ancient enigma of the classical Chinese Book of Changes hexagram sequence. (p ix)

We read in the Preface that the author began to think about these problems in 1983. In the first few years, the main points of his interest were the structure of the 36 invertibility classes (explained below) and the distribution of the hexagrams on seven levels based on the number of yin and yang lines. Because he based his whole theory on these structural elements, it will be useful to show them here in detail. I have collected the contents of paragraphs (1), (2), and (3) from different parts of the book and only concentrate on the essentials.

(1) The hexagram equivalency classes (HEC)

In the Introduction, there is a meaningful and detailed analysis of the ancient Chinese arrangements of the hexagrams and many other source materials. Among them, there are the works of Hu Yigui (1247–?), Lai Zhide (1525–1604), and some others who represented the invertible pairs of the WW sequence by a single hexagram (related to this, see An old Yijing mystery solved).

In this system of representation, the two hexagrams of the invertible pairs are equivalent and form an equivalency class. These classes are represented by one of their hexagrams (usually by the first one in the left-to-right reading of the sequence). The symmetrical (non-invertible) hexagrams remain separate, and each of them is a class in itself. The WW sequence contains 56 invertible hexagrams (28 invertible pairs), and eight symmetrical hexagrams (eight non-invertible hexagrams). Thus, there are 28+8=36 hexagram equivalency classes (HEC) altogether. The HEC form of the WW sequence and the corresponding hexagram numbers can be seen in the next table (of mine).

In this system, for example, HEC No 3 (the 3rd hexagram diagram in the first row) comprises the two invertible hexagrams #3 and #4 (row labelled 'Hex No'), and it is represented by hexagram #3, and so on.

The elements highlighted in orange are the so-called 'neuter HECs': They equally have three yang and yin lines. In the next section, I will come back to them.

Although the earliest known table of the 36 HECs only comes from the 13th century CE, Cook considered this kind of representation to be the primary form of the sequence. He said:

The fundamental unit of the Book of Changes hexagram sequence is not the hexagram, but rather, a special kind of hexagram equivalency class (HEC). (p 29)
(2) The seven levels of the HECs

In ancient Chinese diagrams, the 64 hexagrams are often distributed on seven levels, according to the number of the yang (or yin) lines in them (an example). Cook uses this structure for the HEC elements, as well. In the next diagram, and on each level, the succession of the HECs follows their order in the WW sequence (p 142).

In this table, the ten neuter HECs go to level-3. Their positions in the HEC sequence are as follows: 7, 10, 12, 19, 24, 27, 30, 31, 33, 36 (ie, the ones coloured orange in the previous diagram). The lengths of the nine intervals (the number of the non-neuter HECs) between two adjacent elements are 2, 1, 6, 4, 2, 2, 0, 1, 2 (ie, the number of spaces between the orange boxes). Cook discovered that the five middle numbers (6, 4, 2, 2, 0) agree with the initial section of a doubled Fibonacci sequence, though in reverse order (the corresponding Fibonacci sequence is: 0, 1, 1, 2, 3, …). This 'sequence within the sequence' (referred to further on as 'sequence S') became the key element of his theory on the classical Chinese combinatorics. (See the minimal glossary at the end for an explanation of the Fibonacci sequence and other terms.)

(3) The Cook diagram

There is a third important element in Cook's hypothesis, a 3×18 matrix, which is his own original work (p 34). Thus, I call it the 'Cook diagram'. Here, he distributed the 36 HECs, two in every column, successively from left to right. Each HEC in the columns went in the cell that corresponded to its gender: M (male), N (neuter), or F (female). The male and female genders are like those in the case of the trigrams where the sons have a single yang line, and the daughters have a single yin. In the hexagrams, the minority of yang lines means males, and the females have a minority of yin lines.

* see below. This diagram has some significant features:

Cook says that his extensive and profound search has not turned up any similar figures to this 3×18 matrix in early Chinese works or other publications. Also, the sequence S was unknown. So not having any direct source materials of this nature, he based the whole theory on his own two observations:

Starting from these features, the author has drawn a far-reaching inference about the origin of the WW sequence. His conclusion can be read already flagged up on the first page of the Introduction:

It had become clear that the hexagram sequence demonstrates knowledge of the relation between certain linear recurrence sequences (LRS, a pair of so-called Fibonacci sequences) and rational approximation of division in extreme and mean ratio (DEMR, the Golden Ratio). (p 1)

Note: Here, before getting into the main body of the text, I have to point out that, in the presentation of the theory, the author nowhere follows the usual method of logical reasoning: from premises to the conclusion. He does not use conditionals, arguments, or such words as supposition, assumption, hypothesis, guess, etc. It is not an exaggeration to say that the book only contains statements. These statements are, in many cases, without precedents and/or consequences, and so the reader does not know where they come from or what conclusion can be drawn from them. As a result, it is very difficult to follow the author's train of thought. It is not easy to accept the attitude that he believes in the validity of all his statements and takes them for granted without any reason. This is also the case in the two declarations quoted above: He did not bring forth any argument to prove the knowledge of the HEC in the Zhou era, and the WW sequence does not demonstrate the mathematical knowledge he believes.


In the chapters titled Solution and Derivation, the author sets forth his theory about the origin of the WW sequence. According to him, the person (Wen Wang or others) who arranged the hexagrams in this order knew the recurrence sequences, the golden ratio, and many other mathematical notions, and demonstrated this ancient knowledge in the WW sequence. To establish the validity of this hypothesis, Cook chose a specific method. On p 2 he writes:

The presentation here seeks to mirror the path of discovery that the arranger of the sequence originally followed ...

That is, he imagined himself in the place of the original author (Wen Wang?), having his supposed knowledge, and made up an extraordinary and highly complicated procedure for the transformation of the unordered 64 hexagrams into the received sequence. The description and the realisation of this procedure are detailed on nearly 400 pages and with more than 600 diagrams. Next, in sections (A), (B), and (C), I shall try to put the different ideas in order, and narrow down the contents to the very essentials.

(A) n-gram science and analysis

For the realisation of his conceived plan, Cook invented (not 'reconstructed', as he says in the Preface, p 1) the so-called 'n-gram science', what he considered to be the 'classical Chinese combinatorics' or a part of that.

The subject matter of this science is the definition of n-grams, the generation of them, and the operations with them. An n-gram in the author's interpretation is a combination of the unbroken (yang) and the broken (yin) lines, where 'n' refers to the number of lines in the elements. In the book, he deals with monograms, digrams, trigrams, tetragrams, and hexagrams.

On these n-grams, the author defines and performs many types of operations. There are inverses of hexagrams (zonggua), 'obverses' of hexagrams (cuogua, 'obverse' being an odd word to use for complementary) – see glossary of Yijing terms – and various transformations (trigrams into hexagrams, hexagrams into nuclear tetragrams and nuclear hexagrams, sequences into matrices, circles, and other forms, etc). Frequent operations are the classifications of the n-grams according to invertibility, gender, purity, nuclear elements, and the number of the yin and yang lines. In this way, he created numerous new and already known regular arrangements, which gave him the idea to look for them in the received sequence and the Cook diagram.

For example, the classification of the eight trigrams by gender (male and female) and purity (pure and impure) results in the next groups (pure trigrams are all yin or yang; impure are mixed yin and yang). The two pure trigrams are the male M1: 111, and the female F1: 000. (In the text, because of the technical difficulties, I represent some trigrams by binary numbers, beginning from the bottom line.) The three impure male trigrams (the three sons) can be arranged into six different sequences. The single trigram M1 and the six impure male sequences (M2…M7) are shown in the next table (from p 66).

The female trigrams can be classified similarly.

Cook used the results of these operations for different purposes. He searched the Cook diagram and the different (WW, natural [binary], and HEC) sequences for groups that were identical with the results of the previous classifications. Also, he established the positions of the characteristic elements (subsets, HECs, and hexagrams) in the Cook diagram and the WW sequence.

(B) Linear recurrence sequences, Fibonacci sequences

In the course of the classification of the tri-, tetra-, and hexagrams, some short sequences were brought together from the number of elements in the different groups. It seems that the author intentionally looked for and set up such forms because of their similarity to the sequence S (6, 4, 2, 2, 0).

The next table shows an example of these classifications (p 42). Here, the 16 tetragrams and the 10 tetragram equivalency classes (TEC) are classified according to their invertibility and purity.

The items in the rows are as follows:

  1. The total number of the tetragrams is: 16
  2. The number of TECs: 10
  3. The number of invertible TECs (in the 10 TECs of item 2): 6
  4. The number of non-invertible TECs: 4
  5. The number of impure TECs: 2
  6. The number of pure TECs: 2

According to Cook, the six numbers in the last column (16, 10, 6, 4, 2, 2) constitute a linear recurrence sequence, because they look like the doubled and reversed onset of the Fibonacci sequence.

Note: I think it would have been more exact to call the above sequence and all the similar ones pseudo-recurrence and pseudo-Fibonacci sequences because they do not correspond to the relevant mathematical rules: The numbers are not regularly counted from the previous ones. For example, item 1 and item 2 belong to different classes (tetragrams and TECs), item 3 and item 4 both come from item 2, and item 5 and item 6 both come from item 4.

(C) The derivation

At the centre of Cook's theory, there is a procedure showing how the WW sequence might have been derived from the unordered set of the 64 hexagrams. Here, I only concentrate on the main points of that.

At first, Cook arranged the 64 hexagrams into Shao Yong's natural (binary) order and compressed them into 36 HECs. My question: How did the original author know this sequence?

Note: The binary sequence was set down by the Song dynasty philosopher Shao Yong (1011–1077). There is no evidence for it 2,000 years earlier in the Zhou dynasty.

Then, he distributed the HECs into seven levels and, further on, into nine subsets. The next table shows the nine subsets, marked by the capitals from A to I. Subset A is the union of the two HECs in level-0 and level-6. Differently from the diagram of the seven levels in section (2), here the HECs are arranged in the natural order.

Then, Cook accomplished many complicated n-gram operations, always using the information he had got in the course of the previous examinations, in phase A.

  1. He rearranged the HECs in each of the nine subsets, one after the other, according to their relative order in the WW sequence. (See example 1 below.)
  2. He combined seven subsets into three supersets corresponding to the three genders as follows: Male=D+E+H, Neuter=C, Female=F+I+G.
  3. He created an empty 3×18 frame, the same as that of the Cook diagram. Then, he placed HEC 1 (#1) to the first place of the first row, HEC 36 (#63–64) to the last place of the second row, and HEC 2 (#2) to the first place of the third row. Then, applying the previously (in phase A) established rules of the arrangement, he placed the male, neuter, and female HECs to the corresponding places in the three rows. In the end, he got the original Cook diagram, except for some HECs where the representative hexagrams were changed (the order of the two hexagrams in the HEC was different).
  4. Reading out the above-created table by the columns, the 36 HECs came out in the order of the WW sequence. Still, he reversed the order of the two hexagrams in the irregular HECs.
  5. In the end, he replaced the HECs by the corresponding hexagrams, and the WW sequence of the sixty-four hexagrams came forth.

I show here two examples to demonstrate the methods of the rearrangement (paragraph 1 above).

Example 1:

Given the binary ordered subset D' (sD'), that contains three HECs as follows: subset sD': .

This subset was shown in the above table of nine subsets (the prime symbol on letter D marks the subset is in binary order). Let the three HECs be arranged in their WW order, forming subset D (sD). (The task comes from p 259.)

The three HECs represent three invertible pairs of the hexagrams. The represented hexagrams are #24–23, #7–8, and #15–16.

The solution goes on in the following way:

  1. In subset sD', the three lower trigrams are 100, 010, 001. This arrangement corresponds to the male sequence M5 in the table of the trigram classification (section: n-gram science). The three upper trigrams are pure females (F1).
  2. In the initial phase of the derivation, the order of the upper and lower trigrams of the HECs in every subset of the WW sequence was examined. It was established that in subset D, the three lower trigrams were 010, 001, 100 in succession. This is the male sequence M3 in the table.
  3. Consequently, to get sD from sD', one has to change the lower trigrams from M5 to M3. The upper trigrams remain the pure female F1.
  4. Combining the trigrams of sequence M3 with the trigrams F1, we get the result: subset sD: . Here, the succession of the three HECs agrees with that in the WW sequence: #7–8, #15–16, and #24–23. (In the third HEC, the representative hexagram is #24 instead of the odd #23. The order of two hexagrams will be changed later in the course of one of the following operations.)

Note: Cook applied similar methods to four subsets that had three HECs with pure male or pure female trigrams in them (subsets D, E, F, G). In the case of three other subsets that contained six or nine HECs, he used much more complicated operations (subsets H, I, and C).

Example 2:

Given the binary ordered subset A' (sA'), that contains two HECs as follows: , and .

The two HECs represent two non-invertible hexagrams: they are #2 and #1. Let them be arranged in their WW order, forming subset A (sA). (Please concentrate on the application of the n-gram science, and not on the simplicity of the example.)

To the arrangement of this subset, Cook applied a unique method. He writes on p 249:

Order this subset sA according to the number of pieces in each hexagram. As a yang line has one piece, and a yin line has two pieces, one comes before two, and two comes from one doubled (breaking or doubling the yang line produces a yin line): so too, the class of 6 pieces comes before the class of 12 pieces. … The final ordered sA subset is given as: , and .

Note: In the above example, the basic elements (the two HECs) were arranged on the basis of their graphic representation. In mathematics, it is an unacceptable and mindless method. Though the result is correct, the explanation seems rather strange and unfounded.

Summary and conclusion

The essence of the 'derivation' can be summarised as follows. The process began with the analysis of the WW sequence and the Cook diagram: the author observed the positions of the elements in them and drew conclusions about the regularities. Then, he applied these rules to the natural HEC sequence and got back to the received WW sequence. That is, the final result was already known at the beginning of the derivation procedure. Example 1 demonstrates this general method. In some other cases, this method was not applied to the task. There, the author improvised casual solutions, as shown in example 2. At the end of the book, the author says:

This sequence derivation is not a formal mathematical proof of the kind to which modern mathematicians are accustomed. (p 505)

I can do nothing but agree with him.

In the chapter entitled Conclusion, the author declares his inference:

The structures emerging from the classifications and derivations presented in this study are surprising, and indicate a high degree of mathematical sophistication not previously recognized in a Chinese work of this antiquity, indeed, perhaps in any work of this antiquity, anywhere in the world. (p 414)

Also, more specifically:

The hexagram sequence clearly indicates that the relation between the LRS [linear recurrence sequences] (Fibonacci, Hemachandra, Pingala) and DEMR [division in extreme and mean ratio, golden ratio] (Euclid) was known, and it shows the manner in which this knowledge develops from n-gram classification. (p 502)

At the end of my first reading, when I found the above-quoted sentences, I could not imagine how the author had arrived at this conclusion. I believed that I had overlooked some essential parts of the text. Therefore, I began to read anew, line by line, diagram by diagram, looking for some arguments, but in vain. Here, I set up the way that probably led him to this unbelievable result.

As Cook wrote himself, he began to think deeply about the 36 HECs in 1985 and the seven levels in 1988 (p 1). He did not refer to the 3×18 matrix (the Cook diagram), but certainly it was ready at the same time or soon after. It is easy to understand his ambition for the interpretation of the Cook diagram. The regular positions of the elements in that were certainly worth further investigation. Sixteen years later, he discovered the embedded (pseudo-) Fibonacci sequence (6, 4, 2, 2, 0) in the diagram, in the line of the neuter HECs, and looked at it as if it were the 'long sought key' (p 1). Though, as a matter of fact, it was not a sequence at all since it did not contain any concrete elements (HECs) but the intervals between them. However, believing in the importance of his discovery, Cook based his whole theory on this short, quasi-regular pseudo-sequence.

At first, he arbitrarily assigned the HECs to the fundamental units of the received sequence. He did not give any explanation or argument for this proposition though, and at the end of the book he made some loose remark on this question such as:

The set of 36 HEC arises as a rather natural shorthand for the set of 64 … (p 503)

Then, he made up a long story around the embedded pseudo-sequence. In the role of the original author, he invented the n-gram science in the here-published form. Within the scope of this 'science', he classified the tri-, tetra-, and hexagrams, and the HECs, and arranged them in many regular forms. These formations originated in the binary structure of these elements and did not show up either in the Cook diagram or the WW sequence. Cook, however, regarded his pseudo-sequence as an equivalent of the other regular sequences, and a representative of them in the diagram.

In the course of the process of derivation, Cook created new and arbitrary n-gram operations, and with the help of them regenerated the WW sequence. It is true that, at the end of the procedure, the hexagrams stood in the well-known order, but it was him alone who had arranged them in the 21st century. He did not give any argument for the existence of a predecessor who would have lived circa 3,000 years ago and performed the same procedure in his life. In our time, Richard Cook knew the aim of his work: he wanted to build up the Wen Wang sequence, which was well known by him before. But what might have been the aim of Wen Wang or another, unknown mathematician? What did he intend to tell (if anything) by the sequence or the Cook diagram? These questions have not been answered. Without independent evidence, readers do not get any new information. The process that Cook recounts is only one of the countless possibilities for arranging the hexagrams in this known order, and thus, his story remains only a fiction.

Unfortunately, the author also went astray in the field of mathematics. In the book, it certainly was not possible to give a full description of the rules of the n-gram science. In the absence of that, the mathematical operations, in many cases, seem to be strange and directly adapted to the actual task. For example, some rather bizarre custom-made operations are used in the derivation, such as: combining trigram matrices into hexagrams, reading the hexagram matrices along circular or boustrophedon(!) paths, doubling a yang line to produce a yin (see example 2), and others.

Moreover, there is no connection between this procedure and the ancient Chinese culture. Even the wisdom and the philosophy of the Yijing are not mentioned at all. The personal and objective conditions wherein these mathematical ideas might have been born are missing. I am unable to look at Wen Wang the same way as the author:

… imagine the imprisoned king, the recipient of ancient knowledge, contemplating the loss of that knowledge at the hands of his captors: his thought is not to save himself, but preserve that knowledge, to impart it to someone. (p 502)

In sum, Cook's theory has remained unproved for me. Naturally, I can be mistaken in this opinion, and I accept the responsibility for that. I hope that somebody among the readers of this review will feel like reading the book to discover the hidden essence of it, so I have provided a detailed description here, to make his/her future work more comfortable.

In writing my observations here, I have concentrated on the main line of the theory, the claimed mathematical origin of the received sequence. The chapters Introduction and Conclusion still contain a lot of interesting and useful material that the author collected from old Chinese books and manuscripts, and developed on the basis of his comprehensive knowledge in these fields of the Chinese culture. I can only recommend to read it.

Minimal glossary

Combinatorics: Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterise their properties.

Division in extreme and mean ratio (DEMR): better known as the golden ratio or golden section. It is an irrational number often denoted by the Greek letter Φ (phi) with a value of 1,618033… For further information, see Wikipedia.

Fibonacci sequence: a linear recurrence sequence (0, 1, 1, 2, 3, 5, 8, …) where each term is the sum of the two preceding ones, starting from (0, 1). For further details, see Wikipedia.

Fibonacci number: any number in the Fibonacci sequence.

Linear recurrence sequence (LRS): In a recursive (recurrence) sequence, each term (usually a number) is defined from the earlier terms in the sequence. See Linear Recursive Sequences by Bjorn Poonen (PDF).

n-gram, n-gram science: In the fields of computational linguistics and probability, an n-gram is a contiguous sequence of n items from a given sample of text or speech. In other fields, this knowledge – as science, discipline, or theorem – had not existed before this book (or, I have not found anything about it), though Shao Yong and his followers identified some categorical relationships in the sets of binary figures which can be regarded as the first emergence of that. (When writing this review, I found a rather new article on this subject: N Gua Theory written by Larry Schultz, published on the internet, in April 2018.)


I am very grateful to Steve Marshall for the useful remarks and the good advice he gave me in the course of the final composition of this review. He acted as an expert reader and helped me to make my text understandable.