Modern readers who encounter the Yijing (Book of Changes) most often begin with the "coin method" (tossing three bronze coins to form a hexagram), owing to its simplicity and convenience. Yet when it comes to the most orthodox, the most philosophically rich in cosmogonic thought, and the method most esteemed by the Master (Confucius) and the sages of the pre-Qin era, none surpasses the Great Expansion Milfoil Method (dayan shifa, yarrow stalk divination).

This article was translated from the original Chinese by AI. Nuances may differ from the source.

The Great Expansion Milfoil Method is not merely a divinatory technique; it is a solemn ritual that simulates the cosmos's passage from primordial chaos (Taiji) to the generation of all things (the sixty-four hexagrams). This essay undertakes a comprehensive examination of this ancient art from four dimensions: philosophical principles, ritual procedure, mathematical probability, and historical verification.

I. The Philosophical Foundations of the Number of the Great Expansion

"The number of the Great Expansion is fifty; of these, forty-nine are used. Divide them into two to symbolize the Two (Heaven and Earth); set one aside to symbolize the Three; count them off by fours to symbolize the Four Seasons; return the remainder to symbolize the intercalary month. In five years there are two intercalations; therefore, one returns the remainder twice before completing the cycle." — Xici Zhuan (Great Commentary on the Appended Judgments), Yijing

This passage of fewer than a hundred characters constitutes the core canon of ancient Chinese numerological arts.

1. Why "Fifty"$1

Regarding the origin of "the number of the Great Expansion is fifty," Yijing scholars through the ages have offered a multitude of interpretations. The principal views are as follows:

• The sum of the numbers of Heaven and Earth: The Xici Zhuan states, "The number of Heaven is twenty-five; the number of Earth is thirty; together the numbers of Heaven and Earth total fifty-five." The Song dynasty scholar Zhu Xi held that fifty is the number of the Great Expansion, with five (the generative numbers of the Five Phases) removed and left unused, thus yielding fifty.
• The center of the Hetu and Luoshu: In the Hetu (River Chart), the center contains five and ten, whose product is fifty. In the Luoshu (Luo Document), the center is five, and the ten of the five directions likewise accords with fifty.
• The Pythagorean theorem: With a base of three, a height of four, and a hypotenuse of five, the sum of their squares (9 + 16 + 25) is precisely fifty.

Regardless of which explanation one favors, "fifty" invariably symbolizes the "totality" and the "fixed number" of all things in the universe.

2. Why "Of These, Forty-Nine Are Used"$2

In actual divination, one must take a single stalk from the fifty yarrow stalks and reverently return it to the stalk-holder or place it directly before oneself, never to be touched again throughout the entire process. This single stalk holds a position of supreme eminence.

• The dialectic of substance and function (ti-yong): This one stalk represents Taiji (the Supreme Ultimate). Taiji is the origin of the world — utterly still, yet responsive to all stimuli. Precisely because of Taiji's "stillness," the remaining forty-nine stalks are able to "move."
• The mutual arising of the hidden and the manifest: This is a supremely refined philosophical metaphor — truth (the Dao) is often hidden and invisible. All that we can manipulate and calculate is the functioning (yong) that arises after the Dao manifests, not the essence (ti) itself.

II. The Four Operations and Eighteen Transformations: A Ritual of Communion with the Luminous Spirits

The procedural steps of the Great Expansion Milfoil Method are extraordinarily intricate, earning the name "Four Operations and Eighteen Transformations."

• The Four Operations: Dividing in two, setting one aside, counting off by fours, and gathering the remainders — these four actions constitute one operation (one transformation).
• The Eighteen Transformations: Three operations (three transformations) are required to determine a single line (yao). A hexagram contains six lines, hence $3 \times 6 = 18$ transformations are needed to complete a hexagram.

This process is not merely an exercise in deriving numbers; it is designed to allow the diviner, through the painstaking labor of the procedure, to gather and focus the mind, entering a state of utmost sincerity (cheng).

Preparations

• Instruments: Fifty yarrow stalks (the ancients used serrated grass stems; bamboo sticks or Go stones are commonly used as substitutes today).
• Setting: A clean table or desk. The diviner sits facing south (or north, depending on the lineage), with a heart full of reverence.

Determining the First Line (Requires Three Transformations)

The First Transformation

1. Extracting the Taiji: Remove one stalk from the fifty and set it aside. Forty-nine remain in hand.
2. Dividing in Two (Symbolizing the Two): Divide the forty-nine stalks at random into a left pile and a right pile, symbolizing the initial separation of Heaven and Earth, Yin and Yang.
3. Setting One Aside (Symbolizing the Three): Take one stalk from the right pile and place it between the little finger and ring finger of the left hand. This symbolizes "Man" arising between Heaven and Earth, forming the Three Powers (sancai): Heaven, Earth, and Man.
4. Counting by Fours (Symbolizing the Four Seasons):
   • First, count the left pile: group the stalks in sets of four (symbolizing spring, summer, autumn, and winter). At the end, 1, 2, 3, or 4 stalks will remain (note: if exactly divisible, take 4, not 0). Remove the remainder and place it between the ring finger and middle finger of the left hand (symbolizing the intercalary month).
   • Then count the right pile: likewise group by fours, and place the remainder (1, 2, 3, or 4) between the middle finger and index finger of the left hand (symbolizing the second intercalation in five years).
5. Gathering the Remainders (Combining the Remainders): Gather all the stalks held between the fingers (the 1 set aside + the left remainder + the right remainder) and place them in the upper-left corner of the table. They are set aside and not used again. This is the number of stalks removed in the "First Transformation."

Mathematical Verification: The total number of stalks removed in the First Transformation can only be one of two values: 5 or 9. (At this point, the number of stalks remaining in hand is either 44 or 40.)

The Second Transformation

Take the stalks remaining from the First Transformation (44 or 40) and combine them, then repeat the steps above:

1. Divide in Two: Divide the stalks in hand into left and right piles once more.
2. Set One Aside: Take one from the right pile and hold it.
3. Count by Fours: Count off both left and right piles by fours and take the remainders.
4. Gather the Remainders: Combine the set-aside stalk and the remainders, and place them to one side.

Mathematical Verification: The total number of stalks removed in the Second Transformation can only be one of two values: 4 or 8. (At this point, the number of stalks remaining in hand is 32, 36, or 40.)

The Third Transformation

Take the stalks remaining from the Second Transformation, combine them, and repeat once more:

1. Divide in Two.
2. Set One Aside.
3. Count by Fours.
4. Gather the Remainders.

Mathematical Verification: The total number of stalks removed in the Third Transformation can only be one of two values: 4 or 8.

Deriving the Number for One Line

After the three transformations, take the total number of stalks remaining in hand (those not removed through the gathering process) and calculate as follows. The remaining total can only be one of four numbers. Divide it by four to obtain the strategy number (ce shu) for that line:

• 24 stalks remaining: $24 \div 4 = 6$ — Old Yin (a changing line, notated as X or --x--).
• 28 stalks remaining: $28 \div 4 = 7$ — Young Yang (unchanging, notated as —).
• 32 stalks remaining: $32 \div 4 = 8$ — Young Yin (unchanging, notated as --).
• 36 stalks remaining: $36 \div 4 = 9$ — Old Yang (a changing line, notated as O or —o—).

With this, the first line (the bottom line) is drawn. Repeat the above "three transformations" a total of six times, drawing the lines from bottom to top — first, second, third, fourth, fifth, and sixth — to complete the hexagram.

III. God's Weighted Dice: The Probabilistic Aesthetics of the Great Expansion Milfoil Method

Here lies the most fundamental distinction between the Great Expansion Milfoil Method and the coin-toss method. The coin-toss method is based on a simple binomial distribution, yielding relatively uniform probabilities. The Great Expansion Milfoil Method, through its intricate "set one aside, count by fours" procedure, constructs a markedly asymmetric probability model.

1. Combinatorial Logic of the Variables

In the three transformations, the number of stalks removed each time (set-aside + left remainder + right remainder) follows specific patterns:

• First transformation removal: 5 or 9
• Second transformation removal: 4 or 8
• Third transformation removal: 4 or 8

To obtain the final result, we examine the combinations of remainders:

• Obtaining Old Yang (9): Requires 36 stalks remaining, i.e., $49 - (5 + 4 + 4) = 36$.
  • Condition: All three transformations must remove the minimum number (5, 4, 4).
• Obtaining Old Yin (6): Requires 24 stalks remaining, i.e., $49 - (9 + 8 + 8) = 24$.
  • Condition: All three transformations must remove the maximum number (9, 8, 8).

2. Specific Probability Calculations

Following the derivations of Zhu Xi in his Yixue Qimeng (Introduction to the Study of the Yijing) and those of modern mathematicians, the probabilities of the four line types are as follows:

3. The Philosophical Significance Behind the Data

This is a remarkable discovery: Old Yang (9) appears with a probability three times that of Old Yin (6)! Simultaneously, Young Yin (8) appears with a probability far exceeding that of Young Yang (7).

This asymmetry embodies a profound pre-Qin cosmological vision:

1. Yang is abundant; Yin is scarce: Yang qi (creative force, dynamism) is typically dominant and active. Yet when at its peak and turning to decline (Old Yang), such a transformation occurs more readily than the turning of Yin at its extreme to Yang (Old Yin). This symbolizes that "although accomplishment (Yang) is arduous, Yang qi itself is plentiful."
2. The world's dominant keynote is Young Yin: The most probable outcome, Young Yin (8) at 43.75%, represents a state of stillness, latency, and material existence. This suggests that the phenomenal world is, for the most part, quiescent — preserving rather than transforming, free of violent upheaval.
3. Six is rarely obtained, while Nine comes readily: In Yijing studies, "six" represents Yin at its extreme, symbolizing total collapse or transformation — a condition exceedingly difficult to achieve in the natural world. "Nine" represents Yang at its extreme, where energy released to its fullest occurs comparatively easily.

This renders the Great Expansion Milfoil Method far closer to the complex rhythms of the real world than the coin-toss method — the world is neither black nor white, nor is it a perfectly fair randomness, but rather an evolutionary system possessed of specific tendencies.

IV. Historical Verification from the Zuo Zhuan: The Divinatory Logic of Duke Mu of Qin's Campaign Against Jin

To truly understand how the Great Expansion Milfoil Method was applied in antiquity, we must return to the Spring and Autumn period. The Zuo Zhuan records a wealth of divination cases, and among them, the case of Duke Mu of Qin's campaign against Jin (Zuo Zhuan, Duke Xi, Year 15) is a superb exemplar of pre-Qin hexagram-interpretation logic.

1. Historical Background

In 645 BCE (the seventh year of King Xiang of Zhou), the state of Jin suffered a famine and sought grain from Qin. Duke Mu of Qin generously obliged (the "Flotilla Campaign" to ship grain). The following year, Qin itself suffered a famine and requested grain from Jin. Duke Hui of Jin not only refused but attempted to exploit Qin's weakness by launching an attack. Duke Mu of Qin was enraged and resolved to campaign against Jin.

Before dispatching his troops, Duke Mu ordered his diviner, Tu Fu, to perform the yarrow stalk divination.

2. The Hexagram Obtained

"They performed the divination and obtained Gu (Decay) changing to Dui (Joy). The pronouncement was: 'A thousand chariots shall be thrice repelled; and from the remnant of the thrice repelled, the male fox shall be captured.'"

This involves a key aspect of the ancient method of recording: "obtained Gu changing to Dui."

• The primary hexagram: Shan Feng Gu (Mountain over Wind — Gen above, Xun below), the hexagram of Decay.
• The changed hexagram (zhi gua): Dui (Joy). This formulation is debated. Following the original Zuo Zhuan phrasing "Gu zhi Dui," it usually indicates that the lower trigram transforms to Dui, or that specific lines change.
  • If the initial six (the bottom line) of Gu changes (Yin becomes Yang), the lower trigram Xun would become Qian, yielding Da Xu (Great Accumulation) — which does not match.
  • Under the ancient conventions of the Zuo Zhuan, there are often special usages. Most commentators hold that this indicates Gu underwent a transformation ultimately pointing toward imagery associated with Dui. The mainstream interpretation is that the inner trigram (lower trigram) of Gu changed from Xun to Dui (which would in fact require the first, second, and third lines all to change — an extremely low probability event).
  • An alternative authoritative interpretation (Du Yu's commentary): "Gu zhi Dui" does not mean the entire hexagram transforms into the Dui hexagram, but rather refers to the zhen-hui (inner-outer) relationship of Gu or the imagery of specific line statements.

Let us focus on Tu Fu's pronouncement, for therein lies the essence of pre-Qin Yijing studies.

3. Tu Fu's Brilliant Interpretation

Tu Fu did not simply recite the hexagram judgment from the Yijing (Gu: "Supreme success; it is advantageous to cross the great river"). Instead, he employed the method of "extracting images" (qu xiang):

• Image One: Decay beneath the mountain wind The Gu hexagram has Gen (Mountain) above and Xun (Wind/Wood) below. Wind falling beneath the mountain carries not merely the meaning of "decay" but also the sense of "leaves returning to their roots" and "rectification." Duke Mu of Qin would use this to rectify the perfidious Duke Hui of Jin.
• Image Two: A thousand chariots, thrice repelled This is a deconstruction of the hexagram's structural imagery. In Gu, the lower trigram Xun represents wind, while the upper trigram Gen represents mountain.
  • Tu Fu explained: "A thousand chariots, thrice repelled." This likely derives from an understanding of the Xun trigram, which signifies orderliness, as well as chariots and horses.
  • A more refined explanation lies in the nuclear trigrams (hu gua). The nuclear trigrams of Gu contain shadows of Gui Mei (the Marrying Maiden) or Jian (Gradual Progress) — the third through fifth lines form Zhen (Thunder), and the second through fourth lines form Dui (Joy). Zhen symbolizes the chariot.
  • "Thrice repelled" prophesied the course of the battle: the Jin army would be driven back three times, or the Qin army would cross the river three times.
• Image Three: Capturing the male fox This is the most celebrated pronouncement.
  • The Gu hexagram itself carries the meaning of "bewilderment." Duke Hui of Jin, having broken faith, was confused in judgment — hesitant and irresolute like a fox.
  • From the trigram imagery: Gen (the upper trigram) signifies the young man, and can also symbolize the dog or the hand; Xun (the lower trigram) signifies the elder daughter, and can also symbolize hidden murmuring or the fox.
  • The third line statement of Gu reads: "Rectifying the father's decay" (gan fu zhi gu). But Tu Fu employed the greater image (da xiang): Gen above (Mountain/Stopping) pressing down upon Xun below (Wind/Fox). A mountain pinning down a fox — naturally, this is "capturing the male fox."
  • The "male fox" here directly refers to Duke Hui of Jin.

4. Verification of the Outcome

In the ninth month of that year, the armies of Qin and Jin clashed at the Battle of Hanyuan. The Qin army won a decisive victory and captured Duke Hui of Jin — the very "male fox." The prophecy of "capturing the male fox" was thus fulfilled.

5. Lessons from the Case: How Pre-Qin Yijing Studies Differed from Later Practices

From the case of Duke Mu of Qin's campaign against Jin, we can observe the distinctive characteristics of how the Great Expansion Milfoil Method was applied in the pre-Qin era:

1. Emphasis on "images" over "numbers": When interpreting the hexagram, Tu Fu did not dwell on the mutual generation and conquest of the Five Phases (that was a development of the Han dynasty and later). Instead, he made extensive use of the material imagery of the Eight Trigrams (wind, mountain, fox, chariot) to construct a narrative.
2. A flexible view of line changes: The phrasing "Gu zhi Dui" suggests that the logic of hexagram transformation at that time may have been more intuitive than later "changed hexagram" rules, placing greater emphasis on the flow of hexagrammatic qi toward a particular direction (such as Dui in the west, associated with metal and martial force).
3. Narrative prophecy: The divinatory result was not a simple verdict of auspicious or inauspicious, but a passage of verse rich with narrative ("a thousand chariots, thrice repelled... capturing the male fox"). This was not only prediction but also a form of political rhetoric and psychological warfare.

V. From Yarrow Stalks to Bronze Coins: The Decline and Evolution of Divination Methods

The Great Expansion Milfoil Method flourished from the Shang and Zhou dynasties through the Han. However, as the wheel of history turned, it was gradually marginalized.

1. Complexity Is the Greatest Adversary

Completing a single proper Great Expansion divination takes a swift practitioner fifteen to twenty minutes, and a slower one up to half an hour. Moreover, it is exceedingly easy to miscount during the process — one error and the entire result is invalidated. For the battlefield demanding urgent decisions or the marketplace diviner serving common folk, this was simply too slow.

Thus, from the Tang and Song dynasties onward, the "Fire Pearl Forest Method" (Huo Zhu Lin Fa — the coin method, also known as Six-Line Najia divination) rose to prominence. One need only shake three bronze coins six times, and a hexagram can be formed in a matter of minutes. Moreover, the Five Phases mutual generation and conquest system became more standardized and easier to learn.

2. The Price of "Sincerity of Heart"

The Great Expansion Milfoil Method emphasizes ritual gravity and the attainment of "unity between Heaven and Man" through the prolonged process of counting yarrow stalks. As divination gradually became secularized and commercialized in later periods, people grew more concerned with the accuracy of results than with the psychological cultivation inherent in the process of seeking a hexagram.

3. A Modern Perspective: Why We Still Need the Great Expansion

In an age when computer algorithms can generate hundreds of millions of random numbers in a single second, does the Great Expansion Milfoil Method retain any significance$3 The answer is emphatically yes.

• The wisdom of slowness: In this age of acceleration, the Great Expansion Milfoil Method compels us to slow down. The dividing and recombining of those forty-nine yarrow stalks is, in truth, a process of ordering one's thoughts. By the time you have completed the eighteen transformations, perhaps even before the hexagram appears, the answer already resides within your heart.
• A simulation of chaos theory: As discussed above, the distinctive probability distribution of the Great Expansion Milfoil Method constitutes an ancient simulation of nature's "nonlinear systems." It reminds us that the world's development tends toward inertia (Young Yang, Young Yin), while sudden transformations (Old Yang, Old Yin), though crucial, are rare occurrences.

VI. Conclusion

The Great Expansion Milfoil Method represents a grand attempt by the forebears of Chinese civilization to capture the rhythms of the cosmos through mathematical logic.

The fifty yarrow stalks in their hands were not mere plant stems but the physical embodiment of the "Number of the Great Expansion." Each division in two is a reenactment of Taiji giving rise to the Two Modes; each setting aside of one stalk is a symbol of humanity seeking its place between Heaven and Earth.

When we look back upon the divination performed before the Battle of Hanyuan by Duke Mu of Qin, what we see is not merely the miraculous prophecy of "capturing the male fox," but the courage of an ancient civilization that, facing the unknown tides of fate, sought to cast a net woven of reason (number) and intuition (image) to seize hold of the future.

Today, when we return to the study of the Great Expansion Milfoil Method, it is not necessarily to divine fortune and misfortune, but to feel, in the touch of our fingertips, that reverence and contemplation for the Great Way of Heaven and Earth that has traversed a thousand years to reach us.

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References:

1. Wang Bi (Wei dynasty), annotated; Kong Yingda (Tang dynasty), sub-commentary: Zhouyi Zhengyi (Correct Meaning of the Yijing)
2. Zhu Xi (Song dynasty): Zhouyi Benyi (Original Meaning of the Yijing); Yixue Qimeng (Introduction to the Study of the Yijing)
3. Zuo Qiuming (Spring and Autumn period): Zuo Zhuan, Duke Xi, Year 15
4. Shang Binghe: Zhouyi Gushi Kao (Study of Ancient Divinations in the Yijing)
5. Gao Heng: Zhouyi Gujing Jinzhu (Modern Commentary on the Ancient Classic of the Yijing)