Module 8
Casting the I Ching with three coins
Goal of this module
Actually understand the calculation, rather than memorizing a table of results.
Settle the rules before you start
Two things have to be fixed before you cast, and neither may change partway through.
First, what the question is. The more specific the question, the more the hexagram has to bite on. “How will my life go” is too broad — any hexagram can be draped over it, which amounts to having asked nothing. Module 11 goes into this properly; for now it is enough to know that you think it through before you throw.
Second, which face counts as 3 and which as 2. Heads for 3 or tails for 3 is a matter of convention and is arbitrary in itself. But once fixed, all six throws must use the same one. Switching halfway means the first throws and the last were encoded differently.
Why 3 and 2
3 is odd, 2 is even. That is not an arbitrary pick: odd goes with yang, even with yin — yang is one and whole, yin is two and divided, the same correspondence as the unbroken and broken lines in module 2.
Each of the three coins yields a 2 or a 3, and you add them. The minimum is 2+2+2 = 6, the maximum 3+3+3 = 9, and swapping any single coin moves the sum by 1. So the result is pinned to exactly four numbers: 6, 7, 8 and 9.
Eight combinations — count them yourself
Three coins, two possibilities each, gives 2³ = 8 throws in all (module 3’s arithmetic again). Here they all are:
| Sum | Throws | Ways | Name | Yin/yang | Changes? | Probability |
|---|---|---|---|---|---|---|
| 6 | 2+2+2 | 1 | old yin | yin | changes | 1/8 ＝ 12.5% |
| 7 | 2+2+32+3+23+2+2 | 3 | young yang | yang | stays | 3/8 ＝ 37.5% |
| 8 | 2+3+33+2+33+3+2 | 3 | young yin | yin | stays | 3/8 ＝ 37.5% |
| 9 | 3+3+3 | 1 | old yang | yang | changes | 1/8 ＝ 12.5% |
The “ways” column is the point: 6 and 9 have one throw each, 7 and 8 have three each. So the four numbers are not equally likely. They run 1 : 3 : 3 : 1.
Worth remembering, because it answers a common puzzle: why do so few moving lines turn up? Because any given line has only a 1/8 + 1/8 = 1/4 chance of being one.
Old and young, and why the old ones change
“Old” and “young” in the four names are not about age. They are about degree.
Module 1 established that yin and yang turn into their opposite once they reach the limit. Old yang (9) is yang run to the end, with nowhere left to go but yin; old yin (6) is yin at its end, which can only turn to yang. Young yang (7) and young yin (8) are still on the way, with room left, so they hold.
So “6 and 9 change” is not a rule to be memorized. It is where module 1’s “what reaches its extreme reverses” lands in the numbers.
The “nine” in the text and the 9 you threw
Stop here for a moment. This is the most easily confused point for a beginner, and the least often explained.
Open the source text and you will find line names like “initial nine”, “six in the second place”, “top nine”. Those nines and sixes are names, denoting a yang line and a yin line — the yang line you got by throwing a 7 is written “nine” just the same.
The 9 you threw, by contrast, is a value. It tells you two things at once: this line is yang, and it changes.
The two are genuinely related — it was precisely because changing lines are determined by old yang and old yin that nine and six became the general names for the yang and yin lines. But a shared origin is not a shared meaning. One sentence to keep: the nine and six in a line name are names; the nine and six you throw are values.
The coin method is a simplification
The Zhou Yi’s own procedure is the yarrow-stalk method (the Great Expansion), dividing fifty stalks over and over: three operations per line, eighteen for a whole hexagram. The coin method is a later shortcut, and far quicker.
They are not quite equivalent, and the difference is worth knowing:
| Old yin 6 | Young yang 7 | Young yin 8 | Old yang 9 | |
|---|---|---|---|---|
| Three coins | 1/8 | 3/8 | 3/8 | 1/8 |
| Yarrow stalks | 1/16 | 5/16 | 7/16 | 3/16 |
What is the same: under both methods a single line is yang with probability 1/2 and is a moving line with probability 1/4. So “the yarrow method yields fewer moving lines” is a widespread but mistaken claim — the distribution of how many lines move is identical.
What differs: with three coins, old yang and old yin are equally likely; with yarrow stalks, old yang is three times as likely as old yin. In other words, yarrow makes yang lines far readier to change than yin lines, where the coin method is perfectly symmetric. That is the substantive shift the shortcut introduces.
The yarrow procedure itself is left to a later course. Here it is enough to know that you are using the simplified method, and what the simplification cost.
Recording the six throws
Throw six times and record the first at the bottom, stacking upward. The reason is module 3’s: a hexagram is generated from the bottom up.
Recording them the wrong way round is the classic beginner’s accident, and nothing will flag it — the upside-down figure is usually a real hexagram too, just not the one you threw.
As for throwing by hand versus letting a computer simulate it: mechanically there is no difference, and the distribution of the six numbers is exactly the same. What differs is that throwing by hand forces you to get the question clear — which happens to be the most important step in reading the result.
Exercises
1Three coins, one face worth 3 and the other worth 2. What is the smallest possible total, and the largest?
Right. All three at 2 gives 6; all three at 3 gives 9. In between only 7 and 8 are possible, because swapping one coin moves the total by exactly 1. So the result is pinned to 6, 7, 8 and 9.
That step went astray. Every coin lands on either 2 or 3; there is no “0” on offer. Three coins come to at least 2+2+2 = 6.
That step went astray. 3 is the largest value of a single coin, not the smallest total of three. Take the smallest, 2, three times and the total is 6.
2Which comes up more often, a total of 7 or a total of 6?
Count how many throws produce each.
That step went astray. This is the commonest misreading. The four numbers are not equally likely — 6 has only one throw, “2+2+2”, while 7 has three (any one of the coins can be the 3).
Right. 6 has one throw and 7 has three. Out of eight equally likely outcomes, 6 takes 1 share and 7 takes 3: exactly three times. 8 and 9 mirror them.
That step went astray. The size of the number has nothing to do with it. What sets the probability is how many throws produce it — and 6 has only all-2s.
This is what makes moving lines scarce: old yin and old yang take 1/8 each, so any line has only a 1/4 chance of being a moving line.
3You threw a 9 for the third line. The text reads “nine in the third place”. Are those two nines the same thing?
This is the point in this module most easily muddled.
That step went astray. They look alike, but no. The “nine” in the text is a name: it says this place holds a yang line, regardless of how you threw it. The 9 you threw is a value, saying both “yang” and “changes”.
Right. Right. Both descend from “old yang”. But the “nine” of a line name marks only a yang line (a yang line thrown as 7 is also called nine), while a thrown 9 additionally says the line changes. Same origin, not the same meaning.
That step went astray. Not that either: the connection is real. It was precisely because changing lines are fixed by old yang and old yin (9 and 6) that nine and six became the general terms for yang and yin lines.
One sentence covers it: the nine and six of a line name are names; the nine and six you throw are values. A line thrown as 7 is still called “nine” in the text.
4Where in the figure does the result of the first throw go?
Right. Right. A hexagram is generated from the bottom up, so the first throw gives the initial line. Same rule as the reading direction from module 3.
That step went astray. Writing habits run downward, so plenty of people get it backwards on their first try. Once reversed, the whole figure is upside down — what you look up is very likely a different hexagram, and since that one usually exists, nothing flags the error.
Rules clear? Go cast a hexagram, or carry on to turning six numbers into a primary and a resulting hexagram.
Common mistakes
These are not filler. They are the specific places beginners go wrong.
Thinking the 9 you throw is the “nine” in the text.
The nines and sixes in the source text are names for the yang and yin lines — a yang line thrown as a 7 is still written “nine”. A thrown 9 is a value, carrying the extra information that the line changes. Both descend from old yang and old yin, but they are not the same thing. This is the single most frequent confusion for beginners.
Thinking 6, 7, 8 and 9 are equally likely.
Among the eight equally likely throws, 6 and 9 have one each while 7 and 8 have three each. So it is 1:3:3:1, not even. Assume it is even and you will keep re-throwing, wondering why moving lines never show up.
Thinking the yarrow and coin methods are fully equivalent.
The two do give the same distribution for how many lines move (1/4 per line), but the ratio of old yang to old yin differs: 1:1 with coins, 3:1 with yarrow. The coin method is a simplification, not an equivalent substitute.
Recording the first throw at the top.
A hexagram is generated from the bottom up, so the first throw is the initial line. Get it backwards and the whole figure is inverted — and the inverted figure usually exists too, so nothing warns you. It just quietly hands you a different hexagram.
Switching which face means 3 partway through.
Which face counts as 3 is arbitrary, but once chosen it has to hold for all six throws. Switching midway means two encodings were used, and the six numbers no longer come from one rule.
Thinking 7 and 8 change as well.
Only old yin (6) and old yang (9) change. Young yang (7) and young yin (8) are stable states — “young” is precisely the sense of not yet having reached the limit, with no need to turn.
You have it when
Given any throw of three coins, you can state the sum, the yin or yang, old or young, whether it changes, and which line it belongs to — and you can name one substantive difference between the coin and yarrow methods.
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The prose, examples and exercises on this page are written for this site (L4). Passages quoted from the Zhou Yi and the Ten Wings are L1/L2 and in the public domain; per-passage sourcing is given on the hexagram pages.