Why Six-Line Hexagram Casting Uses Three Coins Rather Than Two

Many people have their own explanations for why Six-Line divination uses three copper coins for casting a hexagram. It is widely acknowledged that three-coin casting produces a uniform probability distribution — but why not two coins?

If two coins were used, one might record outcomes as follows:

Both heads: yang, unchanged
Both tails: yin, unchanged
One heads, one tails: yang changes to yin
One tails, one heads: yin changes to yang

Wouldn't this record a Six-Line hexagram perfectly? Many people are puzzled by this but usually avoid the question.

The fact is that this approach is wrong. The scheme above seems reasonable, but the probabilities are not actually uniform.

If two coins are used, another question arises: when determining whether it is yin-changes-to-yang or yang-changes-to-yin, how should the order be decided?

For example, if left and right are used to determine — left as yin, right as yang — one records it as yin-changes-to-yang. The same could be defined using up and down. Even if two coins stack on top of each other, there is still one on top and one below, and heaven and earth can equally be used to define priority.

So it sounds perfect, but it is actually wrong.

Because once the determination of yin-changes-to-yang or yang-changes-to-yin involves introducing a temporal or spatial distinction to judge order, then when both coins are heads or both tails, a temporal or spatial distinction should also be introduced to judge order.

That is exactly where the problem lies. The very act of having to judge the heads-or-tails orientation of the coins introduces unequal probabilities. Ponder this carefully — isn't it interesting?

For example:

When Coin 1 is on the left and Coin 2 is on the right:
Coin 1 and Coin 2 both heads: record as yang unchanged
Coin 1 and Coin 2 both tails: record as yin unchanged
Coin 1 heads, Coin 2 tails: record as yang changes to yin
Coin 2 heads, Coin 1 tails: record as yin changes to yang

When Coin 1 is on the right and Coin 2 is on the left:
Coin 1 and Coin 2 both heads: record as yang unchanged
Coin 1 and Coin 2 both tails: record as yin unchanged
Coin 1 heads, Coin 2 tails: record as yin changes to yang
Coin 2 heads, Coin 1 tails: record as yin changes to yang

Merging both cases and tallying the counts [many thanks to Johnny for the correction]:

Yang unchanged: 2
Yin unchanged: 2
Yin changes to yang: 2
Yang changes to yin: 2

But in actual random number generation a problem arises. If we organize it, we get the following result:

Coin 1 heads, Coin 2 tails: record as yang changes to yin
Coin 1 heads, Coin 2 tails: record as yin changes to yang

Coin 2 heads, Coin 1 tails: record as yin changes to yang
Coin 2 heads, Coin 1 tails: record as yin changes to yang

In a computer-code implementation using only two random numbers — say producing [0,1] — the result could be either yin-changes-to-yang or yang-changes-to-yin. The two random numbers are thus effectively indistinguishable and cannot map one-to-one onto the yin–yang transformations.

In other words, in practical operation, because the two coins look identical and have no inherent order — and because one cannot pre-assign which is which before casting — there is no way to track and distinguish Coin 1 from Coin 2.

Do not underestimate this problem. In many flawed Six-Line charting programs, casting a hexagram by computer uses random numbers directly, which causes problems in the resulting hexagram itself.

Solving this problem requires defining two coin classes, having each generate its own random number, and pre-defining which is Number 1 and which is Number 2. This introduces too many external conditions and a priori assumptions, which already violates the casting process's own requirement to minimize preset conditions.

It is encouraging to see some Six-Line charting programs beginning to address the issue of true randomness, using the CPU's true random number generator for casting in an attempt to more faithfully simulate real-world hexagram casting.

But many people's actual code simply and crudely uses random numbers without thoroughly simulating the coin-casting process. So while statistical randomness is present, the distribution of the resulting hexagrams still has problems, and many computer-generated hexagrams still do not feel sufficiently "natural" — their accuracy sometimes feels inferior to hand-tossing.

So what is the correct method? Very simple: after generating random numbers in code, strictly follow the coin-casting rules, and the probability distribution will naturally be uniform. For example, with binary random numbers, take every three successive random numbers as one line.

And because 0 is the base state, it should be regarded as heads, while 1 should be tails. Therefore:

Obtaining one 1: young yang (shaoyang), unchanged
Obtaining two 1s: young yin (shaoyin), unchanged
Obtaining three 1s: old yang (laoyang), changes to yin
Obtaining zero 1s: old yin (laoyin), changes to yang

Repeat this six times, from bottom to top, and you obtain the correct main hexagram and its changed hexagram for Six-Line divination. It is very simple.

From the One comes the Six; the One is honored as the ultimate, taking its completed number to know what is formed. Thus the number six is used to mark it — the beginning of the virtue of Water, that is, the number of old yin. Each time a tails is obtained, one is added; this is precisely what is meant by "reversal is the movement of the Dao." Seven is young yang and eight is young yin, whose lines do not change. Nine is old yang and six is old yin, whose lines all change.