On The I-Ching and The Bagua Procedure

The I-Ching

The I-Ching, or the Book of Changes, is one of China's oldest and most ancient surviving texts.  That this text was used for fortune telling is easy to tell from its title -- the study of events in motion or under change. In modern scientific language, it is thus a calculus on dynamic systems, reducing the science of fortune telling to that of predicting how events will change based on certain laws.
In a commentary of the I-Ching named the 說卦傳 (Descriptions of Hexgrams), there is this famous line:

數往者順，知來者逆，是故易逆數也。
The past is positive, the future is negative. Thus, the I-Ching is concerned with negative numbers/calculations.

It is thus clearly stated that the I-Ching is used for making "negative" calculations, or to predict the future. The most authoritative commentary of the I-Ching, the 繫辭 (Bundled Commentaries), has this to claim:

參伍以變，錯綜其數，通其變，遂成天地之文。極其數，遂定天下之象。
Numbers evolve/change and mix with/criss-cross each other (perhaps referring to a procedure or a mathematical formula).
If all such changes are understood (e.g. formulated as differential equations), one can "write the book" on (what will happen in) the heavens and the earth.
As the numbers (formulas) are evaluated/iterated to their end (e.g. fully integrated), all statuses in the world are determined.
生生之謂易... 極數知來之謂占，通變之謂事，陰陽不測之謂神。
The I-Ching is progressive evolution/iteration/change. Fortune-telling is taking numbers (or formulas) and evaluating/iterating them to their ends (e.g. fully integrated) in order to know the future. Events mean understanding all possible changes. That which cannot be predicted by these methods (yin/yang) is God.

This is by far the clearest implication of fortune-telling as a scientific discipline, in a text that appeared perhaps before 1,000BC!  An even more amazing fact is the commentaries' repeated references to the word 數 (numbers, formulas, or referring to mathematics).  The I-Ching itself has no reference to mathematics or numbers in an arithmetical sense; trigrams and hexgrams do not have numbers in them (although one might argue that their formation involves numbers, as described below).

Why, then, should all commentaries attach numbers (or perhaps mathematics) to the I-Ching without any apparent reason at all?  This can only come from the fact that these authors knew the I-Ching to be a mathematical text on the scientific discipline of fortune-telling, and that it outlines a calculus and treats fortune-telling as solving dynamic systems.

The next series of entries will look at the Bagua system of fortune telling outlined in the I-Ching.

The Bagua Procedure

The following procedure is used to form one single gua (or trigram):

1. Start with 50 sticks of yallow grass (大衍之數五十)
2. Throw one away leaving 49 (其用四十有九)
3. The first cycle:
4. Take one stick, and divide the remaining 48 sticks randomly into two piles (分而為二以象兩, 掛一以象三) -- a, b where a+b=48.
5. Take the remainder of each pile divided by 4 (or 4 if zero).  As 48 is divisible by 4, a and b must either both be divisible by 4 or both non-divisible by 4.  Therefore, the two only possible remainders are 4 (both non-divisible) and 8 (both divisible).  (揲之以四以象四時, 歸奇於扐以象閏)
6. The piles that are left, after deducting the remainder, must either total 44 (=48-4) or 40 (=48-8).
7. The second cycle:
8. Repeat steps #4 to #6: divide the remaining 44 or 40 sticks randomly into two piles and take the remainder of each pile divided by 4 (or 4 if zero).  As 44 and 40 are both divisible by 4, the only possible remainders are 4 and 8.  Thus the piles are left must total one of three only possibilities: 32 (=40-8), 36 (=40-4 or 44-8), 40 (=44-4).
9. The third cycle:
10. Repeat steps #4 to #6: divide the remaining 32, 36 or 40 sticks randomly into two piles and take the remainder of each pile divided by 4 (or 4 if zero).  As 32, 36 and 40 are all divisible by 4, the only possible remainders are 4 and 8.  Thus the piles are left must total one of four only possibilities: 24 (=32-8), 28 (=32-4 or 36-8), 32 (=36-4 or 40-8), 36 (=40-4).
11. Divide the resulting pile by 4, yield one of four only possibilities: 6,7,8,9.
12. The even numbers are yin, with 6 being special and marked with an X.
13. The odd numbers are yang, with 9 being special and marked with an O.
14. This concludes the divination of one line out of three that makes up the trigram.  Repeat for two more times to complete the remaining two lines of the trigram.

Diagrammatically, the procedure can be shown as a decision tree:

The Coins Substitute Method

A simpler method was devised subsequently to use three coins instead.  Essentially, each coin of a toss corresponds to each of the three stages in the procedure. A head = -4, tail = -8.  Follow the corresponding branch of the decision tree to reach the ultimate leaf.

Notice that the order of applying the branches (i.e. coin toss results) does not matter -- the same leaf node is reached.  That is because there is only one value that corresponds to any particular combination of -4 and -8.

There are only four possible outcomes of a coin toss regarding three coins:

• Three heads => -4 x 3 = 48 - 12 = 36
• Two heads, one tail => -4 x 2 -8 = 48 - 16 = 32
• One head, two tails => -4 - 8 x 2 = 48 - 20 = 28
• Three tails => -8 x 3 = 48 - 24 = 24

Thus, the coins substitute method is essentially an exact simulation of the trigram-forming procedure.

O and X Lines

In the I-Ching, all divination's are tied specifically to lines marked by O and X (i.e. 36 and 24, or 9 and 6 in terms of multiples of 4).  For example, a hexgram with the fifth line marked by O is named (九五) while that with the fifth line marked by X is named (六五).

Thus, it is seen that only lines that reach 24 or 36 have significance in interpreting a hexgram.

For each line, the probability of reaching 24 is 0.125 (i.e. 1/8), while that of reaching 36 is also 0.125.  Thus, there is a probability of 0.29 to have only one X or only one O in a trigram, or 0.38 to have only one X or only one O in a hexgram.

The probability of having no X (or no O) in a trigram is 0.67, while having no X (or no O) in a hexgram is 0.45.

The probability of having no X and no O in a trigram is 0.42.  The probability of having no X and no O in a hexgram is 0.18.

The probability of having only one X or O in a trigram is 0.42.  The probability of having only one X or O in a hexgram is 0.36.

Thus, the chances of having only one X or O in a hextram is roughly one-third, in addition to 18% chance of having none.  Essentially, the I-Ching deals with this one-third of possibilities.  The fact that the I-Ching does not deal with situations where there are more than one line marked with X or O strongly suggests that there is a method to reduce the n>1 cases to a combination of n=1 cases (where n = number of lines marked with X or O).