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I Ching Permutations

Subject: I Ching Permutations

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Subj:  number of possible readings
Date:  Sat, Feb 17, 1996 12:03 AM PDT

Mike McCormick ( wrote:

Glenn Wallace asked:

> Would the mathematician here on the list help me to
> understand these possibilities?

> 1. How many possible combinations/permutations are 
> there in the whole I Ching system using "no" changing 
> lines?

Answer: 64

> 2. How many possible combinations are there in the whole 
> I Ching system using 1 changing line?

Answer:  64 x 6 = 384

For each of the 64 possible first hexagrams, there are 6
possible second hexagrams if just one line moves.  Therefore
the total number of such readings is 64 x 6.

> 3. How many using 2 changing lines?

Answer: 64 x 15 = 960.  

The reason for this is that there are 15 different ways to
get 2 moving lines.  This is calculated using the
combinatoric "6 choose 2" formula 6!/(6-2)!2! = 15.

> 4. How many using 3 changing lines?

The correct answer is 64 x 20 = 1280. 

There are 20 ways to get 3 moving lines, using the formula

> 5. How many using 4 changing lines?

The answer is 64 x 15 = 960. 

There are 15 ways to get 4 moving lines, using the formula
6!/(6-4)!4!.  It is not a coincidence that this is the same
answer as to question #3; it follows the standard
combinatoric bell curve.

> 6. How many using 5 changing lines?

The answer is 64 x 6 = 384. 

There are 6 ways to get 5 moving lines. This is per the
formula 6!/(6-5)!5! = 6!/5! = 6.  But you don't have to know
math to figure out this one, just use common sense: To count
how many ways to get 5 moving lines, you can just count how
many ways there to get the one line that's *not* moving,
which is 6.

> 7. How many using 6 changing lines ?

Answer: Just 64.  

Forget math, just think about it a moment.  Given some
initial hexagram, if all 6 lines move then there is only one
possible second hexagram.  One reading per initial hexagram
-> 64 possible first hexagrams -> 64 readings.

> 8. How many total possible combinations/permutations in
> the whole I Ching system?

Answer: 4096.  

In general, a I Ching reading can begin with any of 64
hexagrams, and move to any of 64 second hexagrams (counting
the no moving lines case as "moving" to the same hexagram),
therefore the total possible readings is simply 
64 x 64 = 4096.

Just as a sanity check, note that the seven previous answers
I gave do total 4096:

4096 = 64 + 384 + 960 + 1280 + 960 + 384 + 64.

An interesting aside about the magic I Ching number 4096: 
In the octal notation that computer programmers and
mathematicians use, 4096 is written as 10000, and in the 
I Ching the totality of everything in the world is often
called "the ten thousand things".

Hope this helps...
Mike McCormick

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