3. Traditional methods

3.1 Introduction

3.2 Three coins

Materials

Three identical coins. Any coin will do, someone use chinese coins with a square hole inside. Assign 2 to a face and 3 to the the other.

Usage           

Drop the coins and sum face values. Draw the corresponding line using the following table:

6	7	8	9

Repeat other five times way starting from the bottom, moving up.

Comment

The probability for each outcome is:

2+2+2 1/2 * 1/2 * 1/2 = 1/8 6 2+2+3 1/2 * 1/2 * 1/2 = 1/8 7 2+3+2 1/2 * 1/2 * 1/2 = 1/8 7 3+2+2 1/2 * 1/2 * 1/2 = 1/8 7 3+2+3 1/2 * 1/2 * 1/2 = 1/8 8 3+3+2 1/2 * 1/2 * 1/2 = 1/8 8 2+3+3 1/2 * 1/2 * 1/2 = 1/8 8 3+3+3 1/2 * 1/2 * 1/2 = 1/8 9

This means:    Prob( ) = 1/8 = 0.125    Prob( ) = 3/8 = 0.375    Prob( ) = 3/8 = 0.375    Prob( ) = 1/8 = 0.125    Prob( or ) = 4/8 = 0.5    Prob( or ) = 4/8 = 0.5    Prob( -> ) = 1/4 = 0.25    Prob( -> ) = 1/4 = 0.25

3.3 Yarrow Stalks (copied from Clarity)

Materials

Fifty dried stalks of yarrow.

Usage

First take one stalk from the pile and set it to one side; you will not be using this one again while casting this hexagram.
Divide the remaining 49 stalks at random into two piles.
Take one stalk from the right hand heap and place it between the little finger and ring finger of your left hand.
Pick up the left hand heap in your left hand and then remove the stalks four at a time, putting them down separately from the other heap, until you have four or fewer stalks left.
Tuck this remainder between the ring and middle finger of your left hand.
Then pick up the right hand heap and sort it by fours in the same way, tucking the remainder into the next gap between your fingers.
Now add up the number of stalks you are holding in your left hand, not including the one on its own that you picked up at first.
If there are 8, count 2; if there are four, count 3.
Set these stalks aside separately, and gather the other stalks together to repeat the procedure as above only this time, when adding up the remainders, do include the original single stalk.
If there are 8, count 2; if there are four, count three.
Again, set these stalks aside and repeat the procedure a final time with those that are left, counting them in exactly the same way as the second time.
Add up the three numbers you have obtained: the result will be 6,7,8 or 9.
Draw the corresponding line using the following table:

6	7	8	9

Repeat other five times way starting from the bottom, moving up.

Click on the image below to get a visual explanation at Power Press (by Wu Wei):

Comments

Grouping stalks four at a time means we are interested in the following function:

f(n)= if (n mod 4 = 0; 4; n mod 4)

The first time we will have three groups of stalks: A[1] B[48-x] C[x], since 48 mod 4 = 0 we have the following possibilites:

x	A	f(B)	f(C)	sum	value
0	1	4	4	9	2
1	1	3	1	5	3
2	1	2	2	5	3
3	1	1	3	5	3
...	...	...	...	...	...

With x from 0 to 48 we have 48/4*3 times 3 and 48/4+1 times 2.
For the second grouping we can have 40 or 44 stalks, then the groups will be either A[1] B[39-x] C[x] or A[1] B[43-x] C[x], since (43 mod 4) = (39 mod 4) = 3 we will have:

x	A	f(B)	f(C)	sum	value
0	1	3	4	8	2
1	1	2	1	4	3
2	1	1	2	4	3
3	1	4	3	8	2
...	...	...	...	...	...

So we will have 44/2 times 2 and 44/2 times 3 or 40/2 times 2 and 40/2 times 3. The same holds for the third grouping when we will have either 40, 36 or 32 stalks. 

To show evidence of this calculations I've prepared an Excel spreadsheet with all possible case. If you have MS Excel 2000 you can save the html file on your local drive and open as a spreadsheet.

The probability for each outcome is:

2+2+2 13/49 * 1/2 * 1/2 = 0.0663 6 2+2+3 13/49 * 1/2 * 1/2 = 0.0663 7 2+3+2 13/49 * 1/2 * 1/2 = 0.0663 7 2+3+3 13/49 * 1/2 * 1/2 = 0.0663 8 3+2+2 36/49 * 1/2 * 1/2 = 0.1836 7 3+2+3 36/49 * 1/2 * 1/2 = 0.1836 8 3+3+2 36/49 * 1/2 * 1/2 = 0.1836 8 3+3+3 36/49 * 1/2 * 1/2 = 0.1836 9

This means:    Prob( ) = 13/196 = 0.0663    Prob( ) = 85/196 = 0.4336    Prob( ) = 62/196 = 0.3163    Prob( ) = 36/196 = 0.1836    Prob( or ) = 98/196 = 0.5    Prob( or ) = 98/196 = 0.5      Prob( -> ) = 13/98 = 0.1326    Prob( -> ) = 36/98 = 0.3673

Note: Another analisys of the Yarrow stalks method was provided by Ralph Abrahm on his site http://www.ralph-abraham.org. In his analisys for the first subdivision he considers each outcome (1+4+4 1+3+1, 1+2+2 and 1+1+3) as equally probable. Actually there exists 49 possible subdivisions and the 49th is a 2 (1+4+4). This is why our results differ.

You may find another anlysis of yarrow stalks odds in the paper describing the Xiao Guo /Jie method. The differences are due the fact that the author does not consider valid the group [1,0,48] (but he takes into account the group [1,48,0]) and this means you have only 11/48 chance of having 2 instead of 13/49. The same goes for the other odds.