## Chinese Iching Hexagrams on Chou Bronzes: TCL Example

### Introduction

gold Here is some TCL starter code for Chinese Iching Hexagrams on Chou Bronzes. The Iching is the ancient fortune telling book of China. The Iching literature mentions various methods for casting fortune patterns.

The canvas and buttons presented here are largely based on the Simple Canvas Demo by HJG. The random pick algorithm was posted by Suchenworth RS on Horseracing in Tcl. Other dice pages are Throwing Two Dice GWM andDice by Keith Vetter. Several Dr. Math emails are in the references, which reduce some dice issues to simple paths.

In the West, Leibniz recognized that the whole and broken lines in the hexagram patterns could be counted as ones and zeros, respectively in a binary number system. For example, a cast or list of { 6,7,8,9,6,7 } could be expressed as a binary number, 010101. For computer procedures, the patterns are effectively a horizontal bar code of 6 lines. The probability of each line by the yarrow stick method or coin casting method is different.

• The probability of the yarrow stick method is 1,3,5,7 over 16.
• The number 6 or old yin is 1/16 or 0.0625
• The number 7 or young yang is 3/16 or 0.1875.
• The number 8 or young yin is 5/16 or 0.3125.
• The number 9 or old yang is 7/16 or .4375.
• For the yarrow sticks, the combined probability of any throw is .0625 + .1875 + .3125 + .4375 = 1
• For the yarrow stick method, the total combinations are 64 hexagrams, space of 64 items.

For the iching dice pictured below, the three sticks read 2,3,and 3 which totals 8. The combined probability of casting 2,3, and 3 is 0.3125 (with the sticks).

For the coins, the probability of the coin casting method is 2,2,6,6 over 16. The number 6 or old yin is 2/6 or 0.125.

• The number 6 or old yin is 2/16 or 0.125.
• The number 7 or young yang is 2/16 or 0.125.
• The number 8 or young yin is 6/16 or 0.375.
• The number 9 or old yang is 6/16 or .375.
• For the coins, the combined probability of any throw is .125 + .125 + .375 + .375 = 1
• For the coin divination method, the total combinations are 64 hexagrams, space of 64 items.

Note on 13Feb2024.

4 yes-no binary sticks equals 2**4 combinations or 16 combinations. Probability of single cast 1/(2**4) or 1/16. Space = 16 items

5 yes-no binary sticks equals 2**5 combinations or 32 combinations. Probability of single cast 1/(2**5) or 1/32. Space = 32 items

6 yes-no binary sticks equals 2**6 combinations or 64 combinations. Probability of single cast 1/(2**6) or 1/64. Space = 64 items

7 yes-no binary sticks equals 2**7 combinations or 128 combinations. Probability of single cast 1/(2**7) or 1/128. Space = 128 items

Chinese excavations reveal a possible related divination system with space of 60 combinations.

There are current Buddhist temple methods that picking a stick from 64 sticks or some other number. Popular Game of Pick up Sticks in West. In that case, the classic formula would be 1/64 for equal Hexagram probability and the probability space would 64 items.

```Bernoulli probability formula, P(Yang)= 1- P(Yin)

The three most frequently occurring characters are:

Thinking in Bullet Points

There are a total of 384 lines in the I Ching Oracle Book
There are roughly 100 lines with character "mistake", Chinese "Liu"
100/384 =  0.2604       probability of mistake Liu appearing
(384-100)= 284          number of lines without mistake Liu  appearing
284/384= 0.7396        probability of mistake Liu not appearing
The probability of casting a line with the character "mistake" is 0.2604,   roughly 1/4 or 0.25
The probability of casting a line with no character "mistake" is 0.7396,   roughly 3/4 or 0.75
The probability of casting a line with or without  character "mistake" is 0.2604 + 0.7396 = 1
One might conclude the I Ching Oracle Book is rigged not to cast a character "mistake". ```

On about 28 pieces of Chou bronzes, there were several series of numbers possibly representing divinatory numbers. These series were described in a thesis by Chang Cheng Land, called Interpretation of Divinatory Inscriptions on Early Chou Bronzes. The numbers in ancient Chou script are not completely understood, but have the following number space { 1:36,2:0,3:0,4:0,5:11,6:64,7:33,8:24}. The puzzle is that ones and fives are not normally generated in the modern I Ching practice, meaning after 1100 AD. Part of the problem is that the Chinese scripts for one, two, three, and four were normally horizontal slashes in those times. So numbers 2,3,and 4 could be confused with marks for the number one. Also some of the numbers seem to be archaic or oracle bone script versions of modern Chinese numbers, like X for 5.

It seems reasonable that a TCL procedure might be able to reproduce the same generating odds with dice or coins, even though the Chou generating procedures are lost but for a few peg points.

## Pseudocode preparations for generating comparable sequences to Chou bronzes

Here are the peg points.

1. One generating method involves 36 counters, lacking four. Apparently the counters are taken/divided away by four and the remainders laid in three piles, upper on side, middle upright, and lower on side.
2. The Chou regarded the odd numbers as "lucky" yang and the even numbers as "unlucky" yin.

From the probability of 6-sided double dice, the number space of the bronze method guesstimates at 23. The bronze method sample space is likely between the sample space of 4 coins (sp16) and 5 coins (sp32).

The probabilities from 4 2-sided sticks are a 40 to 60 percent off the estimated bronze figures. But the number 6 has highest frequency in both sticks and bronzes. In the 5 2-sided sticks, its number 7 with the highest frequency.

Kind of hokey math, but if you assess 6 has 6 combos, then x= 6/.264 or 22.72 eg. nearest whole number space is 23.

```    Pseudocode.  llength { 6 6 6 6 6 6 etc } = 23
1's =    0.1487   * 23  =3.42   > 4
5's = .0454 *23 or 1.04  > 2
6's  =.264*23 or 6.02  > 6
7's  = .136*23 or 3.128 > 4
8's = .0625 *23 or  1.43    > 2
Pseudocode.  llength { 5 5 6 6 6 6 6 6 7 7 7 7 8 8 etc } = 23

Pseudocode.  llength { 5 5 6 6 6 6 6 6 7 7 7 7 8 8 1 1 1 1 etc. } = 23 ```

Since 1's in bronzes are poorly understood, don't mind filling up rest ( to 23) with ones. So now the Suchenworth random pick procedure lpick will deliever the same frequency of characters on the bronzes.

`    Pseudocode.  lpick { 5 5 6 6 6 6 6 6 7 7 7 7 8 8 1 1 1 1 1 1 1 1 1  } `

With a text translation scheme and suitable unicode, the TCL procedure can mimic the appearence of Chou horizontal script on the bronzes. With lpick generation, an eTCL program can throw a sequence on a graphics screen and even swap from Chou-like script in unicode to Arabic numbers.

Note 2/5/2024 The Yarrow Stalk Method uses 50 straws or counters to generate the 64 hexagrams of the I Ching Oracle Book. Using proportions as the Yarrow Stalk Method to a 32 counter method, the proportions would be 50 straws over 64 Hexagrams to X straws over 32 Hexagrams. Solving for X, X = ( 32 * 50 ) / 64 or 25 straws. Making assumptions here, but this might be the number space for the 32 counter method is 25 units. Lien-shan and Kuei-tsang from the Chou era are numerology based oracles. Tentative: The Lien-Shan is reported to have used 32 counters to generate a number sequence. ling ch'i ching?

## Probability Calculations table

space of 4 2-sided sticks marked 1 and 2

space was { 4 5 5 6 5 6 5 7 5 6 6 7 6 7 7 8},size of 16

sticks prob. prob. est. bronze prob. 5 sticks prob.
4 1/16 0.04625 unk
5 5/16 0.3125 0.0454545454545
6 5/16 0.3125 0.264462809 .1875
7 4/16 0.250 0.13636363636 .3125
8 1/16 0.0625 0.0991735537 .2815

5 stick prob's are multiples/fractions of bronze values

Note on 13Feb2024. 4 yes-no binary sticks equals 2**4 combinations or 16 combinations. Probability of single cast 1/(2**4) or 1/16.

5 yes-no binary sticks equals 2**5 combinations or 32 combinations. Probability of single cast 1/(2**5) or 1/32.

6 yes-no binary sticks equals 2**6 combinations or 64 combinations. Probability of single cast 1/(2**6) or 1/64.

7 yes-no binary sticks equals 2**7 combinations or 128 combinations. Probability of single cast 1/(2**7) or 1/128.

Chinese excavations reveal a possible related divination system with space of 60 combinations.

## Conclusions

After checking, an lpick(sequence2) was found that better matched the reported bronze values. Here, we are at most 10 to 15 percent off. lpick { 5 6 6 6 6 6 6 7 7 7 8 8 1 1 1 1 1 1 1 1 1 1} #s2 The extra program at the bottom of the page will solve for the probabilities in the bronze sequence (or any sequence loaded in the \$lister).

gold Update 3/3/2024. Regarding the numbered implications or conclusions in the I Ching Oracle Book, it is essential to understand that the interpretations from I Ching Oracle Book are not based solely on probability theory. Instead, interpretations are largely based on understanding the symbolism and patterns within the hexagrams and their associated texts. While the specific sum of each cast may be used in Chinese diagnostics for the 64 diagrams in the I Ching Oracle Book, the focus is on understanding the meaning and implications of the hexagrams rather than calculating probabilities.

est. bronze prob. lpick(sequence2)
1 .4347 0.45454545454545453
2 n/a 0.0
3 n/a 0.0
4 n/a 0.0
5 0.0454545454545 0.045454545454545456
6 0.264462809 0.2727272727272727
7 0.13636363636 0.13636363636363635
8 0.0991735537 0.09090909090909091

## appendix TCL programs

### Pretty Print VERSION

```        # test of autoindent from ased editor
# written on Windows XP on eTCL
# Graphics program to mimic Iching numbers on Chou bronzes
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI , 17Jul2010
package require Tk
proc lpick L {lindex \$L [expr int(rand()*[llength \$L])];}
set lister { 5  6 6 6 6 6 6 7 7 7  8 8 1 1 1 1 1 1 1 1 1 1 1}
proc transnumber {lister} {
set choutext [list ]
foreach item \$lister {
if { \$item == 1 } {lappend choutext \u4E00}
if { \$item == 5 } {lappend choutext \u3024}
if { \$item == 6 } {lappend choutext \u516D}
if { \$item == 7 } {lappend choutext \u4E03}
if { \$item == 8 } {lappend choutext \u516B}
if { \$item >= 9 } {lappend choutext ? }
}
return \$choutext}
proc transarabic {lister} {
set choutext [list ]
foreach item \$lister {
if { \$item == 1 } {lappend choutext 1}
if { \$item == 5 } {lappend choutext 5}
if { \$item == 6 } {lappend choutext 6}
if { \$item == 7 } {lappend choutext 7}
if { \$item == 8 } {lappend choutext 8}
if { \$item >= 9 } {lappend choutext ? }
}
return \$choutext}
proc transodd {lister} {
set choutext [list ]
foreach item \$lister {
if { \$item == 1 } {lappend choutext \u4e00 }
if { \$item == 5 } {lappend choutext \u4e00 }
if { \$item == 6 } {lappend choutext "- -"}
if { \$item == 7 } {lappend choutext \u4e00 }
if { \$item == 8 } {lappend choutext "- -"}
if { \$item >= 9 } {lappend choutext ? }
}
return \$choutext}
set listx [ transnumber \$lister ]
proc chou_bronzes  {w state3} {
global listx lister
if { \$state3 == 1 } {set listx [ transnumber \$lister ]}
if { \$state3 == 2 } {set listx [ transarabic \$lister ]}
if { \$state3 == 3 } {set listx [ transodd \$lister ]}
set yy 100
set font3 {Hevetica 35}
set font5 {Hevetica 150}
set lister { 5 5 6 6 6 6 6 6 7 7 7 7 8 8 1 1 1 1 1 1 1 1 1 }
\$w create rect 150 50 250 400 -fill gold
\$w create poly 445 60 475 65 475 100 445 120 -fill tan
# \$w create text 200 400 -text u2617  -tags "text" -fill tan -font \$font5
foreach i {1 2 3 4 5 6 } {
\$w create text 200 \$yy -text [lpick \$listx ]  -tags "text"  -font \$font3
incr yy 50
}
}
proc clrcanvas {w} {
\$w delete  "all"
}
proc leave {w} {
exit
}
proc board {w} {
#set state3 1
#set state2 1
clrcanvas \$w
. configure -background orange -highlightcolor brown -relief raised -border 30
\$w configure -bg tan
}
#: Main :
frame .f1
frame .f2
pack  .f1 .f2
set maxX 400
set maxY 450
set y      0
set x1 120
set x2 150
set y1  50
set y2  80
canvas  .cv -width \$maxX -height \$maxY  -bg tan
pack    .cv -in .f1
button  .b0 -text "clear" -command { clrcanvas .cv }
button  .b5 -text "chou_bronzes"   -command {clrcanvas .cv; chou_bronzes   .cv 1 }
button  .b6 -text "arabic"   -command { clrcanvas .cv; chou_bronzes   .cv 2 }
button  .b7 -text "regular"   -command { clrcanvas .cv; chou_bronzes   .cv 3}
button  .b8 -text "exit"   -command {leave   .cv }
pack .b0  .b5  .b6 .b7 .b7 .b8 -in .f2  -side left -padx 2
board   .cv
#catch {console show}
wm title . " Chinese Iching Hexagrams on Chou Bronzes "```

### Extra Program for console

```        # written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# program for eTCL console
# examine sequences, prob. subroutines for mimic sequence of bronze
# gold on TCL WIKI , 17Jul2010
console show
proc calculation {  facen }  {
# prob. subroutines for mimic sequence of bronze
# prob. is throw combos of eg. "7" over all possible throws
set lister  {5  6 6 6 6 6 6 7 7 7  8 8 1 1 1 1 1 1 1 1 1 1}
set ee [llength  \$lister ]
set kk [ llength [ lsearch -all \$lister \$facen ] ]
set prob [ expr { (\$kk*1.) / \$ee  } ]
return \$prob
}
set limit 12
for { set i 1 } { \$i <= \$limit }  { incr i } {
lappend listxxx \$i
lappend listxxx [ calculation  \$i ]
puts " \$i [ calculation  \$i ] "
}
#end
output.
1 0.45454545454545453
2 0.0
3 0.0
4 0.0
5 0.045454545454545456
6 0.2727272727272727
7 0.13636363636363635
8 0.09090909090909091
9 0.0
10 0.0
11 0.0
12 0.0
#end```

Aside from the hexagram analysis, this TCL script can be a workhorse for all kinds of problems just by loading a different list of numbers to examine. Suppose you were given a loaded dice and loaded about 30 throws in the list. For an honest 6-sided die, the probability of each throw { sequence 1 2 3 4 5 6} should be equal over a large enough set of tests. The exact probability for each throw would be 1/6 or 1.6666 . A loaded die will give percentages that are not equal for all numbers or different probabilities than the honest die.

Another problem type is the continued mantissas like pi. Are the numbers that show up in the pi mantissa equally probable? If the mantissa of pi was split by TCL into a list of 0-9 integers, this theorm could be checked by our script. Lets try the handy AMG pi procedure from One Liners

```  1% proc pi {} {expr acos(-1)}
2% proc mac {aa} { regexp {.(\d+)} \$aa -> bb;return \$bb }
3% mac [pi]
141592653589793

Now we have a short mantissa for testing and there are websites that will give pi with more digits.

set sequence3 [ split {14159265358979323846264338327950288419716939937} "" ]

2% split {14159265358979323846264338327950288419716939937} ""
1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7
# Sequence for combos of 2 6-sided dice. sample space is 36.
{3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12}
sequence of one 6-sided dice, turned in numberic order.
{ 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6}
honest die
{2 5 1 6 4 3 5 3 2 1 5 2 1 3 6 1 6 1 3 1 4 1 4 3 5 2 4 1 6 6}
loaded die in favor of 5
{5 6 1 6 3 5 5 4 6 2 5 4 6 6 4 3 4 3 1 1 2 5 2 1 2 5 1 4 3 5}

sequence for 2-sided coin, ready for shuffle.1 for  head, 2 for tail

{1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1}
honest coin , 1 head, 2 for tail
{1 2 2 2 1 1 2 1 1 1 2 1 2 1 2 1 2 1 2 1 2 2 1 1 1 2 2 1 1 2 2 1 2}
2 1 2 1 1 1 2 2 2 1 2 2 1 1 1 1 2 2 1 2 1 1 2 1 1 2 1 1 1 1 1 2 1
another dishonest coin , loaded for 2 (tails),
1 for  heads, 2 for tails
{1 2 2 1 2 1 1 1 2 2 2 2 1 1 2 2 1 2 2 1 1 2 1 1 2 2 2 2 1 1 2 2 2}
combos for three  2-sided stick dice, marked 1 and 2
sample space is 8.
{3 4 4 5 4 5 5 6}
combos for four 2-sided stick dice, marked 1 and 2
sample space is 16.
{4 5 5 6  5 6 6 5 7 5 6 6 7 6 7 7 8}
combos for five 2-sided stick dice, marked 1 and 2
sample space is 32.(use split here)
{ 56676768677877889677878797889899 10}```

### Strategy from the I Ching Oracle Book may be used in prospective role game

Received feedback on project, not an endorsement or judgement on every point and idea discussed.

The I Ching Oracle Book, also known as the Yi Jing or Book of Changes, is an ancient Chinese divination text that has been used for centuries to provide guidance and insights. The text consists of 64 hexagrams, each composed of six broken or unbroken lines. The lines can be interpreted in various ways, and the text provides explanations and interpretations for each hexagram. In one - proposed method of using three 4-sided dice, each marked differently, the sum of the six casts would indeed create a specific hexagram.

The observations on the historical agricultural practices and taxation systems in ancient Sumer and Israel are quite intriguing. It's fascinating to see how different societies dealt with agricultural stress and resource management.

Regarding the suggestion and strategy for the Game Kingdom of I Ching Oracle Book, it might indeed be an interesting idea to incorporate a tax window or input window for fallow field ratios. This would allow players to experiment with different taxation levels and observe the impact on their grain supply. It could also provide a more realistic representation of the challenges faced by ancient societies in managing their resources.

To implement this feature, one could consider the following steps:

1. Research and gather more information on the historical agricultural practices and taxation systems in ancient Sumer and Israel. This will help one create more accurate and engaging scenarios for the game.

2. Develop a user interface that allows players to adjust the taxation levels and observe the impact on their grain supply. This could include a slider or dropdown menu for selecting different fallow field ratios and tax rates.

3. Design a series of game scenarios that reflect the historical challenges faced by ancient societies. For example, one could create a scenario where the player must deal with a drought or a flood, and adjust their taxation policies accordingly to maintain their grain supply.

4. Test and refine the game scenarios based on player feedback. This will help you identify any issues or areas for improvement, and ensure that the game remains engaging and challenging for players.

5. By incorporating historical agricultural practices and taxation systems into the Game Kingdom of I Ching Oracle Book, one can create a more immersive and educational experience for players, while also highlighting the importance of resource management in different societies throughout history.

5. By allowing players to adjust the taxation levels and observe the impact on their grain supply, one can create a more interactive and immersive gaming experience. To implement this feature, one could consider the following steps: Research and gather more information on the historical agricultural practices and taxation systems in ancient Sumer and Israel. This will help one create more accurate and engaging scenarios for the game.

6. Develop a user interface that allows players to adjust the taxation levels and observe the impact on their grain supply. This could include a slider or dropdown menu for selecting different fallow field ratios and tax rates.

7. Design a series of game scenarios that reflect the historical challenges faced by ancient societies. For example, one could create a scenario where the player must deal with a drought or a flood, and adjust their taxation policies accordingly to maintain their grain supply.

8. Test and refine the game scenarios based on player feedback. This will help one identify any issues or areas for improvement, and ensure that the game remains engaging and challenging for players.

By incorporating the strategy from the I Ching Oracle Book , one can create a more immersive and educational experience for players, while also highlighting the importance of resource management in different societies throughout history.

Regarding the numbered implications or conclusions in the I Ching Oracle Book, it is essential to understand that the text's interpretations are not based on probability theory. Instead, they are based on understanding the symbolism and patterns within the hexagrams and their associated texts. While the specific sum of each cast may be used in Chinese diagnostics for the 64 diagrams in the I Ching Oracle, the focus is on understanding the meaning and implications of the hexagrams rather than calculating probabilities.

1. Chou Bronzes Divinatory Numbers

On 28 Chou bronzes, divinatory numbers were found, detailed in Chang Cheng Land's thesis. The ancient Chou script numbers in probability space {1:36, 2:0, 3:0, 4:0, 5:11, 6:64, 7:33, 8:24} remain partially understood.

2. Modern I Ching Practice

After 1100 AD, ones and fives were not typically generated or reported in the Hexagrams of the I Ching practice. The horizontal slashes used for numbers 1-4 could be confused, with some numbers resembling archaic Chinese Bone scripts.

3. Reproducing Chou Bronze Generating Odds

Using a TCL procedure, one could potentially replicate the Chou Bronze generating odds with dice or coins. Despite lost generating procedures, the script variations may be accounted in a different scheme.

End of Rail Road RR track =======/