**General Tianji'  Horse Race Strategy from China and eTCL demo example calculator, numerical analysis**


This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER  in your comment with the same courtesy that I will give you. Its very hard to reply intelligibly without some background of the correspondent. Thanks,[gold]
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<<TOC>>

**Introduction**

[gold] Here is some eTCL starter  code for General Tianji'  Horse Race Strategy from China in calculator shell. Additional console programs below are used to check or improve subroutines in calculator shell. 

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Both the King and General Tianji  have sets of 3 horses. Each set has low grade horse, middle grade horse and best grade horse. General Sun Bin advised that the General Tianji run the General's  low grade horse against the king's best grade horse, the General's  middle grade horse against the king's low grade horse, and the General's  best grade horse against  the king's middle grade horse. The result was that all the General Tianji 's horses won that day circa  340 BCE. Each round of the horse race won 200 silver dollars. General Tianji  and Sun Bin's share was two wins and a bag of silver, worth expr(2*$200) or 400 silver dollars. When this bit of strategy was explained later to the king, the king promoted General Tianji to chief of all armies and promoted Sun Bin to the military strategy advisor of General Tianji. Not sure an elegant proof can be drummed up, but there are many random horses  in eTCL. The Tianji racehorse problem can be loaded into the  eTCL calculator shell and each horse of three can be simulated with a random horse race probability model. The eTCL calculator from the Random Walk Equation offers some analogies.
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The tale of the General Tianji and  General Sun Bin ( died 316 BCE) was found in the Chinese history book, Shujing. The Shujing history book was literally saved from the flames of the book burning first empire. I have a rare English translation of the Shichih (sic) commentary in my library. Here the particular issue was how the grades of horses were defined in ancient China. The analysis assumed the best grade horse won 50 percent of his races, the middle grade won 35 percent, and the low grade won 15 percent. The eTCL calculator has entries for the grade probabilities to be changed so the effects or rewards of better or lesser random models can be studied.

The Tainji horserace as a computer contest problem is featured in ACM student contests with an average of 6000 Chinese entries ( computer programs ). Most of the entries featured bipartite matching algorithms. For the eTCL calculator, the calculator is returning the results of random models so the random results are not always the same with each solve. Over a number of cases, the cumulative or average results should approach the results or winnings of General Tianji.
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**Pseudocode Section**
======
            # using pseudocode
        proc test1 {   
               # possible problem instances,
               low vs high
               middle vs low
               high vs middle}
        proc test2 {
            initialize algorithm_result = 1.
            initialize terms of tianji combined probability 
        proc test3 {
            combined probability of winning as tianji_win_prob
            tianji_win_prob has terms of unequal probability
        proc test4 {
            use llength(lsearch)/llength  winner list
            to calculate probilities of
            1) prob_1(tianji_low vs king_high)     horses
            2) prob_2(tianji_middle vs king_low)   horses
            3) prob_3(tianji_high vs king_middle)} horses
        proc test5 {
            prob_1 = tianji high probability   * [expr { rand() } ]
            prob_2 = tianji medium probability * [expr { rand() } ]
            prob_3 = tianji low probability    * [expr { rand() } ]
            would need ~100 trials for answer to gel, in iterative solution }
        proc test6 {
             check algorithm
            f(solution) =? initial product
            check_sum = a+b+c+d+e
            set answers and printout with resulting values
            pseudocode: need test cases > small,medium, giant
            pseudocode: need testcases within range of expected operation.
            pseudocode: are there any cases too small or large to be solved?
            pseudocode: limit to size of eTCL list,   ref  versions.   }
                #end of file
======
***Testcases Section***  

In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements in the TCL console. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the   capacity  functions (subroutines). 

**** Testcase 1 ****
%|table 1 |printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 1:|testcase_number | |& 
&| 50.0 :|high grade horse probability percent (a) , usually 50  |   |&
&| 35.0 :|medium grade horse probability percent (b) , usually 35 | |& 
&| 15.0 :|low grade horse probability percent (c) , usually 15 | |& 
&| 1.0 :|optional:| |&
&| 1.0 :|optional: | |&
&| 1.0 :|optional: |  |&
&| 3.0 :|optional: number Tianji wins,  0 to max 3 | random results  |&
&| 600.0 :|answer: total winnings silver dollars | random results |&

**** Testcase 2 ****
%|table 1 |printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 2 :|testcase_number | |& 
&| 50.0 :|high grade horse probability percent (a) , usually 50  |   |&
&| 30.0 :|medium grade horse probability percent (b) , usually 35 | |& 
&| 20.0 :|low grade horse probability percent (c) , usually 15 | |& 
&| 1.0 :|optional:| |&
&| 1.0 :|optional: | |&
&| 1.0 :|optional: |  |&
&| 1.0 :|optional: number Tianji wins,  0 to max 3 | random results |&
&| 200.0 :|answer: total winnings silver dollars | random results  |&
 
**** Testcase 3 ****
%|table 3 |printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 3 :|testcase_number | |& 
&| 50.0 :|high grade horse probability percent (a) , usually 50  |   |&
&| 40.0 :|medium grade horse probability percent (b) , usually 35 | |& 
&| 10.0 :|low grade horse probability percent (c) , usually 15 | |& 
&| 1.0 :|optional:| |&
&| 1.0 :|optional: | |&
&| 1.0 :|optional: |  |&
&| 1.0 :|optional: number Tianji wins,  0 to max 3 | random results  |&
&| 200.0 :|answer: total winnings silver dollars | random results |&

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***Screenshots Section***
****figure 1.****
[General Tianji' Horse Race Strategy from China and eTCL demo example screenshot]
****figure 2.****
[General Tianji' Horse Race Strategy from China and eTCL demo wins of high grade horse]
****figure 3.****
[General Tianji' Horse Race Strategy from China and eTCL demo high grade horse probability]
---- 
***References:***
   * [Horseracing in Tcl] by [RS]
   * Plugging a Math Gap with Harvard mathematician Shing-Tung Yau, 
   * Forbes Magazine 2007, Institute of Mathematical Sciences 
   * in Hong Kong studies betting on horse races and predicting the weather.
   * Tian Ji , The Horse Racing, Peking University, Ying Fuchen,Xu Pengcheng,Xie Di, ACM Programmer's Contest
   * The Ambitious Horse, Ancient Chinese Mathematics Problems, Lawence W. Swienciki, Ph.D. 
   * Game-theoreticanalysis of anancient Chinese horse race problem,  Mingming Leng, Mahmut Parlar,
   * DeGroote School of Business, McMaster University, Hamilton, Ont., Canada L8S 4M4
   * 10 December 2004
   * [Random Walk Equation Slot Calculator Example] 
   * Plugging a Math Gap with Harvard mathematician Shing-Tung Yau, Forbes Magazine 2007,
   * Institute of Mathematical Sciences 
   * in Hong Kong studies betting on horse races and predicting the weather
   * [Sumerian Construction Rates and eTCL Slot Calculator Demo Example]
   * Chabert, J. "Methods of False Position." Ch. 3 in A History of Algorithms: From the Pebble to the Microchip. , 1999. 
   * John Hannah , False position in Leonardo of Pisa's Liber Abbaci,Historia Mathematica Volume 34 issue 3 2007 
   * Eugene C. Boman,False Position, Double False Position and Cramer's Rule, College Mathematics Journal
   * Howard Eves ,On the practicality of the rule of false position, Mathematics Teacher Volume 51 
   * Joy B. Easton, the rule of double false position, Mathematics Teacher Volume 60 
   * Vera Sanford, the rule of false position, Mathematics Teacher Volume 44 issue 5 1951 
   * Karine Chemla, Reflections on the World-wide History of the Rule of False Double Position, or: How a Loop Was Closed 
   * Thomas G. Edwards -- Using the Ancient Method of False Position to Find Solutions, Mathematics Teaching in the Middle School Volume 14 issue 4 2008 
   * [Oneliner's Pie in the Sky] 
   * [One Liners] 
   * [Category Algorithm]
   * [Babylonian Number Series and eTCL demo example calculator]
   * [Brahmagupta Area of Cyclic Quadrilateral and eTCL demo example calculator]
   * [Gauss Approximate Number of Primes and eTCL demo example calculator]
   * Taking a Weighted Average, Dr Math, mathforum.org/dr.math, dated 03/20/2003 
   * google <Weighted Average, Dr Math>, mathforum.org/dr.math
   * Using Weighted Criteria to Make Decisions, Dr Math>, mathforum.org/dr.math, dated 07/18/2008 
   * google <Weighted sum product model Decision >
   * harmonic mean discussed in [Additional math functions]
   * harmonic mean discussed in [Tcl Normalized Extended Benchmarks]
   * see  arithmetic mean on tcl-wiki [Stats] 2011-05-22, [RLE]
   * see similar 1/(1/nth) terms method used in 
   * [Babylonian Combined Market Rates and eTCL demo example calculator, numerical analysis ] 
   * Zhang Baichun and Jürgen Renn (eds.) , Transformation and Transmission:
   * Chinese Mechanical Knowledge and the Jesuit Intervention 
   * Joint Research Group of the MPIWG and its Partner Group at the IHNS of the CAS.
   * Probability models on horse-race outcomes,
   * MUKHTAR M. ALI, Department of Economics, University of Kentucky, USA
   * Probability and Optimization Probability and Optimization Models for Racing Horse Models, 
   * Victor S. Y. Lo University of British Columbia University of British Columbia, Fidelity Investments
   * Designing competitions between teams of individuals, Pingzhong Tanga,Yoav Shohamb,c, Fangzhen Lina,
   * Mentions ACM horse race problems relative to teams of computer students.
   * Sun Bin: Distinguished Military Strategist After Sun Zi,  By David Wu, Epoch Times
   * Lecture 4: Matching Algorithms for Bipartite Graphs
   * Professor: Clifford Stein
   * Lecture notes on bipartite matching
   * 100: “Matchings” Fall 2012
   * Matching (graph theory) From Wikipedia, the free encyclopedia
   * Bayes Theorem,Bayes Theorem in short is a method of analysing data. 
   * Massachusetts Institute of Technology Handout 3 
   * Combinatorial Optimization February 9th, 2009 Michel X. Goemans



**Appendix Code**

***appendix TCL programs and scripts ***

======
        # pretty print from autoindent and ased editor
        # General Tianji' Chinese Horse Race Strategy Calculator
        # written on Windows XP on eTCL
        # working under TCL version 8.5.6 and 1.0.1
        # gold on TCL WIKI, 25feb2017
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        frame .frame -relief flat -bg aquamarine4
        pack .frame -side top -fill y -anchor center
        set names {{} { high grade horse probability percent (a) , usually 50 :} }
        lappend names { medium grade horse probability percent (b) , usually 35:}
        lappend names { low grade horse probability percent (c) , usually 15: }
        lappend names { optional: : }
        lappend names { answers:  check sum all wins =? 3  }
        lappend names { number of King wins, 0 to max 3  }
        lappend names { number of Tianji wins, 0 to max 3 : }
        lappend names { total winnings for Tianji silver dollars :} 
        foreach i {1 2 3 4 5 6 7 8} {
            label .frame.label$i -text [lindex $names $i] -anchor e
            entry .frame.entry$i -width 35 -textvariable side$i
            grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 }
        proc about {} {
            set msg "Calculator for General Tianji's Chinese Horse Race Strategy
            from TCL WIKI,
            written on eTCL "
            tk_messageBox -title "About" -message $msg } 
        proc tianji_horse_race     { xxx1  }   {
            global side1 side2 side3 side4 side5
            global side6 side7 side8 
            set high_grade_prob 50.
            set medium_grade_prob 35.
            set low_grade_prob 15.
            set  tianji_winnings 0.
            set  king_winnings 0.
            # normal 15 percent = low_grade_prob
            # normal 35 percent = medium_grade_prob
            # normal 50 percent = high_grade_prob
            set high_grade_prob $side1
            set medium_grade_prob $side2
            set low_grade_prob $side3
            set tianji_low  [expr $low_grade_prob * [expr { rand() } ] ]
            set king_high [expr $high_grade_prob * [expr { rand() } ] ] 
            set tianji_low 1.0 
            set tianji_medium 1.0 
            set tianji_high 1.0
            set king_low 1.0 
            set king_medium 1.0 
            set king_high 1.0
            set check_sum 0.0
            # 15 percent = low_grade
            # 35 percent = medium_grade
            # 50 percent = high_grade
            set tianji_low [expr $low_grade_prob * [expr { rand() } ] ]
            set king_high [expr $high_grade_prob * [expr { rand() } ] ]  
            if {    [max  $tianji_low $king_high  ] == $tianji_low   } { set tianji_winnings  1. }
            if {    [max  $tianji_low $king_high  ] == $king_high  } { set king_winnings  1.  } 
            set tianji_medium [expr $medium_grade_prob * [expr { rand() } ] ]
            set king_low [expr $low_grade_prob * [expr { rand() } ] ]  
            if {    [max  $tianji_medium $king_low  ] == $tianji_medium   } { set tianji_winnings [+ $tianji_winnings 1.]  }
            if {    [max  $tianji_medium $king_low  ] == $king_low  } { set king_winnings [+ $king_winnings 1. ]   } 
            set tianji_high [expr $high_grade_prob * [expr { rand() } ] ]
            set king_medium [expr $medium_grade_prob * [expr { rand() } ] ]  
            if {    [max  $tianji_high $king_medium  ] == $tianji_high   } { set tianji_winnings [+ $tianji_winnings 1.] }
            if {    [max  $tianji_high $king_medium  ] == $king_medium  } {  set king_winnings [+ $king_winnings 1. ] }
            set check_sum [+ $king_winnings $tianji_winnings ]
            set side5 $check_sum
            set side6 $king_winnings
            return $tianji_winnings }
       proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8 
            global testcase_number 
            incr testcase_number 
            set side1 [* $side1 1. ]
            set side2 [* $side2 1. ]
            set side3 [* $side3 1. ]
            set side4 [* $side4 1. ]
            set side5 [* $side5 1. ]
            set side6 [* $side6 1. ]
            set side7 [* $side7 1. ]
            set side8 [* $side8 1. ] 
            # initialize placeholder answer
            set answer2 [tianji_horse_race  3 ]
            set side7 $answer2 
            set side8 [* 200. $answer2 ]
                    }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$aa"
            .frame.entry2 insert 0 "$bb"
            .frame.entry3 insert 0 "$cc"
            .frame.entry4 insert 0 "$dd"
            .frame.entry5 insert 0 "$ee"
            .frame.entry6 insert 0 "$ff" 
            .frame.entry7 insert 0 "$gg"
            .frame.entry8 insert 0 "$hh" 
             }
        proc clearx {} {
            foreach i {1 2 3 4 5 6 7 8 } {
                .frame.entry$i delete 0 end } }
        proc reportx {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number 
            console show;
            puts "%|table $testcase_number|printed in| tcl wiki format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number | |& "
            puts "&| $side1 :|high grade horse probability percent (a) , usually 50  |   |&"
            puts "&| $side2 :|medium grade horse probability percent (b) , usually 35 | |& "  
            puts "&| $side3 :|low grade horse probability percent (c) , usually 15 | |& "
            puts "&| $side4 :|optional:| |&"
            puts "&| $side5 :|answers: check sum all wins =? 3 : | |& "
            puts "&| $side6 :|number king wins,  0 to max 3 | random results |& " 
            puts "&| $side7 :|number Tianji wins,  0 to max 3 | random results |& "
            puts "&| $side8 :|total winnings for Tianji silver dollars | random results |&" 
            }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 50.  35.  15.  1.0  0.  0.   0.  0.}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 50.  30.  20.  1.0  0.  0.   0.  0. }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 50.  40.  10.  1.0  0.  0.   0.  0. }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .cons -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .cons .about .exit .test4 .test3 .test2   -side bottom -in .buttons
        grid .frame .buttons -sticky ns -pady {0 10}
               . configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
        wm title . "General Tianji' Chinese Horse Race Strategy Calculator"  
           # end of deck 
======
----
*** Pushbutton Operation***

For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console. 

For  testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared on the next solve button.   The command { calculate; reportx  } or { calculate ; reportx; clearx  } can be added or changed to report automatically. Another wrinkle would be to print out the current text,  delimiters,  and numbers in a TCL wiki style table as

======
  puts " %| testcase $testcase_number | value| units |comment |%"
  puts " &| volume| $volume| cubic meters |based on length $side1 and width $side2   |&"  
======
----

** Console program for race output as CVS spreadsheet, includes > equal probability horses < **

======
        # pretty print from autoindent and ased editor
        # initial console program, Chinese horse race problem
        # comma delimited spreadsheet for number of horse races
        # written on Windows XP on eTCL
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # gold on TCL WIKI, 10feb2017
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        console show
        set i 2000
        set horse_race_days 1
        puts "horse race number , trial heat, winner, wins of particular horse, cum. probability, loser,loser, horse_a ,          horse_b ,  horse_c ,  last place "
        while {$i>0} {
            if { $horse_race_days > 180 } { break }
            set horse_a [ expr { rand() } ]
            set horse_b [ expr { rand() } ]
            set horse_c [ expr { rand() } ]
            set trial [ expr { rand() } ]
            if { $trial > 0      &&  $trial < .2     } { set winner_name  "horse_a" ; set loser_name  "horse_b,horse_c"     }
            if { $trial > .2     &&  $trial < .5     } { set winner_name  "horse_b" ; set loser_name  "horse_a,horse_c"      }
            if { $trial > .5     &&  $trial < 1.     } { set winner_name  "horse_c" ; set loser_name  "horse_a,horse_b"      }
            set winner [max $horse_a $horse_b $horse_c]
            set last [min $horse_a $horse_b $horse_c]
            lappend winner_list $winner_name
            lappend loser_list $loser_name
            #set number_wins [ lsearch -all $winner_list  $winner_name ]
            set ee [ llength  $winner_list ]
            set kk [ llength [ lsearch -all $winner_list $winner_name ] ]
            set prob [ expr { ($kk*1.) / $ee  } ]
            if {$winner_name == "horse_c" } {
                puts "$horse_race_days ,$trial ,$winner_name,$kk,$prob,$loser_name,$trial, $winner, $horse_a , $horse_b , $horse_c ,                 $last "}
            incr horse_race_days
            incr i -1
        }
            #end of deck
======
**cvs output for '''> wins only < '''of horse_c**
----
horse race number , trial heat, winner, wins of particular horse, cum. probability, loser,loser, horse_a ,  horse_b ,  horse_c ,  last place 
2 ,0.72403450297379612 ,horse_c,1,0.5,horse_a,horse_b,0.72403450297379612, 0.86678423540051297, 0.13588783151278638 , 0.86678423540051297 , 0.042644376420716001 , 0.042644376420716001 
3 ,0.87722261151169545 ,horse_c,2,0.66666666666666663,horse_a,horse_b,0.87722261151169545, 0.84789148059109765, 0.84789148059109765 , 0.51211429457744317 , 0.104948963087494 , 0.104948963087494 
4 ,0.86264225927304583 ,horse_c,3,0.75,horse_a,horse_b,0.86264225927304583, 0.61519644810594454, 0.48043167706599071 , 0.61519644810594454 , 0.60670331660970267 , 0.48043167706599071 
7 ,0.73498046013292873 ,horse_c,4,0.5714285714285714,horse_a,horse_b,0.73498046013292873, 0.27436126688232704, 0.27436126688232704 , 0.18981249127062619 , 0.17854078541441856 , 0.17854078541441856 
8 ,0.96185003824618176 ,horse_c,5,0.625,horse_a,horse_b,0.96185003824618176, 0.81659345413399553, 0.81659345413399553 , 0.48618363006328402 , 0.28827047361446101 , 0.28827047361446101 
10 ,0.76152973378148381 ,horse_c,6,0.59999999999999998,horse_a,horse_b,0.76152973378148381, 0.95698579864436095, 0.078298148735565204 , 0.95698579864436095 , 0.060317815775199707 , 0.060317815775199707 

----
** Console program for tianji horse race and combined unequal probability**

======
               # pretty print from autoindent and ased editor 
               # subroutine for tianji horse race and combined unequal probability
               # working under TCL version 8.5.6 and eTCL 1.0.1
               # console program written on Windows XP on eTCL
               # written on Windows XP on eTCL
               # working under TCL version 8.5.6 and eTCL 1.0.1
               # gold on TCL WIKI, 2mar2017
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        console show
        proc tianji_horse_race     { xxx1  }   {
            set i 2000
            set horse_race 1
            set trial 1.
            set trialx 1.
            set tianji_low 1.
            set tianji_middle 1.
            set tianji_high 1.
            set king_low 1.
            set king_middle 1.
            set king_high 1.
            set program_result {}
            set winner_list1 {}
            set winner_list2 {}
            while {$i>0} {
            if { $horse_race > 10 } { break }
            set trial [ expr { rand() } ]
            set counter 1
            set tianji_low  [expr 15.* [expr { rand() } ] ]
            set king_high [expr 50.* [expr { rand() } ] ]  
            # 15 percent = low_grade
            # 35 percent = medium_grade
            # 50 percent = high_grade
    if {    [max  $tianji_low $king_high  ] == $tianji_low   } {  lappend winner_list2   " horse_race$horse_race heat$counter "   tianji_low }
     if {    [max  $tianji_low $king_high  ] == $king_high  } {  lappend winner_list2  " horse_race$horse_race heat$counter "   king_high  }                 
            incr counter 
            set tianji_middle [expr 35.* [expr { rand() } ] ]
            set king_low [expr 15.* [expr { rand() } ] ]  
            if {    [max  $tianji_middle $king_low  ] == $tianji_middle   } {  lappend winner_list2   "  horse_race$horse_race heat$counter  "   tianji_middle }
        if {    [max  $tianji_middle $king_low  ] == $king_low  } {  lappend winner_list2  "  horse_race$horse_race heat$counter  "   king_low  } 
            incr counter 
            set tianji_high [expr 50.* [expr { rand() } ] ]
            set king_middle [expr 35.* [expr { rand() } ] ]  
            if {    [max  $tianji_high $king_middle  ] == $tianji_high   } {  lappend winner_list2  " horse_race$horse_race  heat$counter  "   tianji_high  \n}
            if {    [max  $tianji_high $king_middle  ] == $king_middle  } {  lappend winner_list2  "  horse_race$horse_race  heat$counter  "   king_middle \n }       
            incr horse_race
            incr i -1
        } 
           return  $winner_list2   }
           set program_result [  tianji_horse_race  3 ]   
           puts " list of winners  = \n  $program_result  "
           #end of deck
======
**output follows for list of winners in trial races with three 3 heats, > unequal probability horses <**
======
list of winners  = 
  { horse_race1 heat1 } king_high {  horse_race1 heat2  } tianji_middle { horse_race1  heat3  } tianji_high {
} { horse_race2 heat1 } king_high {  horse_race2 heat2  } king_low {  horse_race2  heat3  } king_middle {
} { horse_race3 heat1 } tianji_low {  horse_race3 heat2  } tianji_middle { horse_race3  heat3  } tianji_high {
} { horse_race4 heat1 } king_high {  horse_race4 heat2  } tianji_middle { horse_race4  heat3  } tianji_high {
} { horse_race5 heat1 } king_high {  horse_race5 heat2  } tianji_middle { horse_race5  heat3  } tianji_high {
} { horse_race6 heat1 } king_high {  horse_race6 heat2  } tianji_middle { horse_race6  heat3  } tianji_high {
} { horse_race7 heat1 } king_high {  horse_race7 heat2  } king_low { horse_race7  heat3  } tianji_high {
} { horse_race8 heat1 } tianji_low {  horse_race8 heat2  } tianji_middle { horse_race8  heat3  } tianji_high {
} { horse_race9 heat1 } king_high {  horse_race9 heat2  } king_low {  horse_race9  heat3  } king_middle {
} { horse_race10 heat1 } king_high {  horse_race10 heat2  } tianji_middle {  horse_race10  heat3  } king_middle {
}  
======
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