# Pretty print version from autoindent
               # and ased editor
               # written on Windowws XP on eTCL
               # working under TCL version 8.5.6 and eTCL 1.0.1
               # gold on TCL WIKI , 25may2011
               # relative frequency of indiv. "throw" over all "throws".
               # pi mantissa used here
        package require Tk 
          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 [split {14159265358979323846} ""]
                
                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 0 } { $i <= $limit }  { incr i } {
                lappend listxxx $i
                lappend listxxx [ calculation  $i ]
                puts " $i [ calculation  $i ] "
            }
            #end

set lister [split {14159265358979323846} {}]

set lister [split {14159265358979323846264338327950288419716939937} {}] 

   # test of transmuting integers of pi into words and letters.
   # [Mapping words to integers]
   # Small program for console
   console show 
   source int2wordetc.tcl
   set bid  "14159265358979323846264338327950288419716939937510"
   puts " word  [ int2word   $bid ]"
   #output
   #jzilvrzbjfkfzpwznsqdbwjzsjzeqfxkvop

1. TESTING NORMALITY OF PI – CONCEPT DIAGRAM

+====================================================================================+
| TESTING NORMALITY OF PI DIGITS (Console Example)                                  |
+====================================================================================+
|                                                                                   |
|   Pi (π) digits should behave like truly random numbers if "normal".              |
|                                                                                   |
|   Test Idea:                                                                      |
|   • Take first N digits of Pi (after decimal)                                     |
|   • Count frequency of each digit 0–9                                             |
|   • Compare observed counts with expected uniform distribution                    |
|                                                                                   |
|   Common Statistical Tests:                                                       |
|   • Chi-Square Goodness of Fit                                                    |
|   • Kolmogorov-Smirnov Test                                                       |
|   • Runs Test for randomness                                                      |
|                                                                                   |
|   Goal: Check if Pi digits appear "normal" (uniformly random)                     |
+====================================================================================+

2. eTCL CONSOLE EXAMPLE – BLACK BOX STYLE (Tcl/Tk 8)

+====================================================================================+
| eTCL TESTING NORMALITY OF PI – CONSOLE DEMO (Tcl/Tk 8)                           |
+====================================================================================+
| [ STUDENT / MATH ENTHUSIAST ]                                                     |
|         ↓                                                                         |
|   TCL Console or Simple Script                                                    |
|         ↓                                                                         |
|   proc test_pi_normality {num_digits} {                                           |
|       set digits [get_pi_digits $num_digits]                                      |
|       set freq   [count_digit_frequency $digits]                                  |
|       set chi_sq [chi_square_test $freq $num_digits]                              |
|       set p_value [calculate_p_value $chi_sq]                                     |
|       return [list $freq $chi_sq $p_value]                                        |
|   }                                                                               |
|         ↓                                                                         |
|   Output: Digit Frequencies | Chi-Square Statistic | p-value | Conclusion         |
|         ↓                                                                         |
|   Console: "Digits of Pi appear normally distributed: Yes/No"                     |
+====================================================================================+

3. eTCL SLOT CALCULATOR DEMO – SIMULATION

+====================================================================================+
| eTCL SLOT CALCULATOR DEMO – PI NORMALITY SIMULATOR                              |
+====================================================================================+
|                                                                                   |
|   [ Pi Digits Test Slot Machine ]                                                 |
|                                                                                   |
|   Number of Digits   →  [ 100 | 1000 | 10000 | 100000 ]                           |
|   Test Type          →  [ Chi-Square | KS Test | Runs Test ]                      |
|   Random Seed        →  [ Default Pi | Pure Random ]                              |
|                                                                                   |
|   proc pi_normality_slot {iterations} {                                           |
|       set results [monte_carlo_pi_test $iterations]                               |
|       return [list $results pass_rate average_p_value]                            |
|   }                                                                               |
|                                                                                   |
|   Purpose: Compare real Pi digits against truly random sequences                 |
+====================================================================================+

4. ANALYSIS FLOW

+----------------------------------------------------------------------------------+
| TESTING NORMALITY OF PI – STEP BY STEP                                         |
+----------------------------------------------------------------------------------+
|                                                                                 |
|   1. Generate or load many digits of Pi                                         |
|   2. Count frequency of each digit 0 through 9                                 |
|   3. Calculate expected count = total_digits / 10                              |
|   4. Apply Chi-Square test:                                                     |
|      χ² = Σ (observed - expected)² / expected                                  |
|   5. Compute p-value                                                            |
|   6. Interpret: High p-value → digits appear normally distributed               |
|                                                                                 |
|   eTCL Console Demo makes this quick and educational                            |
+----------------------------------------------------------------------------------+

5. OVERVIEW DIAGRAM – EDUCATIONAL BRIDGE

+====================================================================================+
| TESTING NORMALITY OF PI – FROM DIGITS TO eTCL CONSOLE                           |
+====================================================================================+
|                                                                                   |
|   Mathematical Curiosity     →         Statistical Analysis                       |
|                                                                                   |
|   Pi Digits Sequence         ↔         Frequency Counting                         |
|   Uniform Distribution Test  ↔         Chi-Square / p-value                       |
|   Randomness Question        ↔         Console + Slot Simulator                   |
|                                                                                   |
|   eTCL Features:                                                                  |
|   • Easy digit loading and frequency analysis                                     |
|   • Built-in statistical tests                                                    |
|   • Slot-machine style Monte Carlo comparison                                     |
|   • Clear console output for learning                                             |
|                                                                                   |
|   Goal: Let users test whether Pi digits behave like random numbers              |
+====================================================================================+