#!/usr/bin/perl
use strict;
use warnings;
use Math::BigInt;
use POSIX;
use Tk;
use Tk::Pane;
######################################################################
+##########
#   SOME GLOBAL DECLARATION
######################################################################
+##########
my @tartaglia ;     #AoA used as CACHE
my @tkcache;        #AoA used as CACHE for Tk buttons in the triangles
my $tart_win;       # triangle window
my $ow;             #output window
my $out;            #output var for out_win
my $row_num = 15;   #default row noumber for the triangle
my $dot_after = 3;  # default: instead of '24' it prints '..'
my $debug = 0;      # no debug infos in the output window
my @posible_colors = qw(red royalblue  orange green yellow violet blue
+ pink purple );
my %next_col = (red=>'royalblue',royalblue=>'orange',orange=>'green',g
+reen=>'yellow',yellow=>'violet',
                violet=>'blue',blue=>'pink',pink=>'purple',purple=>'re
+d');
my @colorized;      # array of Tk button yet colorized
my $size_tile = 8;  # size and boldness of various fonts
my $bold_tile = 0;
my $size_help = 10;
my $bold_help = 1;
my $size_out = 10;
my $bold_out = 1;

use subs 'tar_print';
######################################################################
+##########
#   MAIN WINDOW CREATION
######################################################################
+##########
my $mw = MainWindow->new ();
    $mw->Icon(-image => $mw->Pixmap(-data => &tart_icon));
    $mw->geometry("720x900+0+0"); #->geometry("300x450+0+0"); 320+0
    $mw->title(" command ");
    #$mw->optionAdd('*font', 'Courier 10');
    $mw->optionAdd('*Label.font', 'Courier 10');
    $mw->optionAdd( '*Entry.background',   'lavender' );
    $mw->optionAdd( '*Entry.font',   'Courier 10 bold'  );
    my $scrolled_top = $mw->Scrolled('Frame',
                      -background=>'white',
                      -scrollbars => 'osoe',)->pack(-expand => 1, -fil
+l => 'both');

my $fr0 = $scrolled_top->Frame(-borderwidth => 2, -relief => 'groove')
+->pack(-side=>'top',-pady=>10);
    $fr0->Label(-text => "-Tartaglia's triangle properties-" )->pack(-
+pady=>10);

my  $fr1 = $scrolled_top->Frame(-borderwidth => 2, -relief => 'groove'
+)->pack(-side=>'top',-anchor=>'w',-pady=>5); #,-fill=>'x'
    $fr1->Label(-text => "Rows in the triangle: from 0 to ")->pack(-si
+de => 'left');#,-expand => 1, -fill=>'x'
    $fr1->Entry(-width => 3,-borderwidth => 4, -textvariable => \$row_
+num)->pack(-side => 'left', -expand => 1,-padx=>5); #-side => 'left',
+ -expand => 1, -fill=>'x'
    $fr1->Label(-text => "Tiles font size")->pack(-side => 'left',-exp
+and => 1);
    $fr1->Entry(-width => 3,-borderwidth => 4, -textvariable => \$size
+_tile)->pack(-side => 'left', -expand => 1,-padx=>5);
    $fr1->Label(-text => "bold")->pack(-side => 'left',-expand => 1);
    $fr1->Checkbutton( -variable =>\$bold_tile )->pack(-side => 'left'
+, -expand => 1);
    $fr1->Button(-padx=> 5,-text => "introduction",-borderwidth => 4, 
+-command => sub{&help(\&help_intro)})->pack(-side => 'right',-expand 
+=> 1,-padx=>5);#128

my $fr2 = $scrolled_top->Frame(-borderwidth => 2, -relief => 'groove')
+->pack(-side=>'top',-anchor=>'w',-pady=>5);
  $fr2->Label(-text => "Numbers as dot if ")->pack(-side => 'left',-ex
+pand => 1);
  $fr2->Radiobutton(-text => "1",-variable => \$dot_after, -value=>'1'
+)->pack(-side => 'left',-expand => 1);
  $fr2->Radiobutton(-text => "2",-variable => \$dot_after, -value=>'2'
+)->pack(-side => 'left',-expand => 1);
  $fr2->Radiobutton(-text => "3",-variable => \$dot_after, -value=>'3'
+)->pack(-side => 'left',-expand => 1);
  $fr2->Radiobutton(-text => "4",-variable => \$dot_after, -value=>'4'
+)->pack(-side => 'left',-expand => 1);
  $fr2->Radiobutton(-text => "never",-variable => \$dot_after, -value=
+>'9999')->pack(-side => 'left',-expand => 1);
  $fr2->Label(-text => " digits.  Print debug information")->pack(-sid
+e => 'left',-expand => 1);
  $fr2->Checkbutton( -variable =>\$debug,-command => sub { tar_print "
+Debug info ".($debug ? 'enabled' : 'disabled')."\n" })->pack();

my $fr2a = $scrolled_top->Frame(-borderwidth => 2, -relief => 'groove'
+)->pack(-side=>'top',-anchor=>'w',-pady=>5);
   $fr2a->Label(-text => "Size of help texts")->pack(-side => 'left',-
+expand => 1);
   $fr2a->Entry(-width => 3,-borderwidth => 4, -textvariable => \$size
+_help)->pack(-side => 'left', -expand => 1,-padx=>5); #-side => 'left
+', -expand => 1, -fill=>'x'
   $fr2a->Label(-text => "bold")->pack(-side => 'left',-expand => 1);
   $fr2a->Checkbutton( -variable =>\$bold_help )->pack(-side => 'left'
+, -expand => 1);
   $fr2a->Label(-text => "                 Size of output ")->pack(-si
+de => 'left',-expand => 1);
   $fr2a->Entry(-width => 3,-borderwidth => 4, -textvariable => \$size
+_out)->pack(-side => 'left', -expand => 1,-padx=>5); #-side => 'left'
+, -expand => 1, -fill=>'x'
   $fr2a->Label(-text => "bold")->pack(-side => 'left',-expand => 1);
   $fr2a->Checkbutton( -variable =>\$bold_out )->pack(-side => 'left',
+ -expand => 1);

my $fr3 = $scrolled_top->Frame(-background => 'white')->pack(-side=>'t
+op',-pady=>5);
  $fr3->Button(-padx=> 20,-text => "draw triangle",-borderwidth => 4, 
+-command => \&draw_triangle)->pack(-side => 'left',-expand => 1,-padx
+=>5);
  $fr3->Button(-padx=> 20,-text => "delete triangle",-borderwidth => 4
+, -command => \&destroy_tri )->pack(-side => 'left',-expand => 1,-pad
+x=>5);

######################################################################
+##########
#   EXPERIMENTS CREATION FRAME
######################################################################
+##########
my $fr4 = $scrolled_top->Frame(-borderwidth => 2, -relief => 'groove')
+->pack(-side=>'top',-pady=>10);
    $fr4->Label(-text => "-Tartaglia's triangle experiments-" )->pack(
+-pady=>10);

##### BINOMIAL EXPANSION
my $input_bin;
my $color_bin = 'red';
my $title_bin = "Binomial Expansion (a+b)^";
create_experiment (\$input_bin, \$color_bin, $title_bin, \&help_bin, \
+sub { $input_bin=~s/\s+//g;
                                                                &given
+_coord($color_bin,$input_bin." 0-$input_bin");
                                                                &bin_e
+xp($input_bin)});
##### POWERS OF TWO
my $input_p2;
my $color_p2 = 'red';
my $title_p2 = "Powers of 2            2^";
create_experiment (\$input_p2, \$color_p2, $title_p2, \&help_pow2, \su
+b {power_of_two($input_p2,$color_p2)} );

##### POWERS OF ELEVEN
my $input_p11;
my $color_p11 = 'red';
my $title_p11 = "Powers of 11          11^";
create_experiment (\$input_p11, \$color_p11, $title_p11,\&help_pow11,\
+sub {power_of_eleven($input_p11,$color_p11)} );

##### FIBONACCI
my $input_fib;
my $color_fib = 'red';
my $title_fib = "Fibonacci         max row";
create_experiment (\$input_fib, \$color_fib, $title_fib,\&help_fib,\su
+b {fibonacci($input_fib,$color_fib)} );

##### PRIME NUMBERS
my $input_pri;
my $color_pri = 'red';
my $title_pri = "Prime numbers     max row";
create_experiment (\$input_pri, \$color_pri, $title_pri,\&help_pri,\su
+b {is_prime($input_pri,$color_pri)} );

### POLYGONAL NUMBERS
my $input_tri;
my $color_tri = 'red';
my $title_tri = "Triangular numbers    num";
create_experiment (\$input_tri, \$color_tri, $title_tri, \&help_tri, \
+sub {&triangulars($input_tri, $color_tri)});

#### COORDINATES
my $input_coord;
my $color_coord = 'red';
my $title_coord = "Colorize by coordinates";
create_experiment (\$input_coord, \$color_coord, $title_coord,\&help_b
+ycoord, \sub {&given_coord($color_coord ,$input_coord)});

### DAVID'S STAR
my $input_star;
my $color_star = 'red';
my $title_star = "David's star   row col";
create_experiment (\$input_star, \$color_star, $title_star, \&help_dav
+id, \sub {&david_star($input_star, $color_star)});

### CAPELAN
my $input_cat;
my $color_cat = 'red';
my $title_cat = "Catalan's numbers max row";
create_experiment (\$input_cat, \$color_cat, $title_cat, \&help_cat, \
+sub {&catalan($input_cat, $color_cat)});

### MERSENNE AND M PRIMES
my $input_mer;
my $color_mer = 'red';
my $title_mer = "Mersenne numbers  max row";
create_experiment (\$input_mer, \$color_mer, $title_mer, \&help_mer, \
+sub {&mersenne($input_mer, $color_mer)});

### SIERPINSKI
my $input_sie;
my $color_sie = 'red';
my $title_sie = "Sierpinski fractals   num";
create_experiment (\$input_sie, \$color_sie, $title_sie, \&help_sie, \
+sub {&sierpinski($input_sie, $color_sie)});

### COMBINATIONS
my $input_com;
my $color_com = 'red';
my $title_com = "Combinations      row col";
create_experiment (\$input_com, \$color_com, $title_com, \&help_com, \
+sub {&combination($input_com, $color_com)});

### EVALUATION
my $input_eval;
my $color_eval = 'red';
my $title_eval = "Colorize by evaluation";
create_experiment (\$input_eval, \$color_eval, $title_eval, \&help_eva
+l, \sub {&col_eval($color_eval ,$input_eval)});

### HOCKEY STICK PATTERN
my $input_hoc;
my $color_hoc = 'red';
my $title_hoc = "Hockey stick      row col";
create_experiment (\$input_hoc, \$color_hoc, $title_hoc, \&help_hockey
+, \sub {&hockeystick($input_hoc, $color_hoc)});

### PARALLELOGRAM PATTERN
my $input_par;
my $color_par = 'red';
my $title_par = "Parallelogram     row col";
create_experiment (\$input_par, \$color_par, $title_par, \&help_para, 
+\sub {&parallelogram($input_par, $color_par)});

### SUM OF SQUARES
my $input_ssq;
my $color_ssq = 'red';
my $title_ssq = "Sum of squares in the row";
create_experiment (\$input_ssq, \$color_ssq, $title_ssq, \&help_squa, 
+\sub {&sum_squares($input_ssq, $color_ssq)});

tar_print "Welcome to Tartaglia's triangle fun offered by Discipulus a
+s found at www.perlmonks.org";
&draw_triangle;

tar_print "MainWindow geometry: ",$mw->geometry(),"\n";
    tar_print "Triangle geometry: ",$tart_win->geometry(),"\n";
    tar_print "output geometry: ",$ow->geometry(),"\n";

MainLoop;
######################################################################
+##########
#   EXPERIMENTS SUBROUTINES
######################################################################
+##########
sub sum_squares {
    my ($input,$color)=@_;
    if ($input =~ /\s?(\d+)\D/){$input = $1}
    my $col2 = $next_col{$color};
    tar_print "\n\n*** Sum of sqares of rown $input\n\n";
    my @row = tartaglia_row($input);
    my $calc = join ' ** 2 + ',@row;
    tar_print "The sumation of squares of $color tiles in ".$input."th
+ row is:\n$calc = ",eval $calc,"\n";
    given_coord($color, "$input 0-".($input + 1));
    my @double = tartaglia_row($input * 2);
    my $central = $double[ (int $#double / 2 )];
    given_coord($col2, ($input * 2)." ".((int $#double / 2 )));
    tar_print "the central element of $input x 2 (".($input * 2).") ro
+w is $central\n\n";
}
######################################################################
+##########
sub parallelogram {
    my ($input,$color)=@_;
    my ($row,$col)= split ' ',$input;
    tar_print "\n\n*** Parallelogram pattern \n\n";
    given_coord ($color, "$row $col");
    my $wanted = ${[tartaglia_row($row)]}[$col];
    my @parallelogram;

    my $col2 = $next_col{$color};
    $col--;
    my $first = $col;
    my $last = $col;
    foreach my $prow (reverse 0..$row-2){
            my @val = tartaglia_row($prow);
            $first = 0 if $first < 0;
            $last = $col if $last > $col;
            $last = $#val if $last > $#val;
            push @parallelogram, @val[$first .. $last];
            given_coord ($col2, "$prow ".$first.'-'.$last);
            $first--;
            $last++;
    }
    my $sum = join ' + ', sort @parallelogram;
    my $res = eval $sum;
    tar_print "$wanted ($color tile) is equal to the sum of $col2 tile
+s + 1:\n";
    tar_print "$sum = $res\n$res + 1 = ",$res + 1," = $wanted ($color 
+tile)\n";
}
######################################################################
+##########
sub hockeystick {
    my ($input,$color)=@_;
    my ($row,$col)= split ' ',$input;
    tar_print "\n\n*** Hockey stick pattern \n\n";
    my $col2 = $next_col{$color};

    given_coord ($col2, "0-".($row-1)." ".($col-1) );
    given_coord ($color, "$row $col");
    my @hockey;
    foreach my $trow ( 0 .. $row-1) {
            my @val = tartaglia_row($trow);
            defined $val[$col-1] ? (push @hockey, $val[$col-1]) : next
+;
    }
    my $number = ${ [tartaglia_row($row)] }[$col];
    my $sum  =  join ' + ',@hockey;
    tar_print "$number ($color tile) is equal to the sum of $col2 tile
+s:\n$sum = ".eval $sum."\n";
}
######################################################################
+##########
sub triangulars{
    my ($input,$color)=@_;
    if ($input =~ /\s?(\d+)\D/){$input = $1}
    my $col2 = $next_col{$color};
    tar_print "\n\n*** Triangular number $input\n\n";
    given_coord ($col2, "0-$row_num 2");
    given_coord ($color, ($input+2)." 2");
    my @triangulars = map {my $n; my $x = $_; foreach my $i(0..$x) {$n
++=$i};$n    } 1..$input+1;
    tar_print "\nThe $input".'th '."triangular number is: $triangulars
+[-1] ($color tile)\n";
    tar_print "First triangular numbers found ($col2 tiles):\n",(join 
+' ', @triangulars),"\n\n";
}
######################################################################
+##########
sub combination{
    my ($input,$color)=@_;
    my ($row,$col)= split ' ',$input;
    if ($col > $row) {tar_print "Warning column must be lesser or equa
+l to row\n"; return}
    tar_print "\n\n*** Combinations of $col items in a group of $row\n
+\n";
    my $col2 = $next_col{$color};
    my $col3 = $next_col{$col2};
    my $col4 = $next_col{$col3};
    given_coord ($col2, "$row 0-$row_num");
    given_coord ($col3, "0-$row_num $col");
    given_coord ($col4, ($row + $col - 1)." $col");
    given_coord ($color, "$row $col");
    tar_print "There are ",${[tartaglia_row($row)]}[$col]," ($color ti
+le position $row - $col) different combinations (when the order does 
+not matter) of $col items in group of $row.\n";
    tar_print "There are ",${[tartaglia_row($row + $col - 1)]}[$col],(
+ $col > 1 ? " ($col4 tile)" : '')." different combinations with repet
+itions of $col items in group of $row.\n\n";
}
######################################################################
+##########
sub sierpinski{
    my ($input,$color)=@_;
    if ($input =~ /\s?(\d+)\D/){$input = $1}
    tar_print "\n\n*** Sierpinski fractal: show numbers divisible by $
+input\n\n";
    col_eval ($color, '$_ % '.$input.' == 0');
}
######################################################################
+##########
sub mersenne{
    my ($input,$color)=@_;
    my @mersenne;
    tar_print "\n\n*** Mersenne's numbers and Mersenne's primes (max r
+ow $input)\n\n";
    foreach my $row (0..$input){
            my $cur;
            map {$cur += $_ } tartaglia_row($row);
            push @mersenne, $cur-1;
            given_coord($color,"$row 0-".$row);
            $color = $next_col{$color};
    }
    tar_print "\nMersenne's numbers found in first $input rows:\n";
    foreach my $n (@mersenne){
              tar_print "$n ",( check_prime($n) ? "Mersenne prime " : 
+''),"\n";      #check_prime($n)
    }
    tar_print "\n\n";
}
######################################################################
+##########
sub catalan{
    my ($input,$color)=@_;
    my @catalan;
    my $natural = 1;
    tar_print "\n\n*** Catalan's numbers (max row $input)\n\nNote two 
+methods to generate the serie: the first divide the central term of a
+ny odd row ($color tiles) by the correspondant counting number: this 
+gives the right serie: 1 1 2 5 14..\n";
    tar_print "The second method is the central term of any odd row mi
+nus the term two place left, if present ($next_col{$color} tiles). Th
+is gives the rigth serie but without the first '1'.\n\n";
    given_coord($next_col{$next_col{$color}}, "0-".int($input / 2 + 1)
+." 1");
    foreach my $rc (0..$input){
       next if ($rc+1) % 2 == 0;
       my @row = tartaglia_row($rc);
       my $mid = (scalar @row / 2);
       my $two_left = ($mid - 2) >= 0 ? $row[$mid - 2] : 0 ;
       tar_print "$row[$mid] / $natural = ",$row[$mid] / $natural,"\t\
+t$row[$mid] - $two_left = ",$row[$mid] - $two_left,"\n";
       push @catalan, ($row[$mid] / $natural);
       colorize($tkcache[$rc][$mid],$color);
       colorize($tkcache[$rc][$mid - 2],$next_col{$color}) if ($mid - 
+2) >= 0 and defined $tkcache[$rc][$mid - 2];
       $natural++;
    }
    tar_print "\nCatalan's numbers found in first $input rows:\n",(joi
+n ' ', @catalan),"\n\n";
}
######################################################################
+##########
sub david_star {
    my ($input,$color)=@_;
    tar_print ("warning coordinated expected\n") unless $input =~ /\d+
+\s+\d/;
    my ($row, $col) = split /\s/,$input;
    if ($row < 2 or $col == $row or $col == 0){tar_print "warning coor
+dinates must be not on the border\n";return}
    unless ($tkcache[$row][$col]){$debug ? tar_print "skipping $row - 
+$col (outside the triangle)\n" :0;return; }
    my $next_col = $next_col{$color};
    my $other_col = $next_col{$next_col};
    map {&colorize ($_, $next_col)} $tkcache[$row-1][$col-1], $tkcache
+[$row][$col+1], $tkcache[$row+1][$col];
    map {&colorize ($_, $other_col)} $tkcache[$row-1][$col], $tkcache[
+$row+1][$col+1], $tkcache[$row][$col-1];
    &colorize ($tkcache[$row][$col], $color);
    my @above = tartaglia_row ($row-1);
    my @mid = tartaglia_row ($row);
    my @below = tartaglia_row ($row+1);
    tar_print "\n\n*** David's star for number $mid[$col] ( $row - $co
+l, $color)\n\n";
    tar_print "($next_col tiles)\ngreatest common divisor: GCD ($above
+[$col-1], $mid[$col+1], $below[$col]) = ",Math::BigInt::bgcd($above[$
+col-1], $mid[$col+1], $below[$col]),"\n";
    tar_print "product $above[$col-1] x $mid[$col+1] x $below[$col] = 
+",$above[$col-1] * $mid[$col+1] * $below[$col],"\n";
    tar_print "\n($other_col tiles)\ngreateast common divisor: GCD ($a
+bove[$col], $mid[$col-1],$below[$col+1]) = ",Math::BigInt::bgcd($abov
+e[$col], $mid[$col-1],$below[$col+1]),"\n";
    tar_print "product $above[$col] x $mid[$col-1] x $below[$col+1] = 
+",$above[$col] * $mid[$col-1] * $below[$col+1],"\n";
    tar_print "\nProduct of six terms is always an integer perfect squ
+are:\n";
    tar_print "$above[$col-1] x $mid[$col+1] x $below[$col] x $above[$
+col] x $mid[$col-1] x $below[$col+1] = ";
    my $big_prod = $above[$col-1] * $mid[$col+1] * $below[$col] * $abo
+ve[$col] * $mid[$col-1] * $below[$col+1];
    tar_print  $big_prod, "\nsquare root of $big_prod = ", sqrt $big_p
+rod,"\n\n";
}
######################################################################
+##########
sub is_prime{
      my ($input,$color)=@_;
      tar_print "\n\n*** Prime numbers (max row $input)\n\n";
      foreach my $row (0..$input){
              my @vals = tartaglia_row($row);
              foreach my $pos (0..$#vals){
                      next if $vals[$pos] == 1;
                      if (check_prime($vals[$pos]))  {
                          tar_print "$vals[$pos] is prime\n";
                          colorize($tkcache[$row][$pos],$color );
                      }
              }
      }
}
######################################################################
+##########
sub fibonacci{
      my ($input,$color)=@_;
      if ($input > $row_num){$input=$row_num;tar_print "Warning: too m
+any rows specified. Using $row_num\n" if $debug}
      tar_print "\n\n*** Fibonacci's numbers (max row $input)\n\n";
      my @aoa_vals = map {[tartaglia_row($_)]} 0..$input; # why i buil
+d triangle by hockey stick pattern?!?!? argh
      my @fibonacci;
      my $fibonacci;
      my $col_i=0;
      foreach my $row (reverse 0..$input){
              my $cur_pos = 0;
              my $cur_row = $row;
              while ($cur_row >= $cur_pos){
                    next unless $tkcache[$cur_row][$cur_pos]->isa('Tk:
+:Button');
                    colorize($tkcache[$cur_row][$cur_pos], $posible_co
+lors[$col_i]);
                    push @{$fibonacci[$row]}, $aoa_vals[$cur_row][$cur
+_pos];# tar_print "push \$fibonacci[$row], $aoa_vals[$cur_row][$cur_p
+os];\n";
                    $cur_row--;
                    $cur_pos++;
              }
      $col_i++;
      $col_i > $#posible_colors ? $col_i=0 : 0;
      }
      map {  my $sum = join '+',@{$_};tar_print $sum,' = ', eval $sum,
+"\n";$fibonacci.=(eval $sum).' ';} @fibonacci;
      tar_print "\n\nFibonacci's numbers: $fibonacci\n\n";
}
######################################################################
+##########
sub power_of_eleven{
      my ($input,$color)=@_;
      my $big_int = Math::BigInt->new( '11' );
      tar_print "\n\n*** Power of 11:\t11^$input = ", $big_int->bpow($
+input),"\n\n";
      &given_coord($color ,"$input 0-$input");
      my @row =tartaglia_row($input);
      my $level = $input;
      my $cur_dec=0;
      my @final;
      tar_print "row $input: ",join ' ',@row,"\n\n";
      foreach my $num ( reverse @row) { # reverse is not util but..
              my ($dec,$unit,$partial_dec,$tmp);
              if ($num=~/(\d+)(\d)$/){$dec=$1;$unit=$2}
              else{$dec=0;$unit=$num}
              my $pad = '    '.("  " x $level--).' ';
              my $minus = (length ("$dec")+1);
              $pad =~ s/\s{$minus}//;
                    tar_print $pad."$dec|$unit\n";
                    $num+=$cur_dec;
                    if ($num=~/(\d+)(\d)$/){$cur_dec=$1;$num=$2}
                    else{$cur_dec=0; }
                    unshift @final,$num;
      }
      $cur_dec ? unshift @final, $cur_dec  : 0;
      tar_print "\n     ",(join ' ',@final),"\n\n = ",(join '',@final)
+,"\n\n";
}
######################################################################
+##########
sub power_of_two{
      my ($input,$color)=@_;
      my $big_int = Math::BigInt->new( '2' );#tar_print $x->bpow(15);
      tar_print "\n\n*** Power of 2:\t2^$input = ", $big_int->bpow($in
+put),"\n\n";
      &given_coord($color ,"$input 0-$input");
      my $sum = join ' + ', tartaglia_row($input);
      tar_print "$sum = ",eval $sum,"\n\n";
}
######################################################################
+##########
sub bin_exp{ #plagiarized from crazyinsomniac at http://www.perlmonks.
+org/?node_id=68056
    my $n = shift;
    tar_print "\n\n*** Binomial expansion:\t(a+b)^$n =\n\n";
    my @coefficient = tartaglia_row($n);
    for my $j (0 .. $n)
    {
       my $nj=$n-$j;
       tar_print $coefficient[$j];
       tar_print $_ = ($nj!=0)?( ($nj>1)?(' * a^'.$nj):(' * a') ):'';
       tar_print $_ = ($j!=0)?( ($j==1)?(' * b'):(' * b^'.$j) ):'';
       tar_print $_ = ($j!=$n)?(" +\n"):("\n");
    }
    tar_print "\n\n" ;
}
######################################################################
+##########
sub col_eval {
    my $color = shift;
    my $to_eval = shift;
    if ($to_eval =~ /system|exec|`/){tar_print "not safe\n";return}
    foreach my $row (0..$row_num) {
            my @vals = &tartaglia_row($row);
            my $i = 0;
            map {  my $val = $_;
                  ( my $str = $to_eval) =~ s/\$_/$val/e;
                  eval $to_eval ?
                       ( &tar_print ("$str TRUE AT $row - $i\n") and
                         &colorize ($tkcache[$row][$i], $color)
                       ) : 0;
                  $i++;
            } @vals;
    }
}
######################################################################
+##########
#   UTILITY SUBROUTINES
######################################################################
+##########
sub create_experiment{
    my ($input, $color, $title, $help, $sub_ref) = @_;
    my $frame = $scrolled_top->Frame(-borderwidth => 2, -relief => 'gr
+oove')->pack(-side=>'top',-anchor=>'w',-pady=>5);
    $frame->Button(-text => "?",-borderwidth => 2, -command => sub {&h
+elp($help)} )->pack(-side => 'left',-expand => 1);
    $frame->Label(-text => (pack 'A25', $title) )->pack(-side => 'left
+',-expand => 1);
    $frame->Entry(-width => 25,-borderwidth => 4,-textvariable => $inp
+ut)->pack(-side => 'left',-expand => 1);
    $frame->Optionmenu(-options => [@posible_colors],-variable => $col
+or)->pack(-side => 'left',-expand => 1);
    $frame->Button(-text => "Colorize",-borderwidth => 4, -command => 
+$sub_ref)->pack(-side => 'left',-expand => 1);
    $frame->Button(-text => "Clear",-borderwidth => 4, -command => \&d
+ecolorize)->pack(-side => 'left',-expand => 1);
}
######################################################################
+##########
sub tar_print{
    &check_output();
    $out->insert('end', "@_");
    $out->see('end');
    1;  # or col_eval will not call colorizes
}
######################################################################
+##########
sub check_prime { #http://www.perlmonks.org/?node_id=1054405
  my ($i,$j,$h,$sentinel) = (shift,0,0,0);
  # if $i is an even number, it can't be a prime
  if($i%2==0){return 0}
  else{
    $h=POSIX::floor(sqrt($i));
    $sentinel=0;
      # since $i can't be even -> only divide by odd numbers
      for($j=3; $j<=$h; $j+=2){
          if($i%$j==0){
               $sentinel++;
               # $i is not a prime, we can get out of the loop
              $j=$h;
          }
      }
    if($sentinel==0){
        return 1; print "$i \n";
    }
  }
}
######################################################################
+##########
sub decolorize {
                 foreach my $it(@colorized){
                              #tar_print "CLEAR call colorize: $it\n" 
+if $debug;
                              &colorize(  $it,'gray') ;
                 }
                 @colorized=();
                 return;
}
######################################################################
+##########
sub colorize {
    my $ref = shift;
    return 0 unless $ref;
    return 0 unless $ref->can('configure');
    my $color = shift;
    unless ($color eq 'gray'){push @colorized, $ref; }
    $ref->configure(-background =>$color);
    $tart_win->update;
}
######################################################################
+##########
sub given_coord {
    my $color = shift;
    my $to_color = shift;
    my @group = split /,/,$to_color;
    foreach my $pair (@group){
          $pair =~ s/^\s+//;$pair =~ s/\s+$//; $pair =~ s/\s+/ /;
          map  {  my ($x,$y) = split /\s+/,$_;
                  $tkcache[$x][$y] ? &colorize ($tkcache[$x][$y], $col
+or) :
                                    ($debug ? tar_print "skipping $x -
+ $y (outside the triangle)\n" :0);
          } &exp_coord($pair);
    }
}
######################################################################
+##########
sub exp_coord {
    my ($r,$c)=split /\s/,"@_";
    unless (defined $r and defined $c) {tar_print "Both must be define
+d. Received:",map{defined $_ ? "$_ " : 'UNDEF '}($r,$c);return}
    my @r; my @c;
    my @expanded;
    @r = $r=~/^(.*\d)-(.+)$/ ? ($1..$2) : ($r);
    @c = $c=~/^(.*\d)-(.+)$/ ? ($1..$2) : ($c);

    for my $rc (@r) { for my $cc (@c) { push @expanded, "$rc $cc" } };
    return @expanded;
}

######################################################################
+##########
sub destroy_tri {
    if  (Exists($tart_win)) {
        $tart_win->destroy();
        undef @colorized;
    }
    tar_print "MainWindow geometry: ",$mw->geometry(),"\n";
    #tar_print "Triangle geometry: ",$tart_win->geometry(),"\n";
    #tar_print "output geometry: ",$ow->geometry(),"\n";
}

######################################################################
+##########
sub draw_triangle {
  my $scrolledframe;
  if (! Exists($tart_win)) {
    $tart_win = $mw->Toplevel();
    $tart_win->Icon(-image => $mw->Pixmap(-data => &tart_icon));
    my $tart_win_max_width = 1024;
    my $tart_win_max_height = 768
    my $tart_win_width = ($row_num * 50) + 10;
    my $tart_win_height = ($row_num * 30) + 10;
    $tart_win->geometry($tart_win_width . "x" . $tart_win_height . "+7
+20+0");#make this 20 pixels per button + 5 on each side 
    $scrolledframe = $tart_win->Scrolled('Frame',
                      -background=>'black',
                      -scrollbars => 'osoe',
    )->pack(-expand => 1, -fill => 'both');
    $tart_win->title(" Tartaglia's triangle ");
    $tart_win->optionAdd('*Button.font' => 'Arial '.$size_tile.' '.($b
+old_tile ? 'bold' : ''), 20); #'Courier 13 bold'
    tar_print "\nDRAWING a tartaglia's triangle of ".($row_num + 1)." 
+rows (with dots if $dot_after or more digits)\n\n";
  }
  else {
    $tart_win->deiconify( ) if $tart_win->state() eq 'iconic';
    $tart_win->raise( ) if $tart_win->state() eq 'withdrawn';
    return;
  }
  #draw the triangle
  foreach my $row( 0..$row_num ){
          my $frame = $scrolledframe->Frame->grid;
          my ($first,@rest) =  &tartaglia_row ($row);
          my @others;
          foreach my $i (0..$#rest) {
                   my $n = $rest[$i];
                   $tkcache[$row][$i + 1] =
                          $frame->Button(-command => sub{tar_print "HI
+T ($row - ".($i + 1).") VALUE $n\n";},
                                                -text => &shrinkn($n) 
+,
                                                -background => 'gray' 
+);
                   $others[$i] = $tkcache[$row][$i + 1];
          }
          $tkcache[$row][0] = $frame->Button( -command => sub{tar_prin
+t "HIT ($row - 0) VALUE 1\n"}, #print $tkcache[$row][0]->fontActual('
+font'),"\n";
                                              -text => &shrinkn($first
+),
                                              -background => 'gray' )-
+>grid( @others );
  }
 tar_print "\n\n";
}

######################################################################
+##########
#{
# my @tartaglia ; #AoA used as CACHE
  sub tartaglia {
      my ($x,$y) = @_; #tar_print "\t\treceiving ".($y)." $x\t";
      if ($x == 0 or $y == 0)  { $tartaglia[$x][$y]=1 ; tar_print "\tF
+ORCED: 1\n" if $debug;return 1};
      tar_print ""."\tCACHE: ",(defined $tartaglia[$x][$y] ? "$tartagl
+ia[$x][$y]" : ' -not present- '),"\n" if $debug;
      my $ret ;
      foreach my $yps (0..$y){
        #tar_print "\tCACHE:", ( $tartaglia[$x-1][$yps] ? " HIT " : ' 
+-not present- '),"for ".($x - 1)." $yps\n";
        $ret += ( $tartaglia[$x-1][$yps] || &tartaglia($x-1,$yps) );
      }
      $tartaglia[$x][$y] = $ret;
      return $ret;
  }
#}

######################################################################
+##########
sub tartaglia_row {
    my $y = shift;
    my $x = 0;
    my @row;
       tar_print "ROW:".' '.($y)."\n" if $debug;
    $row[0] = &tartaglia($x,$y+1);
    foreach my $pos (0..$y-1) {push @row, &tartaglia(++$x,--$y)}

    return @row;
}

######################################################################
+##########
sub shrinkn {
              my $num = shift;
              my $rex = qr(\d{$dot_after});
              if ($num =~ $rex){ return join '','..' x ($dot_after - 1
+).($dot_after == 1 ? '..' :'')}
              else {return $num;}
}

######################################################################
+##########
sub check_output {
     #my $txt;
     if (! Exists($ow)) { $out = &outwin }
     $ow->deiconify( ) if $ow->state() eq 'iconic';
     $ow->raise( ) if $ow->state() eq 'withdrawn';
}
######################################################################
+##########
sub outwin {
    $ow = $mw->Toplevel( );
    $ow->Icon(-image => $mw->Pixmap(-data => &tart_icon));
    my $chars = 'Courier '.$size_out.' '.($bold_out ? 'bold' : '');
    $ow->geometry("755x429+720+490");
    $ow->optionAdd('*Text.font' => $chars, 20); #'Courier 13 bold'
    $ow->title(" output ");
    my $txt = $ow->Scrolled('Text',
                      -scrollbars => 'osoe',
                      -background => 'black',
                      -foreground => 'green',
                      #NO -data => \$cont,
    )->pack(-expand => 1, -fill => 'both');
    #tie *STDOUT,  $txt, $txt;
    return $txt;
}
######################################################################
+##########
sub help {
    my @helps = @_;
    my $hw = $mw->Toplevel( );
    $hw->Icon(-image => $mw->Pixmap(-data => &tart_icon));
    my $chars = 'Courier '.$size_help.' '.($bold_help ? 'bold' : '');
    $hw->geometry("900x450+0+0");
    $hw->optionAdd('*Text.font' => $chars, 20); #'Courier 13 bold'
    #$hw->optionAdd( '*Text.background'=>   'royalblue', 20 );
    $hw->title(" help ");
    my $txt = $hw->Scrolled('Text',
                      -background=>'white',
                      -scrollbars => 'osoe',
                      -background => 'blue3',
                      -foreground => 'gold2',
                      #NO -data => \$cont,
    )->pack(-expand => 1, -fill => 'both');

    $txt->Contents(map {&{$_}} @helps);
    $txt->Subwidget("yscrollbar")->configure(-background => 'black');
    $hw->update;
}
######################################################################
+##########
#     HELP TEXTS SUBROUTINES
######################################################################
+##########
sub help_eval  {
    return <<'EOH'
* Evaluation *

USAGE: enter valid Perl code. ** USE WITH CARE **

This experiment is dedicated mostly to Perl writer that can evaluate s
+ome code against any number in the triangle. While traversing the tri
+angle numbers '$_' will be the current number.

  $_ == 13

will colorize only 13, while

  $_ == 13 or $_==14

14 too

  $_ % 7 == 0

will show numbers divisible by 7, reveiling some Sierpinski's pattern 
+too.

  $_ > 0

can change the background color of the Tartaglia's triangle.

EOH
}
######################################################################
+##########
sub help_com  {
    return <<EOH
* Combinations *

USAGE: Feed the coordinates of a tile into the form of 'row column'. T
+he row, the column and the tile will be colorized with three differen
+t colors.
The value of combinations with repetition is colorized with another co
+lor, to show the correlation between the two.

The Tartaglia's triangle shows the answer to the question: 'How many p
+ossible groups are ther of a set of X (row) by Y (column)?'.
This is called combination (or k-combination) in mathematics, regardle
+ss of the order of the elements and requiring no repetition of elemen
+ts.
The formula is the binomial coefiicent's one.

               n!
 C(n,k) =  ----------
            k!(n-k)!

If an element can be found more than once, we call the result a 'combi
+nation with repetition' (or k-multicombination). The formula is linke
+d to the binomial coefficient too:

  d                      (n + k - 1)!
C      = C          =    -------------
 (n,k)    (n+k-1,k)      (n-1)!    k!

Speaking in tartaglia's triangle terms, the answer to a combinations w
+ith repetitions as compared to one without repetitions, the result wi
+ll be at the same column but the row will be 'n + k - 1' instead of '
+n'.

EOH
}
######################################################################
+##########
sub help_tri  {
    return <<EOH
* Triangular numbers, polygonal numbers and Figurates Numbers *

USAGE: Put in the entry box a number. The correspondent Triangular Num
+ber will be colorized along with each of the diagonal of Triangulars 
+Numbers as well.

Triangulars Numbers are a subset of Polygonal Numbers which are a subs
+et of Figurate Numbers.

If you can arrange a number of dots forming a regular triangle, then t
+hat number is a Triangular Number. 
In the same way if you can form a square you have a 'Square Number',  
+and a 'Pentaghonal Number' if you can form a pentagon and so on.

                                o
                     o         o o
            o       o o       o o o
     o     o o     o o o     o o o o
o   o o   o o o   o o o o   o o o o o

1    3      6       10          15

Very interestingly, every polygonal number can be calculated using the
+ corresponding Triangular Number.

The nth s-ogonal number P(s,n) is related to the Triangular Number T:


P(s,n)  = (s-2) T(n-1) + n  = (s - 3) T(n-1) + T(n)

For example the 4th exagonal number is:

P(6,4) =  (6 - 2) T(4-1) + 4  =  4 T(3) + 4  =  4 * 6 + 4 =  28

      O O O O
     O       O
    O O O O   O
   O O     O   O
    O O O   O O
     O   O O O
      O O O O

The 6th hexagonal number is, as you can see, 28.

In the II century BC, Ipsicle (http://it.wikipedia.org/wiki/Ipsicle) h
+ad found the relation between polygonal numbers an arithmetic progres
+sions.
A polygonal number with sides n is equal to the summation of all terms
+ of an arithmetic progression with first term 1 and ratio n-2.

For the 4th exagonal number the progression has ratio 6-2  = 4.

1 5 9 13

1 + 5 + 9 + 13 = 28


As you can see, the first column of Tartaglia's Triangle is composed b
+y many 1's.
The second column contains Counting Numbers, while the 3rd contains Tr
+iangular Numbers(2 dimensions).
The 4th column contains Tetrahedral Numbers (3 dimensions) and the 5th
+  Pentatope Numbers (4 dimensions) and so on.

So, if you want to build a pyramid of oranges with a triangular base a
+nd 4 floors you need 20 oranges.
Cubic numbers can be calculated using tetrahedral ones:

Cubic(n) =  Tetrahedral(n-2) + 4 Tetrahedral(n-1) + Tetrahedral(n)

Might I hazzard a guess that counting numbers are Figurate Numbers of 
+1 dimension and the 1's series is a series of 0 dimension Figurate Nu
+mbers?
I think you'll find any Figurate Number of any Regular Shape of any Di
+mension in the Tartaglia's Triangle...

EOH
}
######################################################################
+##########
sub help_squa  {
    return <<EOH
* Sumation of squares of terms in a row *

USAGE: Give the row number of which you want to calculate the summatio
+n of squares.

You'll see the Summation of Squares of the term on row n is equal to t
+he central term of row 2n.

EOH
}
######################################################################
+##########
sub help_para  {
    return <<EOH
* Parallelogram pattern *

USAGE: Give the coordinates of a tile and it will be demondstrated tha
+t the tile number is equal to the summation of all 
numbers in the parallelogram excluded by the two diagonals crossing at
+ the given tile position.


EOH
}
######################################################################
+##########
sub help_hockey  {
    return <<EOH
* Hockeystick pattern *

USAGE: Give the coordinates of tile and it will be demondstrated that 
+the tile number is equal to the summation of 
all numbers in the prior diagonal up to the same position of the given
+ tile number.


EOH
}
######################################################################
+##########
sub help_sie  {
    return <<EOH
* Sierpinski's fractals *

USAGE: Put in the entry box a number. Every tile will be colorized if 
+it's number is divisible by the number given.

Selecting a tile by divisibilty criteria can be drawn as a pattern ten
+ding to a Sierpinsky Triangle (http://en.wikipedia.org/wiki/Sierpinsk
+i_triangle). 
With different numbers you will obtains differnt fractals.
EOH
}
######################################################################
+##########
sub help_mer  {
    return <<EOH
* Mersenne's numbers *

USAGE: just put in the entry box the max row number to be considerated
+ to find Mersenne's numbers from the triangle.

A Mersenne's number is a number which is one less than a power of two.
+ As every row of the Tartaglia's triangle is a power of 2, the sum of
+ every term in a row, minus 1, is a Mersenne's number.

If a number in such sequence is prime it is called Mersenne's prime. S
+uch primes Mp are correlated with perfect numbers: Euclid (4th centur
+y BC) proved that if 2p-1 is prime, then 2p-1(2p - 1) is a perfect nu
+mber. This number is also expressible as Mp(Mp+1)/2
EOH
}
######################################################################
+##########
sub help_cat  {
    return <<EOH
* Catalan's numbers *

USAGE: just put in the entry box the max row number to be considerated
+ to find Catalan's numbers from the triangle.

I have decided to show on the screen two ways to extract Catalan's num
+bers from the Tartaglia's triangle: while the first shows the correct
+ serie (1 1 2 5 ..) the second sequence has only one '1' in the begin
+ning. I choose this way beacause both solutions are really tied with 
+the triangle itself.

EOH
}
######################################################################
+##########
sub help_pri  {
    return <<EOH
* Prime numbers *

USAGE: just put in the entry box the max row number to be considerated
+ to find prime numbers in the triangle.

You'll notice the disposition of primes in the triangle. Also note tha
+t if the 1st number on a row is prime (remember 0th number are always
+ 1) all other entries in that row (until the prime number reappers as
+ penultimate entry)  will be divisible by that prime number.

For example, in the 7th row you have:

1 7(a prime) 21 35 35 21 7(the prime again) 1

And, actually 21 and 35 are divisible by 7.


EOH
}
######################################################################
+##########
sub help_fib  {
    return <<EOH
* Fibonacci's numbers *

USAGE: just put in the entry box the max row number to be considerated
+ to create a Fibonacci's serie.

Fibonacci's numbers are obtained summing all the values present in a d
+iagonal of the triangle.
In this experiment the color choosen is not take in count.

If you enter '12' as max row you'll obtain a colorfull triangle and in
+ the screen:


Fibonacci's numbers (max row 12)

1 = 1
1 = 1
1+1 = 2
1+2 = 3
1+3+1 = 5
1+4+3 = 8
1+5+6+1 = 13
1+6+10+4 = 21
1+7+15+10+1 = 34
1+8+21+20+5 = 55
1+9+28+35+15+1 = 89
1+10+36+56+35+6 = 144
1+11+45+84+70+21+1 = 233


Fibonacci's numbers: 1 1 2 3 5 8 13 21 34 55 89 144 233

EOH
}
######################################################################
+##########
sub help_bycoord  {
        return <<EOH
* Colorize by coordinates *

USAGE: this colorizes by given coordinates, in 'row column' format. Mo
+re coordinates can be given separting pairs with commas. Both row and
+ column can be expressed as interval as in '7 0-7' for entire row 7 o
+r as '0-7 0' for the first 8 elements of the 0th column.

If a too wide range is given (some coordinates values are outside the 
+triangle as  for '0 1') tales outside the triangle are skipped. You c
+an view some worning on screen if you have enabled the 'print debug i
+nformation' control.
EOH
}
######################################################################
+##########
sub help_pow11  {
    return <<EOH
* Powers of 11 *

USAGE: just put in the entry box the power of 11 you want to calculate
+.
It appears that digits of a power of two '11 ^ n' are whom present in 
+the nth row.
While this is evident for row 0-4 you need to displace every quantitie
+s above '9' for row greater than 4.

For example if you insert '8' and hit 'colorize' the 8th row will chan
+ge color and in the screen appears:


Power of 11:    11^8 = 214358881

row 8: 1 8 28 56 70 56 28 8 1

                   0|1
                 0|8
               2|8
             5|6
           7|0
         5|6
       2|8
     0|8
   0|1

     2 1 4 3 5 8 8 8 1

 = 214358881


Please note, i'm too lazy to show it, that this is true for every sum 
+of two distinct powers of 10.

Id est: this procedure is valid for these three sums: (10+1), (100+1) 
+e (10+0,1):

          1                   1                       1         1
        1   1                11                   1.001        10,1
      1   2   1             121               1.002.001       102,01
    1   3   3   1         1.331           1.003.003.001     1.030,301
  1   4   6   4   1      14.641       1.004.006.004.001    10.406,0401
1   5  10  10   5   1   161.051   1.005.010.010.005.001   105.101,0050
+1

In the same way, if you write the Tartaglia's triangle not in base 10 
+but in base 'c' you'll be able to read the powers of every sum of two
+ distinct power of 'c'.

EOH
}
######################################################################
+##########
sub help_david  {
    return <<EOH
* David's start  *

USAGE: feed cordinates of a tile not in the border of the triangle and
+ seven tiles will be colorized: the given one of the color specified,
+ the surrounding other six ones in two different, alternate colors fo
+rming a David's star pattern.

On the screen will appear three different properties of such pattern a
+s calculation: the two terns share the Greatest Common Divisor and th
+e result of the product of their three terms. Also the product of all
+ six surrounding terms is always an integer perfect square. The last 
+one is obvious: as the product of two terns are equal their product w
+ill be a square.

EOH
}#####################################################################
+###########
sub help_pow2  {
    return <<EOH
* Powers of 2 *

USAGE: just put in the entry box the power of 2 you want to calculate.
It appears that a power of two '2 ^ n' is equal to the sum of every el
+ement in the nth row of the triangle.
The corrispondent row will be colorized with choosen color and the res
+ulting addition will be printed on the screen.

For example if you insert '13' and hit 'colorize' the 13th row will ch
+ange color and in the screen appears:



Power of 2:     2^13 = 8192

1 + 13 + 78 + 286 + 715 + 1287 + 1716 + 1716 + 1287 + 715 + 286 + 78 +
+ 13 + 1 = 8192


Note for this and others experiment: when te result appears two times,
+ as above for 8192, the first time is calculated directly, while the 
+second time is evalueted from the operation just created (in this cas
+e a 14 terms addition).
EOH
}
######################################################################
+##########
sub help_bin  {
    return <<EOH
* Binomial expansion *

USAGE: just put in the entry box the power you want to calculate for t
+he biomial (a + b)
The corrispondent row will be colorized with choosen color and the ful
+l expansion will be printed on the screen.

For example if you insert '5' and hit 'colorize' the 5th row (remember
+ the first row is the 0th) will change color and in the screen appear
+s:


Binomial expansion:     (a+b)^5 =

1 * a^5 +
5 * a^4 * b +
10 * a^3 * b^2 +
10 * a^2 * b^3 +
5 * a * b^4 +
1 * b^5

Binomial expansion describes also the 'Heads and Tails' game, when you
+ trow a coin.
If you trow a coin three times you can have these results:

        HHH
    HHT HTH THH
    TTH THT HTT
        TTT

Id est: 1 time 3 heads, 3 times 2 heads and 1 tail, 3 times 2 tails an
+d 1 heads, 1 time 3 tails.
This is the sequence 1 3 3 1, the 3th row of the triangle, the coeffic
+ients of the cubic expansion of (a+b).


Incredibly for me, binomial expansion describes also the geometrical '
+points in a circle' scenario.
Given a circle draw points on it for 1 to any number you want and draw
+ all the possible lines between them: you'll see segments, or if you 
+put 3 or more point, some polygons. The number of each type of geomet
+rical shape are binomial coefficients as shown by the Tartaglia's tri
+angle.
Id est: skipping the first diagonal (all 1s),if  the second one (count
+ing numbers) holds how many points you drawn on a circle then others 
+numbers in the row are how many segments, trinagles, quadrilaters, pe
+ntagons, hexagons, heptagons ... are possible with all vertices on th
+e circle.


  points
in a circle   segments    triangles   quadrilaters    pentagons   hexa
+gons

    1            -            -            -               -          
+-
    2            1            -            -               -          
+-
    3            3            1            -               -          
+-
    4            6            4            1               -          
+-
    5            10           10           5               1          
+-
    6            15           20           15              6          
+1
EOH
}
######################################################################
+##########
sub help_intro  {
    return <<EOH
* Introduction *

In Italy, the arithmetic triangle is called Tartglia's triangle, becau
+se exposed in the "General trattato di numeri et misure" written in 1
+556 by Niccolò Fontana (1499 ca, Brescia 13 December 1557, Venice), 
known also as Tartaglia.

In 1512 when the French invaded Brescia, a French soldier sliced Nicco
+lò's jaw and palate with a saber. This made it impossible for Niccolò
+ to speak normally, prompting the nickname "Tartaglia" ("stammerer"),
+ which he adopted.

Known as Pascal's triangle (but Pascal drew it as right triangle) in m
+any other countries was known by Halayuda, an Indian commentator in 1
+0th century, studied around 1100 by Omar Khayyam, a Persian mathemati
+cian, known in China as early as 1261 and so studied in India, Greece
+, Iran, China, Germany and Italy before Pascal.

About the program: Keep in mind that I'm not a mathematician, I was si
+mply impressed by the huge amount of things you can see in the triang
+le and I want to show these things to others.

Many useful things about the Tartaglia's Triangle are shown using the 
+Experiment Panel, some others are listed at the end of this introduct
+ion.

When you click on a tile of the displayed triangle, it's coordinates a
+nd it's numerical value are displayed in the Output Window.

Remember that the first row is 0 and the first column is also 0. The t
+riangle is constructed by summing the values of two adiacent position
+s in row and putting the result below them in the middle. The general
+ formula to calculate any given result in the triangle, given the til
+es coordinates, is also known as "n choose k" ...

               n!
 C(n,k) =  ----------
            k!(n-k)!


...where n is the row and k is the position, both counting from 0.



* Experiments Panel *

At the top you have the Properties Configuration: This allows the user
+ to determine and/or display:
 * How many rows to draw 
 * A button to display this introduction
 * At what point large numbers should be subtituted with dots (to buil
+d the shape of the triangle acceptable)
 * The abilty to enable debug information to be displayed on the scree
+n 
 * The size and boldness of both output displays
 * The Help Window 
 * The main creation or distruction control

Consider that building a bigger triangle requires bigger calculations:
+ You can draw a 127 (or more) row triangle in few seconds on a modern
+ calculator if you want. 
If this is the case consider that the values of any element in rows ar
+e cached by the main Perl program, so that following calculation will
+ use cached values with no speed penalty.

The next part is a number of experiments you can do with the aritmetic
+ triangle. 
The experiments looks very similar: 
 * All have some help associated (button '?')
 * A short description
 * An entry field
 * A color chooser
 * And the colorize/clear buttons



* Other properties of the triangle *

-The triangle is symmetrical.
-Some of the numbers in Tartaglia's triangle correlate to numbers in L
+ozanic''s triangle
-Imagine each number in the triangle is a node in a grid which is conn
+ected to the adjacent numbers above and below it. Now for any node in
+ the grid, count the number of paths there are in the grid (without b
+acktracking) which connect this node to the top node (1) of the trian
+gle. The answer is the number associated to that node.
-The only number that appears once is 2.
-All entries in row n are odd if and only if the binary representation
+ of n consists of 1s.
-If p is a prime, then every internal entry in row p ^ n (with n as an
+y positive integer) is divisible by p.



* Further readings and credits *

This software is written in Perl and would not be possible without the
+ aid of the community of www.perlmonks.org (just plagiarized some bit
+ from crazyinsomniac, Anonymous, helped by ambrus and many others).

If you want learn even more properties of the Tartaglia's triangle (se
+ems impossible but there are more) consider worth a visit to:

http://mathforum.org/mathimages/index.php/Pascal%27s_triangle
http://www.cut-the-knot.org/arithmetic/combinatorics/PascalTrianglePro
+perties.shtml
http://ptri1.tripod.com/
http://www.mathsisfun.com/pascals-triangle.html
http://mathworld.wolfram.com/PascalsTriangle.html

EOH
}
######################################################################
+##########
sub tart_icon {
    return <<EOI
/* XPM */
static char * Icon_xpm[] = {
"32 32 4 1",
"     c #000000000000",
"g    c #00FF00",
"X    c #FF0000",
"D    c #FFFF00",
"                                ",
"                                ",
"                                ",
"                                ",
"                                ",
"                                ",
"        XXXXXXXXXXXXXXXX        ",
"        XXXXXXXXXXXXXXXX        ",
"                                ",
"                                ",
"               XX               ",
"               XX               ",
"              XXXX              ",
"              XXXX              ",
"             XXXXXX             ",
"             XXXXXX             ",
"            XXXXXXXX            ",
"            XXXXXXXX            ",
"           XXXXXXXXXX           ",
"           XXXXXXXXXX           ",
"          XXXXXXXXXXXX          ",
"          XXXXXXXXXXXX          ",
"         XXXXXXXXXXXXXX         ",
"         XXXXXXXXXXXXXX         ",
"                                ",
"                                ",
"                                ",
"                                ",
"                                ",
"                                ",
"                                ",
" Discipulus as in perlmonks.org ",};
EOI
}

__DATA__
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