Beefy Boxes and Bandwidth Generously Provided by pair Networks
Perl-Sensitive Sunglasses
 
PerlMonks  

comment on

( [id://3333]=superdoc: print w/replies, xml ) Need Help??
       Dedicated to my father who studied the other Tartaglia
After more then one month of sparetime works and 35 subversion i'm very happy to present you:

16 fun experiments with the Tartaglia's triangle

This is a Perl Tk program that shows many of the properties of such incredible triangle: you can modify the aspect of the triangle itself and of the output window and of the help pages too.

In Italy the name of the arithmetic triangle is dedicated Tartaglia so I want to present with this name.

I'm not a mathematician and the math used in the code is something late Middle Age, but works.

If someone wish to improve this program i will be very happy: inernal math used, better explication in output windows, or even typos spot(i'm not english native, as you can guess) or suggestion are welcome. In fact i wish this program to be used in educational context.

Have fun!

L*


Update 1/07/2014: commented lines 188-190 and 555 (printing debug info for windows dimensions and positioning).

L*

#!/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 = 2; # 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 = 13; my $bold_help = 1; my $size_out = 13; 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("688x861+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 12 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)); $tart_win->geometry("300x450+708+0"); $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+708+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 in the form of 'row column'. The + row, the column and the tile will be colorized with three different +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 g +roups are possible grouping a set of X (row) by Y (column)?'. This is called combination (or k-combination) in mathematic, id est no + matter of the order of the elements and no repetition of elements. The formula is the binomial coefiicent's one. n! C(n,k) = ---------- k!(n-k)! If an element can be found more times we speack of 'combination with r +epetition' (or k-multicombination). The formula is linked to 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, in respect to one without repetitions, will be at th +e same column but the row will be not 'n' but 'n + k - 1'. EOH } ###################################################################### +########## sub help_tri { return <<EOH * Triangular numbers, polygonal numbers and figurates ones * USAGE: just put in the entry box a number. The correspondent triangula +r number will be colorized and all the diagonal of triangulars number +s too. Triangulars numbers are a subset of polygonal numbers that are a subse +t of figurates ones. If you can arrange a number of dots forming a regular triangle, then s +uch number is a triangular one. In the same way if you can form a squ +are you have a 'square number', 'pentaghonal number' if you can form +a pentaghon 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 +correspondent triangular one. The nth s-gonal number P(s,n) is related to the triangular one 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. Ipsicle, in the II century BC, had found the relation between polygona +l numbers an arithmetic progressions. 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 the Tartaglia's triangle is compose +d by many 1. The second column contain counting numbers, while the 3th triangular o +nes(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 triangular base and +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) Can I hazard that counting numbers are figurate numbers of 1 dimension + and the 1s serie is a serie of 0 dimension figurate one? 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 sumation + of squares. You'll see that the sumation of squares of 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 tile and will be demondstrated that thi +s number is equal to the summation of all numbers in the parallelogra +m excluded by the two diagonals crossing at the given position. EOH } ###################################################################### +########## sub help_hockey { return <<EOH * Hockeystick pattern * USAGE: give the coordinates of tile and will be demondstrated that thi +s number is equal to the summation of all numbers in the prior diagon +al up to the same position of the given number. EOH } ###################################################################### +########## sub help_sie { return <<EOH * Sierpinski's fractals * USAGE: just put in the entry box a number. Every tiles will be coloriz +ed if divisible by number given Selecting tile by divisibilty criteria can draw a pattern tending to a + Sierpinsky triangle. With different numbers you will obtains differn +t 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 * The arithmetic triangle is called in Italy Tartglia's triangle, becaus +e exposed in the "General trattato di numeri et misure" written in 1 +556 by Niccolò Fontana (1499 ca, Brescia 13 December 1557, Venice), k +nown 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 drow it as right triangle) in m +any other countries was known by Halayuda, an Indian commentator, in +10th century, sudied 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 it mind i'm not a mathematician, i was only im +pressed by the huge amount of things you can see in the triangle and +i want to show them. Many useful things about the tartaglia's triangle are shown using the +experiment panel, others are enumered at the end of this introduction +. When you click a tale of the triangle it's coordinates and it's numeri +cal value are printed on the output window. Remember that the first row is 0 and the first column is also 0. The t +riangle is constructed summing the values of two adiacent position in + row and putting the result, below them, in the middle. The general f +ormula to calculate any given number in the triangle given the coordi +nate 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 top you have the properties configuration: how many rows to draw, a + remainder to this introduction, when subtitute big numbers with dots + (to build the shape of the triangle acceptable), the possibilty to e +nable debug information to be printed on the screen, size and boldnes +s of both output and helps windows and the main creation or distructi +on control. Consider that build a bigger triangle requires bigger calculation: you + can draw a 127 (or more) rows triangle in few seconds on a modern ca +lculator, if you want. If this is the case consider that values of an +y element in rows are cached by the main Perl program, so that follow +ing 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 ass +ociated (button '?'), a short description, an entry space, a color ch +ooser 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__

There are no rules, there are no thumbs..
Reinvent the wheel, then learn The Wheel; may be one day you reinvent one of THE WHEELS.

In reply to Tk Tartaglia's triangle fun - Pascal's triangle fun by Discipulus

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post; it's "PerlMonks-approved HTML":



  • Are you posting in the right place? Check out Where do I post X? to know for sure.
  • Posts may use any of the Perl Monks Approved HTML tags. Currently these include the following:
    <code> <a> <b> <big> <blockquote> <br /> <dd> <dl> <dt> <em> <font> <h1> <h2> <h3> <h4> <h5> <h6> <hr /> <i> <li> <nbsp> <ol> <p> <small> <strike> <strong> <sub> <sup> <table> <td> <th> <tr> <tt> <u> <ul>
  • Snippets of code should be wrapped in <code> tags not <pre> tags. In fact, <pre> tags should generally be avoided. If they must be used, extreme care should be taken to ensure that their contents do not have long lines (<70 chars), in order to prevent horizontal scrolling (and possible janitor intervention).
  • Want more info? How to link or How to display code and escape characters are good places to start.
Log In?
Username:
Password:

What's my password?
Create A New User
Domain Nodelet?
Chatterbox?
and the web crawler heard nothing...

How do I use this?Last hourOther CB clients
Other Users?
Others taking refuge in the Monastery: (4)
As of 2024-04-20 10:10 GMT
Sections?
Information?
Find Nodes?
Leftovers?
    Voting Booth?

    No recent polls found