Here is the code, commented and with follow the progress prints. There are ~35 code lines, the rest are comments, prints or whitespace.
As hinted at in my short poem, the secret (such as there is one) is to analyse the problem.
THREE must be a five digit number, but also it must be a 5 digit number that
ends with the same two digits. With this we remove 90% of the possibles
for the letters THRE, in fact there are a mere 27 in all as you can see if you run the code.
As we go we just push the possibilities for our letters into an array, we then split the letters out as required.
TEN uses the same T and E as found in THREE so we only
need to search for N within the constraint of Ts and Es found initially. Once we have
our possibles that satisfy THREN we look for ONE. Once again we only have to
look for O within the constraint of the Ns and Es already generated.
At this stage there are only six possible combinations of digits for THRENO.
We get rid of 4 as they contain the same digit for two or more letters to leave just two.
We then brute force the possibilities for SIX from the remaining 4 digits (only 4*3*2 = 24 cases)
to get the answer.
Tachyon
#!/usr/bin/perl w
use strict;
my $time = time();
my (%tri,@pos,@pos1,@pos2,@pos3);
# make hash of tiangular numbers 5 digits or less
# the 447th has 6 digits so we don't map past 446
map{$tri{.5*$_*($_+1)}=1}1..446;
# find all possible matches for 'three'
# these are 5 digits long, but last two digits are the same
# this allows us to limit the search
for my $key(keys %tri){
push @pos,$1 if $key =~/(\d\d\d(\d)\2)/;
}
# let's see how many possibilities we have
print "Initially we have ".@pos." possibles for \$t\$h\$r\$e\$e\n";
print "$_\n" for @pos;
# find all possible matches for 'ten' within constraint
# of $t and $e possibilities generated above, we are looking for 'n'
for (@pos) {
my($t,$h,$r,$e)=split'',$_;
for my $n(0..9) {
push @pos1, "$t$h$r$e$n" if defined $tri{"$t$e$n"}
}
}
# let's see how many possibilities we have left
print "\nNext we have ".@pos1." possibles for \$t\$h\$r\$e\$n\n";
print "$_\n" for @pos1;
#now look at 'one' in same way, we are looking for 'o'
for (@pos1) {
my($t,$h,$r,$e,$n)=split'',$_;
for my $o(0..9) {
push @pos2, "$t$h$r$e$n$o" if defined $tri{"$o$n$e"}
}
}
# let's see how many possibilities we have left
print "\nNow we have ".@pos2." possibles for \$t\$h\$r\$e\$n\$o\n";
print "$_\n" for @pos2;
# remove dulicates where digits for $t$h$r$e$n$o are not unique
# I'm sure there is something more elegant but this works
for (@pos2) {
$_ =~ /(.)(.)(.)(.)(.)(.)/;
push @pos3,$_ if $_=~m/[^$2$3$4$5$6][^$1$3$4$5$6][^$1$2$4$5$6][^$1
+$2$3$5$6][^$1$2$3$4$6][^$1$2$3$4$5]/;
}
# let's see how many possibilities we have left
print "\nAfter removing cases where we have duplicate digits\n";
print "we have ".@pos3." possible matches for \$t\$h\$r\$e\$n\$o\n";
print "$_\n" for @pos3;
# find the solution
for my $pos(@pos3) {
# get the remaining digits available for 'six'
# we erase the 6 digits we are currently using
# for t h r e n o
my $remaining = '0123456789';
for (split'',$pos) {$remaining =~ s/$_//;}
# look at the remaining cases
print "\nBrute forcing\n";
print "If \$t\$h\$r\$e\$n\$o\ is $pos then \$s\$i\$x must come fro
+m $remaining\n\n";
# brute force possibilities for six, it's only 4 digits
my @rem = split'',$remaining;
for my $s(@rem){
i: for my $i(@rem){
next i if $i==$s;
x: for my $x(@rem) {
next x if $x==$i or $x==$s;
if (defined $tri{"$s$i$x"}){
my($t,$h,$r,$e,$n,$o)=split'',$pos;
# prove we are right!
print "\nfound solution\n";
print "###################################\n";
print "one $o$n$e " if defined $tri{"$o$n$e"};
print "three $t$h$r$e$e " if defined $tri{"$t$h$r$
+e$e"};
print "six $s$i$x " if defined $tri{"$s$i$x"};
print "ten $t$e$n\n" if defined $tri{"$t$e$n"};
print "###################################\n\n";
} else {print "No match \$s\$i\$x > $s$i$x\n"}
}
}
}
}
$time = time()$time;
print "\nElapsed $time seconds\n";
If anyone has an elegant way of deleting the cases where we have duplicate characters I would love to see it.
My 2 line regex is functional, but fairly agricultural!
Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
Read Where should I post X? if you're not absolutely sure you're posting in the right place.
Please read these before you post! —
Posts may use any of the Perl Monks Approved HTML tags:
 a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)

For: 

Use: 
 &   & 
 <   < 
 >   > 
 [   [ 
 ]   ] 
Link using PerlMonks shortcuts! What shortcuts can I use for linking?
See Writeup Formatting Tips and other pages linked from there for more info.