in reply to Re^2: Dueling Flamingos: The Story of the Fonality Christmas Golf Challenge in thread Dueling Flamingos: The Story of the Fonality Christmas Golf Challenge
It's unfortunate that those golfers did feel so very disadvantaged about not being 'in the know' about Ton's discovery. But I have to agree with Ton's words on this:
Nothing stops anyone from thinking of the magic formula for himself.
At this point, it might be worth mentioning that there are vastly many Tonlength or nearTonlength formulae, waiting to be had by anyone who had thought to look for one. My (poorly written) brute forcer found several almost immediately:
"12e$&"%109082 # I=0, V=[89], X=[347]
"12e$&"%109083 # I=0, V=7, X=[649]
"12e$&"%109086 # I=0, V=4, X=[35]
"14e$&"%87498 # I=0, V=2, X=6
...
the last of which is equal in length to Ton's. Of course, the most difficult part of finding a magic formula is not writing the search algorithm, but knowing what to search for in the first place. Given that you know that you will be transliterating digits for numerals (admittedly, a logical step that I was spared), a few things should be immediately apparent:
 The base of the expression should increase by a factor of 10 for each digit. This follows directly from the regularity (the adding of one digit) expressed from 03, and again from 58 (Ton's expression increases by a factor of 10 plus a constant, but the result is similar).
 The modulo should fall between the value for 3, and the value for 4. This follows directly from the fact that 4 breaks the above mentioned regularity.
 The value for 5, after the modulo, should be smaller than the modulo/1000. This follows directly from the fact that the regularity beginning with 5 needs to continue until 8.
The most obvious formulation of this is "XXe$&"%YYYYYY. Had none of these worked (instead of the vastly many that do), I'm not sure if I would have thought of string repetition as a suitable candidate as Ton did, but probably not. At least, not in any short period of time.
As an aside, there are no other solutions of the form that Ton used (1x$&*XX where Ton's XX is 40). It seems that his solution truly is a one of a kind!
Re^4: Dueling Flamingos: The Story of the Fonality Christmas Golf Challenge by eyepopslikeamosquito (Chancellor) on Dec 18, 2012 at 10:12 UTC 
As an aside, there are no other solutions of the form that Ton used (1x$&*XX where Ton's XX is 40).
It seems that his solution truly is a one of a kind!
In addition to the one used in the 2006 Fonality golf challenge:
s!.!y$IVCXL426(:$XLMCDIVX$dfor$$_.=5x$&*8%29628
don't forget about Ton's original one
of equal length:
s!.!y$IVCXL9180$XLMCDXVIII$dfor$$_.=4x$&%1859^7
used in the 2004 Polish golf tournament.
Update: Here is a test program to verify that all four magic formulae are correct:
use strict;
use Roman;
sub ton1 { my $t = shift; my $s;
($s.=4x$_%1859^7)=~y/IVCXL9180/XLMCDXVIII/d
for $t=~/./g; return $s }
sub ton2 { my $t = shift; my $s;
($s.=5x$_*8%29628)=~y/IVCXL426(:/XLMCDIVX/d
for $t=~/./g; return $s }
sub pmo1 { my $t = shift; my $s;
($s.="32e$_"%72726)=~y/CLXVI609/MDCLXVIX/d
for $t=~/./g; return $s }
sub pmo2 { my $t = shift; my $s;
($s.="57e$_"%474976)=~y/CLXVI09/MDCLXIXV/d
for $t=~/./g; return $s }
for my $i (1..3999) {
my $r = uc roman($i);
my $t1 = ton1($i);
my $t2 = ton2($i);
my $p1 = pmo1($i);
my $p2 = pmo2($i);
print "$i: $r\n";
$r eq $t1 or die "t1: expected '$r' got '$t1'\n";
$r eq $t2 or die "t2: expected '$r' got '$t2'\n";
$r eq $p1 or die "p1: expected '$r' got '$p1'\n";
$r eq $p2 or die "p2: expected '$r' got '$p2'\n";
}
print "all tests successful\n";
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I tried many magic formulas, but this happens to be one of the first
ones I tried since the $m x $& tends to multiply the result by 10 each
time, so getting a result that's one longer each time, about the only
somewhat regular pattern in roman numerals.
 Ton, remarking on how he found his magic formula back in 2004
I suppose if I had done my research better, I would have known that Ton had reached his result based on exactly the same methodology that I posted above. But seriously:
4x$&%1859^7
xor 7? Now that's just crazy.
 [reply] [d/l] 
