Some more micro-optimization, proably getting into foolish optimization at this point. Reduced unnecesary copying of variables, performing some calculations used many times outside the loop, added a special check for sequences that start with n.

The line checking for sequences starting with n needs some explanation. It is probably the most obscure.

There are two givens for this algorithm.

- If ($next == $length) then ($level + 1 == $length).
- $maxflips DOES NOT get reset between runs.

So: if the first character is n then the minimum $maxflips{n} is $maxflips{n-1} + 1. It is impossible to have a sequence to take less than length - 1 flips, so if $length - 1 is less than $maxflips{n-1} then it impossible for a sequence that starts with n to have more than $maxflips{n-1} + 1. Therefore, no need to check them.

Rather than compare $length to $maxflips + 1, since $level == $length - 1, I save the calculation in the loop and compare $level to $maxflips.

`sub fannkuch {
my ( $copy, $level, $split ) = ( @_, 0, 1 );
my ( $index, $next ) = ( $level, $level + 1 );
my $length = length($copy);
if ($next == $length) {
($index, $split) = ($split - 1, $level);
substr($copy, $index, 0) = chop($copy);
}
do {
if ($next == $length) {
unless ( ord($copy) == $length and $level < $maxflips )
{
my $q = $copy;
my ( $k, $flips );
for ( $flips = 0; ( $k = ord( $q ) ) != 1; $flips++ )
+{
substr( $q, 0, $k ) = reverse substr( $q, 0, $k );
}
if ( $flips >= $maxflips ) {
if ( $flips == $maxflips) {
push @max_sequence, $copy;
}
else {
$maxflips = $flips;
@max_sequence = ($copy);
}
}
}
}
else {
fannkuch( $copy, $next, $split );
}
substr( $copy, $index - 1, 2 ) = reverse substr( $copy, $index
+ - 1, 2 );
$split = $next if $index == $split;
} while $index--;
return $maxflips;
}
`

Side by side with your code: It is only about a 4-6% increase, but it is repeatable.

`Your sub My sub
Pfannkuchen(1) = 0 for: Pfannkuchen(1) = 0 for:
1 1
0.000351 elapsed seconds. 0.000274 elapsed seconds.
Pfannkuchen(2) = 1 for: Pfannkuchen(2) = 1 for:
21 21
0.000196 elapsed seconds. 0.00016 elapsed seconds.
Pfannkuchen(3) = 2 for: Pfannkuchen(3) = 2 for:
231 231
312 312
0.000275 elapsed seconds. 0.000225 elapsed seconds.
Pfannkuchen(4) = 4 for: Pfannkuchen(4) = 4 for:
2413 2413
3142 3142
0.000348 elapsed seconds. 0.000288 elapsed seconds.
Pfannkuchen(5) = 7 for: Pfannkuchen(5) = 7 for:
31452 31452
0.000701 elapsed seconds. 0.000634 elapsed seconds.
Pfannkuchen(6) = 10 for: Pfannkuchen(6) = 10 for:
365142 365142
415263 415263
416523 416523
456213 456213
564132 564132
0.00376 elapsed seconds. 0.003357 elapsed seconds.
Pfannkuchen(7) = 16 for: Pfannkuchen(7) = 16 for:
3146752 3146752
4762153 4762153
0.025629 elapsed seconds. 0.023704 elapsed seconds.
Pfannkuchen(8) = 22 for: Pfannkuchen(8) = 22 for:
61578324 61578324
0.226504 elapsed seconds. 0.209445 elapsed seconds.
Pfannkuchen(9) = 30 for: Pfannkuchen(9) = 30 for:
615972834 615972834
2.209246 elapsed seconds. 2.088572 elapsed seconds.
Pfannkuchen(10) = 38 for: Pfannkuchen(10) = 38 for:
59186210473 59186210473
24.218115 elapsed seconds. 23.077981 elapsed seconds.
`

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