By the way, you can improve your analyze function quite a bit. This piece in particular stood out to me:

   for ( 0 .. 9999 ) {
        my $k = 0 x ( 4 - length $_ ) . $_;
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Even if you had to use a for loop, you could have just written it as for ( '0000' .. '9999' ) instead. Isn't Perl great? You don't even need the for though. I wrote my analyze function like this:

sub analyze {
    my $number = shift;
    my $length = length $number;
    my $duplicates = 0;
    my %seen;

    while (length $number >= 4) {
        $seen{substr($number, -4)}++ and $duplicates++;
        substr($number, -1, 1, '');
    }

    my $missing = 10000 - keys %seen;
    return ($length, $missing, $duplicates)
}
[download]

-sauoq
"My two cents aren't worth a dime.";