Thanks for posting this inspiring problem. Similar to what tybalt89 has posted, one can significantly reduce the search space by checking whether for a given number in a given position the remaining numbers would still fit. Below some code to do that based on positions counting from 0 to 5 and each can be vertical or horizontal. Given the output it is nearly trivial to solve the puzzle manually.

use strict;
use warnings;

my @numbers = qw( 113443 143132 241131 321422 323132 331222
                  341114 412433 414422 431331 443112 444313 );
                
# find possible positions
my %positions;
for my $n (@numbers) {
    $positions{$n} = [];
    # frequency of digits in current number
    my @nfreq = (0) x 5;
    $nfreq[$_]++ for split //, $n;
    # check which position is possible
    for my $p (0..5) {
        # frequency of digits in current position w/o current number
        my @freq = (0) x 5;
        $freq[substr( $_, $p, 1 )]++ for @numbers;
        $freq[substr( $n, $p, 1 )]--;
        # check if position is feasible
        # ie enough of each digit available
        my $possible = 1;
        $freq[$_]<$nfreq[$_] and $possible = 0 for 1..4;
        push @{$positions{$n}}, $p if $possible;
    }
}

for my $n (sort { scalar(@{$positions{$a}}) <=> scalar(@{$positions{$b
+}}) } @numbers) {
    print "Number $n can be at positions @{$positions{$n}}.\n";
}
[download]