Here, I figured out what was wrong.
The $added variable was unnecessary:

use strict;
use warnings;
my $i;
my $j;
my $cubed;
my @num;
my @numbers;

for $i(100..999){
    undef($cubed);
    undef(@num);
    @num=split(//, $i);
    for $j(@num){
        $cubed+=$j**3;
        }        
    if($cubed==$i){
        push @numbers, $i;
        }
    }
print "$_\n" for @numbers;
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Thanks for the help!