Let's see... if I use the 5, then I have to form a sum of 16 with the remaining numbers. But if I don't use the 5, then... (is anything starting to click?)

Not that this is going to help you, but here is naive approach using modules available from CPAN. The idea is simply to iterate through all the subsets of the values finding those which satisfy the summation constraint.

use Algorithm::Combinatorics qw(subsets);
use List::Util qw(sum);

my @data = (5, -6, 8, 10, 12, 3, 10);
my $iter = subsets(\@data);
while (my $subset = $item->next) {
  if (sum(@$subset) == 21) { print "found a solution: @$subset\n" }
}
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