Use backtracking, aka the Perl regex machine. Give your set of number as command line arguments to the following program:

#!/usr/bin/perl

use strict;
use warnings 'all';
use re 'eval';

use vars '%seen';

my $regex   = <<'--';
   (?{ local $x = 0 })
   (?{ local @x = (-1) x @ARGV })
--

my $i = 0;
foreach my $number (@ARGV) {
    $regex .= "(?:(?{ local \$x = \$x + $number; local \$x [$i] = $i }
+)|)\n";
    $i ++
}

$regex .= <<'--';
 (?(?{ $x }) fail | )
 (?(?{ grep {$_ >= 0} @x }) | fail)
   (?{ local $str = join " + " => @ARGV [grep {$_ >= 0} @x] })
 (?(?{ $seen {$str} ++ })     fail | )
 (?(?{ print "$str = 0\n" })  fail | )
--

"" =~ /$regex/x;
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Abigail