Code seems to work, but it is better in PERL way as below.

use strict;
use warnings;

print "RegEx Engine 1.0\n________________\n";
print "Gimme a string: ";
my $str = <STDIN>;
print "Gimme a RegEx: ";
my $pattern = <STDIN>;
my $answer = ($str =~ /$pattern/);
if ($answer) {
    print "Yes!";
} else {
    print "No.";
}
print "\nkthxbye\n";
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Code Change at line "my $answer = ($str =~ /$pattern/);" will look for the pattern in $pattern in the string $str and will result in yes or no.