if ($num =~ /^[^\D1]\d+$/) {
   print "does not begin with a 1\n";
}
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The key to understanding this is that you are looking for anything that is not a non-digit or a 1. When you negate the negations in the above phrase, you'll see that it means you're matching 0, 2-9.

uh, update: that doesn't quite work, hang on a sec...

if ($num =~ /^(?:[^\D1]|1\d)\d*$/) {
   print "not a 1\n";
}
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2nd update: now it works. It is instructive to see how the pattern had to be modified to exclude 1, but allow 10, 19, 1111, ...