  Random $K$-satisfiability ($K$-SAT) is a paradigmatic model system for
studying phase transitions in constraint satisfaction problems and for
developing empirical algorithms. The statistical properties of the random
$K$-SAT solution space have been extensively investigated, but most earlier
efforts focused on solutions that are typical. Here we consider maximally
flexible solutions which satisfy all the constraints only using the minimum
number of variables. Such atypical solutions have high internal entropy because
they contain a maximum number of null variables which are completely free to
choose their states. Each maximally flexible solution indicates a dense region
of the solution space. We estimate the maximum fraction of null variables by
the replica-symmetric cavity method, and implement message-passing algorithms
to construct maximally flexible solutions for single $K$-SAT instances.
