  Solving for the bound state eigenvalues of the Schr\"odinger equation is a
tedious iterative process when the conventional shooting or matching method is
used. In this work, we bypass the eigenvalue's dependence on the eigenfunction
by simply trying out all eigenvalues to a desired accuracy. When the eigenvalue
is known, the integration for the eigenfunction is then trivial. At a given
energy, by outputting the radial distance at which the wave function crosses
zero (the hardwall radius), this method automatically determines the entire
spectrum of eigenvalues of the radial Schr\"odinger equation without iterative
adjustments. Moreover, such a spherically symmetric "hardwall" can unmask
"accidental degeneracy" of eigenvalues due to hidden symmetries. We illustrate
the method on the Coulomb, harmonic, Coulomb+harmonic, and the Woods-Saxon
potentials.
