  A benchmark problem for numerical relativity has been the head-on collision
of two black holes starting from the ``Misner initial data,'' a closed form
momentarily stationary solution to the constraint equations with an adjustable
closeness parameter $\mu_0$. We show here how an eclectic mixture of
approximation methods can provide both an efficient means of determining the
time development of the initial data and a good understanding of the physics of
the problem. When the Misner data is chosen to correspond to holes initially
very close together, a common horizon surrounds both holes and the geometry
exterior to the horizon can be treated as a non-spherical perturbation of a
single Schwarzschild hole. When the holes are initially well separated the
problem can be treated with a different approximation scheme, ``the
particle-membrane method.'' For all initial separations, numerical relativity
is in principle applicable, but is costly and of uncertain accuracy. We present
here a comparison of the different approaches. We compare waveforms, for
$\ell=2$ and $\ell=4$ radiation, for different values of $\mu_0$, from the
three different approaches to the problem.
