  We have built a code to numerically solve the Riemann problem in ideal
magnetohydrodynamics (MHD) for an arbitrary initial condition to investigate a
variety of solutions more thoroughly. The code can handle not only regular
solutions, in which no intermediate shocks are involved, but also all types of
non-regular solutions if any. As a first application, we explored the
neighborhood of the initial condition that was first picked up by Brio & Wu
(1988) and has been frequently employed in the literature as a standard problem
to validate numerical codes. Contrary to the conventional wisdom that there
will always be a regular solution, we found an initial condition, for which
there is no regular solution but a non-regular one. The latter solution has
only regular solutions in its neighborhood and actually sits on the boundary of
regular solutions. This implies that the regular solutions are not sufficient
to solve the ideal MHD Riemann problem and suggests that at least some types of
non-regular solutions are physical. We also demonstrate that the non-regular
solutions are not unique. In fact, we found for the Brio & Wu initial condition
that there are uncountably many non-regular solutions. This poses an intriguing
question: why a particular non-regular solution is always obtained in numerical
simulations? This has important ramifications to the discussion of which
intermediate shocks are really admissible.
