An exploratory computational study of the reflection of an inward-facing conical shock wave from its axis of symmetry is presented. This is related to more complex practical situations in both steady and unsteady flows. The absence of a length scale in the problem studied makes features grow linearly with time. The ensuing flow is related to the Guderley singularity in a cylindrical imploding shock. The problem is explored by making a large number of computations of the Euler equations. Distinct reflection configurations are identified, and the regions of parameter space in which they occur are delineated.