<p>In recent work, we have studied the motion of a point mass on a horizontally vibrating frictional table. For a specific open-loop planar motion of the table, the point mass spirals in towards the table center. Superposed on the spiraling-in motion of the point mass are slow circular oscillations of moderate amplitude and fast oscillations of small amplitude. In our prior work, the fast oscillations were averaged out using the method of multiple scales, yielding a nonlinear system with slow parametric forcing. In this work, we present an informal and numerically-aided reduction of the slow dynamics of the point mass to the smallest possible number of states, namely two. Our approach uses two main analytical tricks. One is an initial generalized harmonic balance approximation wherein the number of states is increased, but their solutions have usefully slow variations. The second trick is to identify slowly varying state variables such their rates of change can be approximated as zero, to obtain lower order dynamics. Since these reductions are made without appealing to any predefined small parameters, the method rests on finding coordinate transformations that reveal slowly varying quantities. These coordinate transformations are themselves prompted by numerical solutions. Although some intermediate expressions are long, the final reduced model is compact, has the smallest possible number of states, and matches full numerics well.</p>