Summary: As a sequel to the previous two papers of the second author [Czech. Math. J. 41(116), No. 2, 249-259 (1991; Zbl 0738.20061) and Contributions to general algebra 9, 133-150 (1995; Zbl 0889.20043)], we investigate the structure of medial idempotent groupoids by \(p_n\)-sequences. To complete the series of research, this paper has three purposes. First, we summarize some results in the previous papers so that this paper can cover the materials presented there. Secondly, using earlier results, we prove a few theorems which show the importance of the medial law in controlling the growth of \(p_n\)-sequences of groupoids. Finally, we state some problems and conjectures raised during the series of research.