The paper once again deals with the mobility analysis methodology of mechanisms based on constraint screw. It is a very important subject in mechanism theory lasting about 150 year. Firstly, the paper introduces some background and then the methodology. The paper focuses on analysis of some “paradoxical mechanisms”, including classical mechanism and modern parallel mechanisms, such as Bennett mechanism, Delta robot, Tsai’s CPM mechanism and so on. All the results prove that for the difficult mobility problem the analysis procedures are quite simple and convenient, and will easily be held by numerous mechanical engineers. The complicated problem is solved only by using a pencil, a sheet of paper and just for a few minutes.