The complex and articulated ball is one of the popular artifacts for collections and is highly expandable and collapsible. This paper investigates the mechanism structure of this magic ball by hypothetically decomposing the mechanism into kinematic loops and chains and subsequently into basic kinematic subchains, leading to the analysis of the mobility of the ball mechanism. Dismantling those kinematic chains which do not contribute to the mechanism mobility, the magic-ball mechanism is decomposed into a number of kinematic chains, and further disintegrated into two distinct types of elementary platforms with three and four legs respectively. The theory of the relationship of a screw system and its reciprocal system is then used to identify both common constraint and virtual constraint in each of the elementary platforms and to examine the mobility of those platforms. The analysis is then extended to the mobility analysis of the closed-loop circular kinematic chain and supplementary chains. A systematic analysis is hence produced in mechanism decomposition and in the analysis of virtual constraints. The paper produces a theoretical basis for the mobility analysis of the mechanism using mechanism decomposition and screw system analysis.