A puzzle assembly is shown and comprises a multiplicity of geared links that are arranged such that their turning axes are oriented along the edges of a polyhedral shape so that when a single link is turned, all links turn synchronously. In the assembly, there are two end positions of the geared links. In each end position, a set of link groups are formed, each of those groups corresponding to one face of the polyhedral shape so that each link group forms a circular track that is inscribed on that particular face. When the puzzle assembly is moved to a second end position, a new set of new link groups is formed corresponding to a different a different set of faces of the polyhedral shape. The puzzle assembly further comprises a number of "petal elements" equal to the number of geared links. The petals have a faceted form and a base piece that extends from that form for providing a sliding contact with the geared links. Each petal element is retained by a geared link. When the puzzle assembly is in one end position, a group of petals corresponding to a polyhedral face touch each other to form a pyramidal shape. Thus, one-half of their surfaces are exposed and one-half of their surfaces are hidden. When the puzzle assembly is "flipped"--the geared links are turned to their other end position--the petals recombine to hide the previously visible surfaces and to reveal the previously invisible surfaces. Importantly, in either end position, the group of petals corresponding to a polyhedral face may be twisted around the center point of that face, thereby changing the correspondence between the geared links and the petal elements.