The direct kinematics of several planar mechanisms, including 3RPR planar platform robots, reduce to the problem of assembling a 3RR planar structure, also known as the pentad mechanism. A general pentad has at most six isolated real assemblies, which are the solutions of a system of polynomial equations. As the parameters of the mechanism are specialized, the nonsingular roots can merge to form singular roots of multiplicity up to six. While classical results concerning four-bars can be used to derive 3RR mechanisms with multiplicity as high as four, the results presented here are the first to provide a complete solution to the case of multiplicity six, thereby solving a problem formulated by the second author. In such a configuration, the idealized mechanism still has an isolated root but, in practice, when small deformations or joint clearances are allowed, the mechanism can move through a large displacement.