The wedge paradox for stress–free boundaries is the wedge loaded by a concentrated moment at the vertex (Carothers problem). The paradox consists of the fact the for wedges smaller than a half–space the solution exists, while for wedges bigger than a half–space as shown by Sternberg–Koiter the solution does not exist. We examine the possiblity of a paradox for clamped–clamped boundary conditions. We obtain the solution for a clamped–clamped wedge loaded by a concentrated couple at an interior point and show that for wedges smaller than a half–space the limit is zero as the moment approaches the vertex (i.e. the clamped wedge absorbs the concentrated couple), for a half–space it is a constant, and for angles bigger than a half–space the limit does not exist (paradox). Moreover, from the solution for the clamped wedge, taking the limit as Poisson's ratio tends to one, the solution for the stress–free wedge is retrieved (Carothers) according to the correspondence between cavities and rigid inclusions established by Dundurs in 1989 and Markenscoff in 1993.