A toy for one year olds consists of a series of holes and a bunch of objects to push through those holes. The holes come in shapes of circle, square, triangle, etc. And the objects are cylinder/prism versions of the holes. Usually when a child tries to push an object through the hole he or she usually twists the object until it goes through, although each object can be pushed through without twisting if the correct orientation is found.

What if instead of nice cylindrical objects, they were irregularly shaped but still convex. Assume that it is possible to get these irregular objects through the holes by twisting and manipulating them. Is it still possible to find an orientation that will allow the irregular objects through without twisting?