DNA has been used to build a variety of devices, ranging from those that are controlled by DNA structural transitions to those that are controlled by the addition of specific DNA strands. These sequence-dependent devices fulfill the promise of DNA in nanotechnology because a variety of devices in the same physical environment can be controlled individually. Many such devices have been reported, but most of them contain one or two structurally robust end states, in addition to a floppy intermediate or even a floppy end state. We describe a system in which three different structurally robust end states can be obtained, all resulting from the addition of different set strands to a single floppy intermediate. This system is an extension of the PX-JX(2) DNA device. The three states are related to each other by three different motions, a twofold rotation, a translation of approximately 2.1-2.5 nm, and a twofold screw rotation, which combines these two motions. We demonstrate the transitions by gel electrophoresis, by fluorescence resonance energy transfer, and by atomic force microscopy. The control of this system by DNA strands opens the door to trinary logic and to systems containing N devices that are able to attain 3(N) structural states.