We describe our approach that derives reliable 6-DOF information including the translation and the rotation of a rigid marker in a 3D space from a set of insufficient 5-DOF measurements. As a practical example, we carefully constructed a prototype and its design and evaluated it in our 3D dexterous motion-tracking system, IM6D, which is our novel real-time magnetic 3D motion-tracking system that uses multiple identifiable, tiny, lightweight, wireless, and occlusion-free markers. The system contains two key technologies; a 6-DOF computation algorithm and a marker design for 6D marker. The 6-DOF computation algorithm computes the result of complete 6-DOF information including translation and rotation in 3D space for a single rigid marker that consists of three LC coils. We propose several possible approaches for implementation, including geometric, matrix-based kinematics, and computational approaches. In addition, we introduce workflow to find an optimal marker design for the system to achieve the best compromise between its smallness and accuracy based on the tracking principle. We experimentally compare the performances of some typical marker prototypes with different layouts of LC coils. Finally, we also show another experimental result to prove the effectiveness of the results from the solutions in these two problems. Concepts: Human-centered computing ! Graphics input devices; Virtual reality;.