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Selection of co-belonging fragments from the numerous ceramic findings of an archaeological excavation remains a difficult process of questionable effectiveness, based exclusively on the experience and patience of the conservators. While the screening of the fragments is a central prerequisite and the most important stage of the process of vase reconstruction, established methods based on scientific criteria and guaranteed efficiency for the detection of co-belonging ceramic fragments suggested in the bibliography, do not exist. On the contrary, for methods dealing with the assembly of vases from co-belonging fragments, which is a secondary process that can be done more easily and effectively in an empirical way, there exist numerous studies based on fragment morphology. However, even these are also not implemented because of the time requirements, sheer volume and complexity of the proposed methods, in order for them to be applicable in practice. The proposed methods in this paper are based on thermoremanent magnetization (A/m), which is calculated from the weak magnetic field measurements by a fluxgate-sensor/magnet apparatus forming a three-dimensional orthogonal system. Experimental measurements from fragments of 6 vases show that the magnetization magnitude of co-belonging fragments display similar values, despite the magnetic anisotropy of the ceramic material, since these belong to vases that are made of the same clay and fired under the same conditions. This is the criterion for finding ceramic fragments of the same vase from archaeological excavations. The thermoremanent magnetism directionality of fragments, which is aligned along the geomagnetic field at the same place and time during the vase firing process, as it is configured by their rotational symmetry, defines the position of the fragments on the body of the 6 vases. The shape of the original vase can be reconstructed when only a few non adjacent fragments are available. The proposed measurement apparatus can be used for the construction of a useable portable magnetometer specialized for ceramic surface measurements to achieve the above objectives.