Revisiting symbolic addition: a step-by-step introduction to manual direct methods

Since the mid-1980s, direct methods have become the standard approach for solving crystal structures. However, because the software is now highly automated, the underlying concepts are often less transparent to beginners. This paper therefore revisits the symbolic addition method as a pedagogical framework for understanding phase determination in crystallography. The method is introduced by providing a brief explanation of triplet phase relationships derived from Sayre's equation, as well as their probabilistic interpretation within the formalism of direct methods. This is followed by three fully worked, two-dimensional examples of increasing complexity, which illustrate the method's underlying principles. Starting from subsets of strong, normalized structure factors, the examples describe the complete symbolic addition procedure, including fixing the origin, propagating the phase algebraically through triplet relationships and evaluating trial phase sets using Fourier synthesis. The shown examples provide a transparent, step-by-step demonstration of how phase relationships lead from diffraction intensities to interpretable Fourier maps. Thus, this method and the provided materials can be used as an effective teaching tool to show the principles behind modern, direct methods for determining the atomic structure of crystals.