The mathematician Herbert Hauptman took part in 5 consecutive Lindau Chemistry Meetings, but only gave lectures at the first four. These lectures all concern the so-called phase problem of X-ray crystallography, the problem on which Hauptman had worked since around 1950, partly together with the physical chemist Jerome Karle. Together they had published a set of texts describing a way of handling this problem practically and it was for this work that they received the 1985 Nobel Prize in Chemistry together. The phase problem of X-ray crystallography was thought to imply that the direct inversion of experimental data into crystal structure is strictly impossible from a fundamental mathematical viewpoint. Therefore scientists historically used different methods to try to overcome this difficulty. For small simple crystals, it has been often been enough to extract certain crystal parameters out of the experimental data. For larger and more complex crystals, methods of changing the crystal structure by insertion of heavy atoms and comparing diffraction data with and without insertions have been (and are still) used. What Hauptman and Karle showed, is that the knowledge that crystals are made up of atoms, is enough to overcome the phase problem. Using this knowledge, they developed a probabilistic method which is particularly suited to medium complex crystals and which relies heavily on the use of computer calculations. In all his four Lindau lectures, Hauptman gives clear and pedagogical presentations and it is really a pity that we don’t have his equations. But Hauptman’s Nobel Lecture given in Stockholm in 1985 concerns the same phase problem and can be found on the web site of Nobelprize.org. If you are seriously interested in Hauptmans’s lecture, I recommend that you to look it up!