• maximum likelihood.

Maximum-likelihood methods have now been applied to most areas of macromolecular crystallography, including data reduction, molecular replacement, experimental phasing and refinement. However, students of macromolecular crystallography are predominantly taught only traditional crystallographic methods and therefore have little understanding of the methods underlying the modern software that they routinely use in structure determination. This situation arises, at least in part, because maximum likelihood is considered to be too difficult to be taught to students who lack substantial mathematical training within the limited time frame of undergraduate/graduate courses. A method of introducing maximum-likelihood concepts with the help of dice is described here and it is then shown how these concepts can form the core of understanding maximum-likelihood refinement, molecular replacement and experimental phasing. Within the framework described, the crystallographic maximum-likelihood techniques are all reduced to the same basic concepts and become easier and less time-consuming to teach than traditional methods, which rely on disparate concepts.