Open Problems and Exercises on Words and Languages *

The goal of this presentation is to collect several open problems on words and (finite) languages. The problems are not intended to be new. On the other hand, all of these problems are very easily stated and in that respect extremely natural. Moreover, the choice of these problems reflects the interests of the author – each of those have some connections to the work of the author. In other words, these belong to the list of favourite open problems of the author. The number of problems discussed is sixteen. They belong in a rather natural way to four different groups. Namely, they are either problems on morphisms of free semigroups, problems on equations on words, problems on integer-valued matrices, or problems on finite or regular languages. Some history of the problems is discussed, and in particular several solutions of special cases of the problems are stated as exercises. Attempts are made to choose these exercises as nice, rather easily provable results which, however, are not trivial. In a few cases these subproblems – which look as harmless exercises – are, in fact, open problems. We interpret this as a challenging nature of our problems.