Construction problems—why and how?

Abstract

This paper discusses the nature of construction problems and their pedagogical value from a modern point of view. The main observation is that a solution of a construction problem is really a constructive proof of a geometrical existence theorem. We explain this idea in the first four sections, and we show, with detailed examples, how to translate any construction problem into a modern language, without even mentioning rulers and compasses. According to this translation, construction problems are logically more complex than the usual mathematical claims of high school but less complex than the ordinary theorems of analysis. They can be used, therefore, for bridging the gap between the two levels. Another possible use, as an example of a non‐numeric algorithm, is suggested in §5, where a different presentation, this time in the language of computer science, is given. The close connection between the two presentations is then emphasized. Section 7 gives some further operative conclusions, while § 8 discusses the different meanings of the ‘numbers of solutions' concept, as well as the difficult problems connected with it. Finally, the last section suggests practical methods for solving construction problems, which are strongly related to basic ideas of AI, such as, reducing a goal to subgoals and search.