It is well known that several classical geometry problems (e.g., an- gle trisection) are unsolvable by compass and straightedge construc- tions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to an intuitive idea of a geomet- ric construction as a special kind of an “algorithm” using restricted means (straightedge and/or compass). However, the formalization is not obvious, and different descriptions existing in the literature are far from being complete and clear. We discuss the history of this notion and a possible definition in terms of a simple game.