This paper aims to show how the Logic of Game Theory can facilitate the structuring of games for the learning of mathematical concepts, in a way which is cognitively resonant with students' attitudes and epistemologically sound from the mathematical standpoint. We propose a kind of game, based on an inquiry approach to mathematics, called Digital Inquiry Game (DIG), the aim of which is to foster students' positive beliefs about their mathematical capabilities with regards to problem solving and to improve the way students are able to grasp the epistemic aspects of the mathematical knowledge in question. The main issues surrounding the theoretical background and the inspiring key constructs of the DIG are explained. The design of a DIG is validated through a case study concerning some properties of integers and a general divisibility criterion. Finally, some issues for further researches are considered.