The paper reports a small-scale, long-term pilot project designed to foster strategic and reasoning abilities in young primary school pupils by engaging them in a number of computer games, mainly those usually called mind games (brainteasers, puzzlers, etc.). In this paper, the objectives, work methodology, experimental setting, and tools used in the project are outlined, together with an analysis of some findings. In particular, we perform a brief analysis of some of the cognitive processes involved in playing with the computer games considered; we then discuss software features that, in our experience, help children tackle different cognitive tasks. The quantitative data collected during the pilot allow us, also, to take account of children's performance according to a number of different parameters, such as their level of achievement, the game's degree of difficulty and the type of data handled. Moreover, we reflect on the general impact of the project on children's reasoning abilities. The extent and duration of the study mean that, whilst the findings are not generalizable, they do offer insights into mechanisms underpinning basic strategic and reasoning skills as well as the educational potentialities offered by some of the existing computer games; they also point to some areas for further research.