A friend of mine recently asked me for some suggestions for games or other activities for children that would help promote quantitative reasoning or mathematical skills, while remaining fun to play (i.e. more than just homework-type questions poorly disguised in game form). The initial question was focused on computer games (and specifically, on iPhone apps), but I think the broader question would also be of interest.
I myself have not seriously played these sorts of games for years, so I could only come up with a few examples immediately: the game “Planarity“, and the game “Factory Balls” (and two sequels). (Edit: Rubik’s cube and its countless cousins presumably qualify also, due to their implicit use of group theory.) I am hopeful though that readers may be able to come up with more suggestions.
There is of course no shortage of “educational” games, computer-based or otherwise, available, but I think what I (and my friend) would be looking for here are games with production values comparable to other, less educational games, and for which the need for mathematical thinking arises naturally in the gameplay rather than being artificially inserted by fiat (e.g. “solve this equation to proceed”). (Here I interpret “mathematical thinking” loosely, to include not just numerical or algebraic thinking, but also geometric, abstract, logical, probabilistic, etc.)
[Question for MathOverflow experts: would this type of question be suitable for crossposting there? The requirement that such questions be "research-level" seems to suggest not.]