I don't have any data to back this up, but I've always thought that the bulk of the benefits from learning a game -- improving problem solving, pattern recognition, strategic thinking -- come late in the beginning of the process, just after the rules are internalized. If that's true (and maybe even if it isn't), you might be able to extend that period of intense learning by modifying a game so that old rules are seen in new ways.
Case in point, Gliński's hexagonal chess.
Gliński's chess variant is hugely popular in Europe (more than 100,000 sets have been sold). You can get the rules at my Kruzno site, but you can probably figure most of them out for yourself. The only pieces that might give you trouble are the bishops and,to a lesser extent, the knights.
Bishops come in three colors, which points out an interesting topological feature of a hexagonal grid which I'm betting you can spot for yourself.
It's a strange and intriguing game and yet another reason why every house should have a hexboard.