I spent time with my nephew over the weekend, he's 5 and we just, as of yesterday, taught him how to play checkers.
It was extremely interesting from a psych/cognitive developmental point of view. You could watch him learn a rule, try and apply the rule all the time, and learn its limits. He also practiced taking turns and figure ground discrimination, but that was just the regular stuff.
I got to witness him talking out some of his first experiences with more advanced theory of mind. Theory of mind is the idea that your thoughts are yours alone, and that other people have their own cognitions. Kids take some time to figure this out. That's why really little kids don't tell you that they know where your keys were the whole time. Because they know, they assume you know, and vice versa. It takes some time to figure this isn't the case. The most overt sign of this taking hold is when kids learn to lie. They figure out that they can try and tell you a false hood, and have a chance of not getting caught.
My Nephew started phase two of this recently, modeling other people's thoughts and using them to plan. This is a huge step and is what makes playing games possible. Every time you play checkers, chess, backgammon, poker, etc. You need to take the information in front of you, make your plan, then create a mental model of what your opponent will do, then modify your plan to take this into account.
He did really well for his first few games of checkers; he ended up being able to predict what would happen about a move ahead of time. 2, and 3 moves is still beyond his grasp but he's on the road.
It's great to see, because this is also a huge factor in developing good relationships. You need to be able to accurately predict what they other person will do in response to what you say or do to/near/with them. Accurate predictions tend to show better functioning which is usually quite nice :-) . It's always great to see people develop themselves, and I gotta say, I take a whole lot of extra pride in seeing my little nephew do it too :-D