There's no such thing as a "brainy game", really. The great strength of videogames, after all, is that they're as challenging as you want them to be. Bored to tears by the prison escape sequence in Call of Duty: Black Ops? Try completing it with just a knife, or without using cover, or while reciting the Faerie Queene backwards. See: instant smartypants.
Nonetheless, there are games that cater to brainy play more effectively than others, games that wear their self-awarded intellectual credentials on their sleeves. Look at Dishonored, where the simple act of getting past a dozy guard can tie the experience down for half an hour. Or XCOM: Enemy Unknown, a fierce old Victorian schoolmaster of a turn-based strategiser, compound eyes vengefully peeled for the slightest lapse in coordination.
In tribute to the latter two stunners, here's a list of Xbox 360 and Xbox Live Arcade releases that are almost too clever for their own good, though thankfully not too expensive. Buy them all, and we'll see if we can't mail you an honorary doctorate.
1. Civilization Revolution
Second only to Enemy Unknown, Revolution is a gorgeously elastic world domination game that marries the broad strokes of the old PC ruleset to an interface that never leaves you in any doubt about what to do next. There's not quite enough material to pad out the endgame, and some may be disappointed by the shortage of city management options given the majesty of XCOM's base. Still, if you've ever wanted to shepherd a society from inception to destruction, this is the most entertaining way to do it with a control pad.