One of the most common expository metaphors of discrete mathematics is "playing a game". Sometimes this is taken quite literally, as with the algebro-geometric algorithm jeu de taquin, the enumerative object named for an 8-bit Nintendo game, and the patience sorting algorithm, which American audiences might prefer to call the Klondike sorting algorithm. It shouldn't be surprising that some discrete games have been a rich source of interesting mathematical questions, and that a certain game with a long history still poses academic challenges.
And speaking of games, an excellent source of casual games is J. Bibby's site Jay is Games. I discussed one of their recommended games earlier, although I came across it independently of JiG.
In Planarity, I often have planarized a subset of the vertices and want to move them all at once. If only the interface would let me manipulate more than one vertex at a time! J. Han has put together a device implementing a multi-touch interface in his lab at the Courant Institute; in addition to running silly graph-theoretic games, it augurs what may very well be the next paradigm in human-computer interaction. In case the video on that page isn't enough to make you covet the tenth-generation iMac, take a look at this live demo. He mentions while illustrating the "puppet" application that cutting-edge mathematics and computational science make those dancing drawings possible. We are finally reaching the stage where the massive computational power sitting on the desktop can be harnessed to make computers behave in a way that is convenient to us, rather than the other way around. This is true innovation.