If you're looking for one of the absolute best board-game experiences on mobile, you have to play Carcassonne. It won iPhone Game of the Year a few times, and it's extremely easy to play.
In Carcassonne, you build medieval roads, cities, and more in a typical board-game style. The logic and rules make it easy to play, yet still challenging enough for people of all ages and skill levels. They even have Solitaire in there too, if you're interested. You can do local or online human opponents, battle tough AI, and enjoy all that it has to offer.
On Reddit, this game is the #1 suggestion when people ask, "what iPhone game should I play with my girlfriend". Not to mention all of the reviews on iTunes say it's "our" favorite game, and not "my" favorite game. Which means people are playing it together, and loving every minute of it.
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