Like Brain Age, after you complete the daily exercises you can test your progress and record it. Unfortunately, unlike Brain Age where you can view your progress over time, the progress chart in Left Brain Right Brain only shows your current levels, which is disappointing.
Left Brain Right Brain does offer a card-sharing multiplayer mode where two players can compete in three different minigames, but it's definitely not the main attraction here. Most of the minigames aren't that fun on their own, and are really only good for training your weak hand. Since handedness doesn't really come in to play in the multiplayer mode, it really just devolves into balloon popping and safe-opening races, which gets old quickly. (You can select which hand to play with at the start, but you're obviously going to choose your better hand since it's a competition.)
