Lumosity and Lumosity Mobile are iPad and iPhone versions of the same app; you can also play Luminosity's games on the company's website, like Fit Brains. Lumosity's games are easy to understand and fun, but they seem limited. Unlike Elevate or Fit Brains, you can't see all the available games and choose a specific function to train.
Lumosity uses a Lumosity Performance Index to track your progress, but with the free version of the game you can't compare your brain with others. (It's $60 a year to have full access.) However, only 12 games are available on mobile devices; another 40 are web-only, so if you want to play this kind of game on your iPhone or iPad, this may not be the best choice.
The New York Times Crossword
One thing I found when playing these brain-training games is that they're all a bit twitchy; they all have timers, and you have to react quickly. You might as well just play Guitar Hero if you're looking to improve reactivity.
Me, I like to take my time and think when I'm exercising my brain. My game of choice is the New York Times crossword puzzle. For $40 a year, you get access to daily puzzles and a full archive, and the app is compatible with both iPad and iPhone (though it's pretty cramped on the smaller displays).
If you're not familiar with the New York Times crossword, it increases in difficulty from Monday through Saturday, and offers a larger, Thursday-level puzzle on Sundays. I can generally finish Monday to Wednesday puzzles, and if I have the time, I can solve most Sundays. But when I need a fix, I can go back through the archive (more than ten years of puzzles!) and choose another puzzle at the level I want. Crossword puzzles only help you keep the language part of your brain spry, but that kind of thinking is certainly helpful.
Peak Brain Training is an iPhone only app, with a limited free version and a $35 annual subscription after that. In my trial, the games were a lot more difficult than the other apps. There's a sliding block puzzle, where you have to connect a line (something I've never been good at), a Boggle-type word-find puzzle, and a variety of word and math puzzles. I found Peak's games to be the least interesting to play, and the app gives you no information other than the points you've scored.
All these apps may just be ways to separate you from your money, but if you find them enjoyable, why not play? Having fun is probably a good way to keep your brain active, so try some of them out and see if you want to play them regularly. One word of advice: try them out, and, if you plan to subscribe, go for a one-month subscription, rather than a year, to see if you get bored.
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