What are the benefits over just playing a normal game? Sites such as Lumosity claim they are training your brain and making you smarter using "what seems like games".
Is there any scientific evidence from independent studies?
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There's probably a different answer for different games.
In 2008 there was a study that showed that a specific game called Dual-n-Back improves fluid intelligence. There are Android and iPhone versions of the game.
As far as Brain Age for the Nintendo DS goes, Nintendo doesn't claim that it based on sound science. They see the game primarily as entertainment.
According to a statement released by the Stanford University Center on Longevity and the Berlin Max Planck Institute for Human Development, there is no solid scientific evidence to back up the promise that brain training video games make you smarter.
Signed by 70 of the world’s leading cognitive psychologists and neuroscientists, the statement minces no words:
The strong consensus of this group is that the scientific literature does not support claims that the use of software-based “brain games” alters neural functioning in ways that improve general cognitive performance in everyday life, or prevent cognitive slowing and brain disease.
The statement also cautions that although some brain training companies:
present lists of credentialed scientific consultants and keep registries of scientific studies pertinent to cognitive training…the cited research is [often] only tangentially related to the scientific claims of the company, and to the games they sell.
Additional Information: See this.