Exercising your body seems to have all sorts of disease-delaying benefits. Why would exercising the mind be necessarily different? After all, when you're thinking hard you're burning extra calories.
(Calories are a bad metric. There's so much chemical complexity they intentionally avoid. Not all "calories" are spent the same way, especially when you are talking about brain chemistry. Optimizing for how much heat a brain generates is directly optimizing for the wrong thing because when the brain is literally smoking, that is a stroke and it is dying.)
When I think really hard, I get depressed that can't solve the problem and binge eat junk food for comfort.
To that end, playing sudoku, but only staying at the "1 star" difficulty puzzles, I don't know why I would expect brain growth, as it were. Learning the strategies and modeling that is needed to solve the advanced ones, though, is a very different thing.
1. People imagine without any evidence, but only a plausible story, that X is good for Y (say for instance, vitamin C for colds, because vitamin C is involved in the immune system)
2. Study after study is done finding no benefit, but each study you quote, someone finds some nit pick about it "well maybe the dosage wasn't high enough" or "maybe they did/didn't take it with calcium" and then continue to believe there must be some way to derive benefit
That said, I should be clear that I'm not pushing for learning some of these "brain games" for their own sake. I am more throwing out that equivalent of the Tyler Durden quote regarding chickens. :D