>What's better for the brain, solving Sudokus, or writing a Sudoku solver?
Both can be good. Chances are that your first Sudoku solver uses some bruteforce approach, while your manual solution certainly does not. Then the intuitions that you get from many manual games can drive the design of your solver. Of course you can implement some established algorithm but I don't see that challenging.
I have mathematician friend who won't ever play Nine Men's Morris because it's a solved game, so it's not longer interesting for him. It's a shame, you can learn a lot even playing Nim variants if you don't already know the winning strategy, it's a good learning material for kids too.
What's better for the brain, solving Sudokus, or writing a Sudoku solver?