Puzzles can be a real aid in helping you improve your mind. That's the story, anyway, but what is the reasoning behind it?
It depends on what kind of puzzles you do. Word searches are fine, but they are really done to relax, and not to mostly sharpen your mind.
You really have to get beyond the easy puzzles to reap the greater mental rewards of doing puzzles.
The "gold standard" of crossword puzzles in the United States is the Sunday New York Times Crossword Puzzle. Most people do not know this, but the Friday and Saturday NYT Crossword puzzles are harder than the Sunday puzzle. Sometimes much harder.
One thing that I've found on many or the New York Times Sunday crossword puzzles, at least over the last few years since I started doing them, is that there will often be a mention of something that appears in the NYT that day. Often it's from the magazine section.
This is a very cool thing the NYT does. It reinforces what you've read that day. And this is one of the things that I believe puzzles help your mind with. Any time you cross-reference information in your mind, you are building connections. These are sometimes referred to as "neural pathways."
Doing puzzles is a perfect way to develop thinking habits. Often you think you may have an answer, because it seems like it fits, but if you aren't sure, and you can't corroborate the answer, it's best to pencil it in lightly (or not at all, and just keep it in the back of your mind.) Filling in the answer because you "think it's right" can mess you up later by preventing you from coming up with the correct answers to the clues that cross the answer that you filled in wrongly.
That's sort of like math. If you are doing an equation, and you're not sure of part of the answer, but "it looks ok, I guess..." then you will probably be screwing things up more and more as you base your reasoning on a false assumption. Puzzles and math help you develop much clearer and cleaner thinking habits.
As much as I like puzzles, I've come to think of math as about the ultimate puzzle. Not in the sense of it being confusing (because by its nature it aims at reducing confusion and creating clarity), but in the sense that it has logical rules but requires creativity to understand and use in any meaningful sense.
If you like to sharpen your mind, make yourself more mentally resilient, learn something useful, and have a good time doing it, I don't think you can do better than learn math and mathematical philosophy.
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