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There are many good "mind games" for exercising your brainpower. Here are some of the more common ones and how they may benefit you. (The ones that I regularly play are chess and Scrabble, although I have tried all of the games listed here.)
This classic word game can really help you learn a lot of new words, especially if you play with an open-dictionary policy. To win more often, though, remember that it really isn't just a vocabulary game. There is a lot of strategy involved in figuring out where to place your words. A player with a good understanding of how to score the most points for each play can easily win against one with a larger vocabulary.
This is historically THE thinker's game. Contrary to what many think, though, it's definitely not just a left-brain analytical exercise. It requires both hemispheres of the brain to play well. The left-brain part is all about analyzing possible moves and predicting responses. Grasping the spatial relationships on the board and having an intuitive grasp of good positioning come from the right-brain. Interestingly, many top players are also musically talented, and handling music is right hemisphere function.
Long ago people noted that older people who regularly did crosswords seemed to stay mentally sharper as they aged, compared to those who didn't exercise their brains. Some recent research has supported this idea. One of the primary advantages of this as a mental exercise, is that it's a mind game you can play alone.
This game has recently become much more popular. It isn't really about math skills as much as it is about deductive reasoning. It also has the advantage of being a one-person activity, and a book of these number puzzles can be carried with you.
It's not as popular as it used to be, but you can still buy a Boggle game in most department stores. This word game is based on speed, and on observation. Trying to quickly find words in the letters is great for training your mind to observe and look for patterns. You see an "s" for example, and watch every noun thereafter to see if the "s" can be connected to it for another word. It is of course a vocabulary game in part, but pattern recognition skills will win you more games.
Poker is one of the best of the mind games because it requires so many different mental skills. You have to learn about probabilities and how to analyze a hand in relation to all the other information you gather. You have to train yourself to be observant to gather that information. Reading other people is a big part as well. The whole process is directly beneficial to real life decision making, because the poker develops your intuitive grasp of what a "good bet" is.
Playing dominoes is a very analytical process if you make it so. Gin rummy is another good game for developing pattern-recognition skills. Monopoly (or the pricier "Cash Flow" game by Robert Kiyosaki) is not only a good mind game, but a good way to learn basic real estate investing principles.