Strategy games are similar (but less complicated) to chess, where you have to think about what move you're going to make. There are different styles of Strategy games.
Disgaea, Utawarerumono, FFTactics are good examples of strategy where you have a grid, and you have to move a character mush like a chess piece, choose if you attack or defend, execute the action and then let the enemy have it's turn.
https://youtu.be/sfFSmlG9aZU
A video of a tactics battle! ovo I recorded it not long ago just for moi... then I realized I had Disgaea 4 videos I coulda used... Dunno where my Disgaea 5 playlist got to.. >:x coulda swore I streamed Disgaea 1 as well.. =/
Most Final Fantasy games, where it's turn based, but no moving around, is still strategy. You still have to think and use the appropriate moves to efficiently take out the enemy.
War games and FPS games tend to require strategy as well. I don't play them so I can't give any insight. o3o
Puzzle games are where you have to solve riddles, get through mazes, or simple match the same image to progress.
Tetris, Candy Crush, Homescape, Gardenscape, Puzzle & Dragons. The most basic of puzzle games.
Chocobo's Dungeon, make your way through maze like dungeons to find what you need. .Hack IMOQ's dungeons can also place them and the GU games into the puzzle game category.
Amnesia Dark Descent and other investigative games, like Phantasmagoria are also considered puzzles.
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