PixlBit

I included a video of Intelligent Qube because it's kind of impossible to describe. Not only is it a puzzle game, but it also requires quick thinking, smart moves, and precision. Released for the PSONE, it's actually a pretty cool game, and will boggle your mind and test your reflexes at the same time. 

I remember playing this in San Francisco in the late 90s. My roommate was really obsessed with it. The story isn't revealed until you progress through the game a little, but apparently mankind is being tested by some sort of supreme extraterrestial intellligence, and the puzzles you solve make up that test. As you progress through the game, this is explained. 

The player character starts out as a man, but I think changes to a woman and then a child eventually. Later in the game, the number of blocks, in width and length, as well as the speed they move at, increases a lot. 

Great game.