It’s been awhile since I’ve read the books, but I think I can recall the important plot points.
2010:
[spoiler]Dr. Heywood Floyd goes on a joint American-Russian mission on the Leonov to Jupiter, where they will attempt to salvage the Discovery, find out what happened to Bowman and co., and investigate TMA-2 (the monolith).
Along the way, the Chinese launch their own mission to Jupiter, race to Europa (one of Jupiter’s moons), and land to refuel. Turns out they’ve got a water-propelled ship, and can use Europa - an ice-planet - as a cosmic fueling station. However, their lights attract a seaweed-like thing, which pulls the spaceship under the ice, killing all but one crewmemeber. The sole survivor uses the transmitter on his spacesuit to beam a distress call to the Leonov to tell everyone the bad news about his ship, and that there is life on Europa.
Meanwhile, Bowman-as-star-child travels around, eventually coming back to Jupiter. He finds primitive life on Europa, where lava creates oases under the ice. When the lava cools, entire mini-civilizations are wiped out. On Jupiter, there is also primitive life, but everything floats or flys. The pressure is so great on the planet that nothing can survive. In fact, the pressure is so great that the core of Jupiter is a huge diamond about the size of Earth.
The Leonov finally reaches the Discovery, and the crew re-educates HAL with the help of HAL’s creator (also a crew member). A couple weeks before they’re really ready to leave, Floyd gets a message from Dave Bowman stating they must leave in a couple weeks. The Leonov piggybacks on the Discovery to boost away from Jupiter, then cuts free. The Discovery will eventually drift back to Earth - along with HAL’s creator, who snuck over to the Discovery to be with her “child.”
As they are leaving, TMA-2 disappears. Oooh, spooky. Then someone notices a black dot on Jupiter’s surface. The spot is growing, and it turns out it’s composed of monoliths. In a few days, the completely cover Jupiter, and start sucking up all the gas on the planet. HAL starts broadcasting a message to Earth (on Bowman’s orders), stating that the moons of Jupiter are “yours”, except Europa, so don’t try to land there. Jupiter explodes, the Discovery is vaporized, and there’s a bright, shiny Sun where Jupiter used to be.
This new Sun, Lucifer, melts the ice on Europa and enables evolution there. As the life on Jupiter could never live on the surface, they could never achieve real evolution, so they were sacrificed to give Europa a chance.
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I was much less impressed by 2061, the third book, so I remember a lot less about it. The short and skinny is:
[spoiler]Dr. Floyd is exploring Haley’s Comet when his ship gets a distress call. A ship has been hijacked and crash landed on - you guessed it - Europa. His grandson just happens to be on board. Why would anyone hijack a spaceship and force it down on Europa? Becuase there is a giant diamond mountain on it - a fragment of Jupiter’s core.
Dr. Floyd hightails it over to Europa, beaming a message around the cosmos asking Bowman’s pardon for having to land there. They land, rescue the crew members, check out the diamond mountain, and investigate a nearby monolith. At the foot of the monolith is a village of (absent) amphibians. Floyd’s grandson is greeted by the ghost-like image of Floyd, who was somehow copied into the monolith (the real Floyd knows nothing about it). After they leave, Dave and HAL-as-star-child (having been copied in the previous book) talk with Floyd-as-start-child about. . .something. Obviously, it’s not too important, because I can’t remember it.
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