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The thing is, Earth is doomed because of strife and lack or resources, so the government sends a few (six? I think) people on a voyage toward "a new home for mankind", but actually there isn't any habitable planet at the destination, and the whole thing is an experiment by the government, trying to make the astronauts find clever and innovative solutions to, because people with fewer resources get better ideas, or something like that.

Eventually it backfires on the government, because the people on the spaceship become super-smart. It's all delivered by messages that Earth receives from the spacecraft.

Rings a bell?

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    This reminds me of the recent SyFy miniseries "Ascension", though obviously it is a TV show in conventional format rather than a short story written in the form of messages.
    – Random832
    Commented Mar 19, 2015 at 16:08
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    One of our most duplicated ID targets. Apparently the story makes a subtle impression on people. Not stunning enough to be regularly remembered, but it gets it's hooks in an hangs on. Commented Mar 19, 2015 at 21:20

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"The Gold at Starbow's End" by Frederik Pohl, expanding into a full-length novel in Starburst.

In the story, the Americans launch eight astronauts in a spaceship on a ten-year journey to Alpha Centauri to land on its habitable planet called “Alpha-Aleph”. The trouble is, the planet does not exist and the spaceship has only sufficient fuel to reach the star system. More damningly, the American President and a very small circle of science and political advisors have sanctioned this one-way suicide mission, fully aware that the planet does not exist.

The rationale for proceeding with this 40-billion dollar hoax is the strong desire to advance fundamental knowledge, backed by the theory that if you throw together a small group of very intelligent people in extended isolation, they will learn to use the meagre tools at their disposal along with raw brain power to make new discoveries in fundamental sciences, especially if such thought process is not aimed at making specific discoveries.

In the end:

The story ends up with the twist that the US starts going up in flames while the astronauts develop god-like powers to come back for the rescue.

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  • Yes, that seems like a closer match - it's a very similar premise, though!
    – jonrsharpe
    Commented Mar 19, 2015 at 10:55
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Could this be "Destination: Void" by Frank Herbert?

In the future, humankind has tried to develop artificial intelligence, succeeding only once, and then disastrously. A transmission from the project site on an island in the Puget Sound, "rogue consciousness!", was followed by slaughter and destruction, culminating in the island vanishing from the face of the earth.

The current project is being run on the moon, and the book tells the story of the seventh attempt in a series of experiments to create an artificial consciousness. For each attempt the scientists raise a group of clones. These clones are kept isolated and raised to believe that they will be the crew of a spaceship that will colonize a planet in the Tau Ceti solar system (Tau Ceti has no habitable planet; its choice - should they manage to reach it - is part of the planned frustration of the crew). The spaceship will take hundreds of years to reach the system and the crew will spend most of their time in hibernation. Along with the crew of six, the ship carries thousands of other clones in hibernation, intended to populate the new colony and, if necessary, provide replacements for any crew members who die along the way.

Per this summary:

The ship’s chaplain, though, is aware that this is really an experiment in high-pressure environments to create brilliance.

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