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First, what “many objects along the way”? There are basically none. To do this requires many detours that significantly increase the distance you’re having to travel.

Second, the faster you’re going, the less useful a gravitational slingshot is because you spend less time in the object’s gravity well.

It’s a fun and useful trick within our solar system, but I don’t think the physics plays out well for interstellar travel.

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Stars is what I meant. Although upon further thought I think you're right. I guess the intervening stars are unlikely to be moving at helpful velocities for such a game of pinball, metric expansion being what it is.
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Slingshots require very specific conditions and configurations of orbiting bodies that won't really work in deep space - all the Kuiper Belt/Oort Cloud bodiees are far too small and too far away to be usable.

But assume for example that you don't care about how long it is going to take (you are an immortal cyborg, an upload living in the computer or a popsicle in hibernation & the AI is driving. Then you can just "island hop" between Kuiper Belt/Oort Cloud objects, never gaining any high % of the speed of light but always having the next body in your sights, not farther than a few decades or a couple hundred years of flight at most.

You reach the body, refuel, repair, maybe upgrade and then off you go, to the next one. And eventually, you might be at you destination, with a fully fueled and repaired ship. It might just take like 20000 years for start to finish, but hey, you can use it for something productive like learning to play Go or something. :)

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You're right, I overlooked some important slingshot requirements. As for the longevity, I figure the ship can build a facility for growing settlers when it arrives. With no humans on board it can sustain heavier accelerations.
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This is quite nicely described in the engenerator article on Orions Arm (including possible problems): https://www.orionsarm.com/eg-article/4a3d78dc77e6f

In this setting this was first done when a bunch of immortal vyborgs was on a colony ship, about hundred years from arrival but there was a probe with a matter printer already in the destination. So their got the idea to send the probe both blueprints for a humanoid body printer as well as their mind backups. That got their colony going quite a bit faster but might have resulted in some funny/awkward moments once they met in person later. :)

Presumably you could also run this withou a mind upload and let he individual develop normally.

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