What it is not

It is not unsolvable. The path exists, it is unique, and it can be computed to as many digits as you are willing to pay for. What does not exist is a closed-form formula of certain kinds — algebraic conserved quantities, single-valued analytic ones — and a usable one in general. 'No formula' and 'no answer' are different claims and only the first one is true.

Chaos is not randomness. The same start gives the same path every time, exactly. What chaos adds is that nearby starts part company at an exponential rate, so a prediction's useful life is finite. A dice roll has no path; a three-body system has one and you cannot get at it.

Three bodies are not automatically unstable. Lagrange's triangle holds its shape forever. The figure eight repeats forever. Jupiter's Trojan asteroids have sat at L4 and L5 for billions of years, and the Sun–Earth–Moon arrangement is older than anything that has ever looked at it.

Computers did not solve it. They step it, which is a different thing, and the stepping has to be audited. Some chaotic triples are not reproducible at ordinary sixteen-digit precision — the figure on the numbers page — so a simulation is a hypothesis with a method attached.

More bodies is not strictly worse. A galaxy of a hundred billion stars is described well by smooth approximations, because the crowd averages out. Three is the awkward middle: too many for a formula, too few for statistics. Which is why it took two hundred and fifty years and is still producing papers.

And the novel is a novel. Liu Cixin's The Three-Body Problem takes the name and the chaos seriously enough, and then goes where fiction goes. The planet with three suns in it would be governed by the mathematics on this site; the rest is the book's own.