physicists can solve anything

there's a persistent urban legend out there, that the three body problem can not be solved

this, of course, is not true:

so when XKCD promulgates the myth, the time comes when a stand must be taken

click to embiggen

sigh.

The classical problem of the solution of the motion of three bodies, moving under the influence of Newtonian gravitational dynamics only, is an old one.
The difficulty of the problem was quickly realised in the context of predicting the orbit of the Moon - just the Earth-Moon-Sun problem, though well posed, is extremely hard to even get to the point of having a formalism for a solution.

Poincare understood the problem, in particular that the solutions may be chaotic in a rigorous sense, though it took Lorenz and his better sense of public relations and framing to convey the whole "chaos" concept to the general, public decades later: the intrinsic chaos of the system is well illustrated by the Sitnikov problem, and the more general nature of the ergodicity is demonstrated by KAM theorem for the restricted three body problem.

A very well known solution to the general threefour body problem for a particularly interesting bound-free asymptotic incoming state. ooops, what can I say, I like this plot...

Of course a general and exact solution to the three body problem was found by Karl Sundman in 1912 (Sundman, K. E.: Memoire sur le probleme de trois corps, Acta Mathematica 36 (1912): 105-179.).

Ok, he was a mathematician, it was published in french, and it does not apply to a subset of the problem of measure zero, but it was a good start.

Pedants might object that Sundman's general solution is more descriptive than prescriptive, and is rather slowly convergent - but going to 10¹⁰⁰ terms in a power series solution is really more of an engineering issue than anything fundamental.
It exists!
And, don't forget, nature solves these things in O(1) time, so clearly a faster solution algorithm exists for any given posed subset of the three body problem, by construction.

Now, that does not mean we're done with the three body problem, and indeed a number of PhD theses focused on various aspects of the classical and relativistic three body problem have been done in the last few decades.

One of my favourites is:
Dynamics of neutron stars and binaries in globular clusters or, menages a trois: Revitalizing burnt out degenerates through partner swapping

For example: initially bound, negative energy three body states are generally not stable, but a fraction can last a very long time, these generally decay by ejection leaving a more tightly bound pair; another process much studied, as it tends to produce unusual systems, is exchange, which is significant for negative total energy states, and leads to ejection of one body and replacement with another, often after a prolonged chaotic interaction.
These processes have been well quantified for a broad range of interesting initial conditions. For extended bodies, collisions of course also become important, but that can get messy real quick - still soluble, in the sense of producing a well defined final state, just messy.

sadly this was only the second best thesis title, with "Persistence of Charm in the Relentless Decay of Beauty" the clear winner - what can I say, these rocket scientists are such romantics.

So, I am happy to tell you that the classical three body problem is well under control by physicists; well, astrophysicists anyway, I can't speak for the atomic and quantum types, they seem to struggle with even simple bound states.

Not only that, Starlab is free, online, and will solve any classical three body problem.
You just provide the initial conditions (caution, actually solving the problem may, or may not, take an infinite amount of time and resources, but it will be solved, except for a subset of measure zero).

Of course, in relativity the two body problem is not solved, although much progress has been made numerically in recent years, to the point where mergers can be consummated without singularities crashing everything.
Gradual mergers, not just plunges! And not just equal masses!
Even with spin.
Even misaligned spin!

click to embiggen

Interestingly, progress has also been made for a number of subsets of the relativistic three body problem, to the point where there is at least some predictive power subject to falsifiable experiments, even if no general solution is available. At least semi-classically.

Of course, in quantum gravity not only is the two body problem not solved, neither is the one body problem, or for that matter the zero body problem.
In string theory I think they are convinced that either a solution exists, or maybe all possible solutions exist, but they don't yet agree on what constitutes a "body" per se.

But, to say that physicists can't solve the three body problem...
well, some physicists can, thank you very much.

Another, less well known solution to a bound-free three body problem...

ok, I didn't say it was a simple and practical solution, in general, but then life is just like that

PS: don't forget there are an infinite number of weakly perturbative solutions, where an initially bound pair is adiabatically invariant to rapid flyby of single bodies.
This process dominates the bound-free cross-section even for negative energy initial states.

Not to mention the near trivial case of three initially unbound bodies with net positive energy, which only leads to bound states for rare very close encounters.
Assumning no dissipation of course, extended bodies require a bit more work.