system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: What is a Three Body Problem?
gpt
There are some problems in physics where you can come up with an exact solution or an equation that gives you a whole set of solutions.  For example, if you are driving at a constant speed then the distance you have traveled after a given time is distance = speed \* time.  Similarly, if you drop a ball from a given height and can ignore air resistance then the position of the ball at a given time is height = initial height - 1/2 \* g \* time^(2) (where g is the acceleration due to gravity)

Other problems have no such solution available.  For example, if we weren't allowed to ignore air resistance on the falling ball then there's no nice equation that tells us where the ball is at every instant.  You can still calculate it, but you have use an iterative approach--if you know where the ball is at time T then you can compute where it'll be a short while later and be pretty accurate.  By using sufficiently short time steps you can get as accurate of an answer as you'd like, but there's no simple formula.

In orbital mechanics the "two body problem" is where you have just two things in a hypothetical universe, one orbiting the other (or really both orbiting their mutual center of mass, but often we choose one object to be so massive that it barely moves; often this is a planet and moon, a star and planet, a planet and satellite, etc).  The two body problem does have a number of nice equations to describe where you'll find each of the objects at any given time, starting from some given starting conditions.  An astronomer by the name of Johannes Kepler worked that out from observations, then later Newton came along and formulated how gravity works well enough to prove Kepler's work mathematically.  Similar to how you can ignore air resistance on a ball and wind up with an accurate enough result most of the time, in orbital mechanics you can ignore all other celestial bodies except the one you're looking at and the biggest thing it's orbiting and usually get a pretty good solution.  Sometimes, however, you do have to consider all of the forces on an object. For that we turn to the three body problem.

The three body problem is similar to the two body problem, but instead of a hypothetical universe with just 2 objects we now have 3 (and could continue on to 4, 5, etc, all the way up to the real universe with an enormous number of objects).  Unlike the 2 body problem there is not a nice equation that will tell you where to find any given object at any given time, but similar to our ball with air resistance we can take an iterative approach, simulating paths with shorter and shorter time steps to get more and more accurate results.

Note that the phrase "three body problem" is sometimes lifted from its physics application to describe social situations where the addition of a third person makes the social situation similarly difficult to solve, whether that's a couple plus a single friend who gets in the way of couples activities or a trio of people where one has feelings for both of the other two.