  The problem of $N$ particles interacting through pairwise central forces is
notoriously intractable for $N\geq3$. Some quite remarkable specific cases have
been solved in one dimension, whereas higher-dimensional exactly solved systems
involve velocity-dependent or many-body forces. Here we show that the guiding
center approximation---valid for charges moving in two dimensions in a strong
constant magnetic field---simplifies the three-body problem for an arbitrary
interparticle interaction invariant under rotations and translations and makes
it solvable by quadratures. This includes a broad variety of special cases,
such as that of three particles interacting through arbitrary pairwise central
potentials. A spinorial representation for the system is introduced, which
allows a visualization of its phase space as the corresponding Bloch sphere as
well as the identification of a Berry-Hannay rotational anholonomy. Finally, a
brief discussion of the quantization of the problem is presented.
