  We present a method for solving trapped few-body problems and apply it to
three equal-mass particles in a one-dimensional harmonic trap, interacting via
a contact potential. By expressing the relative Hamiltonian in Jacobi
cylindrical coordinates, i.e. the two-dimensional version of three-body
hyperspherical coordinates, we discover an underlying ${\rm C}_{6v}$ symmetry.
This symmetry simplifies the calculation of energy eigenstates of the full
Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian
eigenstates. Particle superselection rules are implemented by choosing the
relevant representations of ${\rm C}_{6v}$. We find that the one-dimensional
system shows nearly the full richness of the three-dimensional system, and can
be used to understand separability and reducibility in this system and in
standard few-body approximation techniques.
