  We solve the three-boson problem with contact two- and three-body
interactions in one dimension and analytically calculate the ground and excited
trimer-state energies. Then, by using the diffusion Monte Carlo technique we
calculate the binding energy of three dimers formed in a one-dimensional
Bose-Bose or Fermi-Bose mixture with attractive interspecies and repulsive
intraspecies interactions. Combining these results with our three-body
analytics we extract the three-dimer scattering length close to the dimer-dimer
zero crossing. In both considered cases the three-dimer interaction turns out
to be repulsive. Our results constitute a concrete proposal for obtaining a
one-dimensional gas with a pure three-body repulsion.
