  We use an effective field theory (EFT) which contains only short-range
interactions to study the dependence of a variety of three-nucleon observables
on the pion mass. The pion-mass dependence of input quantities in our
``pionless'' EFT is obtained from a recent chiral EFT calculation. To the order
we work at, these quantities are the 1S0 scattering length and effective range,
the deuteron binding energy, the 3S1 effective range, and the binding energy of
one three-nucleon bound state. The chiral EFT input we use has the inverse 3S1
and 1S0 scattering lengths vanishing at mpi_c=197.8577 MeV. At this
``critical'' pion mass, the triton has infinitely many excited states with an
accumulation point at the three-nucleon threshold. We compute the binding
energies of these states up to next-to-next-to-leading order in the pionless
EFT and study the convergence pattern of the EFT in the vicinity of the
critical pion mass. Furthermore, we use the pionless EFT to predict how doublet
and quartet nd scattering lengths depend on mpi in the region between the
physical pion mass and mpi=mpi_c.
