  A major challenge in modern physics is to accurately describe strongly
interacting quantum many-body systems. One-dimensional systems provide
fundamental insights since they are often amenable to exact methods. However,
no exact solution is known for the experimentally relevant case of external
confinement. Here, we propose a powerful ansatz for the one-dimensional Fermi
gas in a harmonic potential near the limit of infinite short-range repulsion.
For the case of a single impurity in a Fermi sea, we show that our ansatz is
indistinguishable from numerically exact results in both the few- and many-body
limits. We furthermore derive an effective Heisenberg spin-chain model
corresponding to our ansatz, valid for any spin-mixture, within which we obtain
the impurity eigenstates analytically. In particular, the classical Pascal's
triangle emerges in the expression for the ground-state wavefunction. As well
as providing an important benchmark for strongly correlated physics, our
results are relevant for emerging quantum technologies, where a precise
knowledge of one-dimensional quantum states is paramount.
