  We study the production of low atom number Fock states by reducing suddenly
the potential trap in a 1D strongly interacting (Tonks-Girardeau) gas. The
fidelity of the Fock state preparation is characterized by the average and
variance of the number of trapped atoms. Two different methods are considered:
making the trap shallower (atom culling [A. M. Dudarev {\it et al.}, Phys. Rev.
Lett. {\bf 98}, 063001 (2007)], also termed ``trap weakening'' here) and making
the trap narrower (trap squeezing). When used independently, the efficiency of
both procedures is limited as a result of the truncation of the final state in
momentum or position space with respect to the ideal atom number state.
However, their combination provides a robust and efficient strategy to create
ideal Fock states.
