  We show that four narrow zigzag dice lattice ribbons, which have the minimal
widths among their separate categories, constitute a unique collection of
systems to study physics related to one-dimensional Dirac cones and flat bands.
In zero magnetic field, all three combinations, including only Dirac cones,
only flat bands, coexisting Dirac cones and flat bands, are realized in the
low-energy band structures of one or two of the four ribbons. In particular, we
identify flat bands and Dirac cones corresponding to the edge states of wide
ribbons. In a perpendicular magnetic field that gives half a flux quantum per
elementary rhombus, two of the four minimal ribbons have fully pinched
spectrum, and dynamical evolutions from initially localized wave packets always
lead to compact Aharonov-Bohm (AB) cages. The experimental realizations of
these narrow zigzag dice lattice ribbons, and the opportunities of exploring
novel single-body and many-body physics therein are discussed.
