  Recently there has been a resurgence of intense experimental and theoretical
interest on the Kondo physics of nanoscopic and mesoscopic systems due to the
possibility of making experiments in extremely small samples. We have carried
out exact diagonalization calculations to study the effect of the energy
spacing $\Delta$ of the conduction band on the ground-state properties of a
dense Anderson model nanochain. The calculations reveal for the first time that
the energy spacing tunes the interplay between the Kondo and RKKY interactions,
giving rise to a zero-temperature $\Delta$ versus hybridization phase diagram
with regions of prevailing Kondo or RKKY correlations, separated by a {\it free
spins} regime. This interplay may be relevant to experimental realizations of
small rings or quantum dots with tunable magnetic properties.
