  This paper investigates the synchronization of three identical oscillators,
or clocks, suspended from a common rigid support. We consider scenarios where
each clock interacts with the other two, achieving synchronization through
small impacts exchanged between oscillator pairs. The fundamental outcome of
our study reveals that the ultimate synchronized state maintains a phase
difference of $\frac{2\pi}{3}$ between successive clocks, either clockwise or
counter-clockwise. Furthermore, these locked states exhibit an attracting set,
which closure encompasses the entire initial conditions space. Our analytical
approach involves constructing a nonlinear discrete dynamical system in
dimension two.
  These findings hold significance for sets of three weakly coupled periodic
oscillators engaged in mutual symmetric impact periodic interaction,
irrespective of the specific oscillator models employed. Lastly, we explore the
amplitude of oscillations at the final locked state in the context of two and
three interacting Andronov pendulum clocks. Our analysis reveals a precise
small increase in the amplitude of the locked-state oscillations, as quantified
in this paper.
