  In 1665, Christiaan Huygens reported the observation of the synchronization
of two pendulum clocks hanged on the wall of his workshop. After
synchronization, the clocks swung exactly in the same frequency and $180^{o}$
out of phase -- anti-phase synchronization. Here, we propose and analyze a new
interaction mechanism between oscillators leading to exact anti-phase and
in-phase synchronization of pendulum clocks, and we determine a sufficient
condition for the existence of an exact anti-phase synchronization state. We
show that exact anti-phase and in-phase synchronization states can coexist in
phase space, and the periods of the synchronized states are different from the
eigen-periods of the individual oscillators. We analyze the robustness of the
system when the parameters of the individual pendulum clocks are varied, and we
show numerically that exact anti-phase and in-phase synchronization states
exist in systems of coupled oscillators with different parameters.
