  We present a scenario in $1 + 1$ and $3 + 1$ dimensional space time which is
paradoxical in the presence of a time machine. We show that the paradox cannot
be resolved and the scenario has {\em no} consistent classical solution. Since
the system is macroscopic, quantisation is unlikely to resolve the paradox.
Moreover, in the absence of a consistent classical solution to a macroscopic
system, it is not obvious how to carry out the path integral quantisation.
Ruling out, by fiat, the troublesome initial conditions will resolve the
paradox, by not giving rise to it in the first place. However this implies that
time machines have an influence on events, extending indefinitely into the
past, and also tachyonic communication between physical events in an era when
no time machine existed. If no resolution to the paradox can be found, the
logical conclusion is that time machines of a certain, probably large, class
cannot exist in $3 + 1$ and $1 + 1$ dimensional space time, maintaining the
consistency of known physical laws.
