do two watch indicate the same time ?

Answer 1

Unlikely, since Chaos theory states that sensitivity to initial conditions would soon have them reporting different times after a short...well, time!

Answer 2

Yes. Here's why - we wish to show that two watches both report exactly the same time at any given instant. Note that many watches are either digital or have second hands that move discontinuously (i.e. in discrete ticks), and in either case do not distinguish between fractions of a second. We also note that there are more than 86400 of these watches. Since there are only 86400 seconds in a day, these watches can have at most 86400 distinct values, but there are more than 86400 of them, therefore by the pigeonhole principle at least two watches must indicate the same time at any given time of day. Q.E.D