which $2^n \ge 1000$ is $10$. if you think about it in this fashion from the beginning, you will probably realize that each prisoner can sample from more than one bottle in the process, because that's the only way this method can really work.
there is a similar problem in which you have two days to figure out which bottle out of 27 is poisoned, which only requires three slaves instead of the five it would require if you only had one day. why is this?
because each slave now has three possible states - not dead, dead in one day, dead in two days. so $3^3 = 27$, as required. but 