You have an infinite supply of crystal balls that you take into a building with 1000 stories.
There is a critical height at which if you drop the ball from there, the ball would break. If the ball is dropped from the floors under that critical height, it would not break, but if it's dropped from the floors above, obviously, it breaks.
How many trials minimum do you need in order to determine the critical height?
Advanced version #1: If you only have a supply of $k$ balls, how many trials minimum do you need?
Advanced version #2: What is the strategy that would minimize the expected number of trials, given that you only have $k$ balls, if we treat the critical height as a random variable drawn uniformly from (1,...,1000)?
Note that the two advanced versions are two different questions and thus beget two different strategies.
Clarification: When I say "how many trials minimum do you need in order to determine the critical height" it formally means: find the smallest integer $N$ such that it's always possible to determine the critical height within $N$ trials, regardless of where the critical height is. Naturally, this $N$ will be expressed in terms of $k$ in the case of advanced version #1.
Solution to advanced version #1: