I randomly pick a number from 1 to 1000. After you make your first guess at the number I will only tell you whether it is warmer (closer) or cooler (farther) or neither, to a 'zero^{th} guess' of 500.

For successive guesses I will tell your warmer, colder, neither, to the closest of the previous guesses.

What is the minimum number of guesses that would guarantee that you win with an optimal strategy?

What is this strategy?