The “guess the number” game for arbitrary rational numbers?

Question

I once got the following as an interview question:

I\'m thinking of a positive integer n. Come up with an algorithm that can guess it in O(lg n) quer

Answer 1

You can sort rational numbers in a given interval by for example the pair (denominator, numerator). Then to play the game you can

1. Find the interval [0, N] using the doubling-step approach
2. Given an interval [a, b] shoot for the rational with smallest denominator in the interval that is the closest to the center of the interval

this is however probably still O(log(num/den) + den) (not sure and it's too early in the morning here to make me think clearly ;-) )