I think of an integer $n$ with $1 \leqslant n \leqslant 10^{6}$. Explain how to find $n$ using twenty questions (or less) of the form 'Is it true that $n \geqslant m$ ?' to which I answer yes or no.

I have watched a horse race with 15 horses. Is it possible to discover the order in which the horses finished by asking me twenty questions to which I answer yes or no?

Roughly how many questions of the yes/no type are required to discover the order in which $n$ horses finished if $n$ is large?

[You may assume that I answer honestly.]