So I am studying information theorem, and have come across an extension to a problem that I can't work out if it is right or not. The original problem is the one where you have 12 coins (one which is heavier than all the others), and a scale, and you want to use the scale an optimal number of times to find out which coin is the heaviest. I have worked out that this is 3, by using different groups of the coins each time.
Now, I have a more complex version of optimisation. If we have $X$ which is uniformly distributed over a finite set $\chi$ with $|\chi|=2^n$ for some $n\in\Bbb{N}$, and we can choose a sequence of subsets $A_1,A_2,...$ and each time we can ask a question of the form $X\in A_1$, $X\in A_2$,....The question is then how many such questions do we need to determine the value of $X$?
My intuition tells me we need $n$, as we can let $A_1$ be the lower subset of $\chi$ and then let $A_2$ be the higher subset, and keep dividing the subsets; if $X$ was not in $A_1$ then we know it must be in $A_2$, and then if $A_3$ and $A_4$ are the divided sections of $A_1$ we know we can discard them, and just see if $X$ is in $A_5$. This means we have to ask $n$ questions. But is this optimal? I was wondering if there was some method where I drew a rooted tree, like in the formation of Huffman codes; this then I guess would be optimal, as Huffman codes are optimal? Any help appreciated.