This came up in an algebra class today, but I'll phrase it a bit differently.
Let's say Alice and Bob are playing a game. Alice thinks of an integer partition, and tells Bob the sum of the partition's parts $n$. Bob can then ask Alice some questions to determine which partition she's thinking of.
For example, suppose that Alice says $n=4$. Then the possible partitions are $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. If Bob asks "What is the largest part of your partition?", then Alice might answer 2 - in this case Bob cannot uniquely determine which partition Aice is thinking of as there are two possibilities, $2+2$ and $2+1+1$. Bob can then ask "How many parts does your partition have?" and complete the game in two questions for $n=4$ (this is not necessarily minimal). Note that the same strategy won't work for $n=7$, as $3+3+1$ and $3+2+2$ are both partitions with three parts and maximum part three.
So now I ask, given $n$, what is the minimum number of questions needed to determine Alice's partition?
(By "question" I mean a question relating to the partition or its parts, returning a nonnegative integer answer. So Bob could say "How many odd numbered parts are in your partition?", but he can't say "I have a list of partitions here, what number is your partition on this list?".)