**Suppose** we have a multiset $M$ of positive rational numbers. Sum of $M$ equals $1$. We'll call this multiset *$n$-distributable* for some $n\in \mathbb{N}$, if there exists a partition $M_1 \sqcup ... \sqcup M_n$ of the $M$ such that the sum of each (multi)subset $X_i$ equals $\frac{1}{n}$. If the multiset is $n$-distributable and $m$-distributable for some $n,m\in\mathbb{N}$, we will call it *$(n,m)$-distributable*, and so on.

**The problem is** to find for some fixed $a_1, \ldots, a_k \in \mathbb{N}$ the minimal possible cardinality of a $(a_1, ..., a_k)$-distributable multiset.

**Real-world analogy.** You are organizing a party. You know that the number of guests to attend your party can be anything from $a_1, \ldots, a_k \in \mathbb{N}$. In order to be prepared you cut the cake beforehand into smaller pieces, not necessarily of equal size. The requirement is that, no matter how many guests come, you will be able to give each of them some pieces of the cake without having to cut the cake any further so that everybody will get the same amount of cake. What is the minimum number of pieces of your cake you will have to cut it into?

**The question.** Formulated like this, is it a solved problem? If it's not, what specific cases are discussed and where could we read about it? If it is solved, well... basically the same, what mathematical branch is it and where to read about it? I tried to ask it on MSE here: https://math.stackexchange.com/questions/1381042/dividing-the-whole-into-a-minimal-amount-of-parts-to-equally-distribute-it-betwe. After that some other gentleman asked essentially the same question: https://math.stackexchange.com/questions/1383406/minimum-cake-cutting-for-a-party. The latter even has a bounty which is running out, but noone seems to know the answer (or maybe we chose wrong tags). I tried to dig into this problem by myself, but being an amateur, all I wind up with is a lot of specific made up terminology, some useless properties, couple of toy model cases solved and a constant feeling that I'm trying to invent a bicycle. Any insight would be greatly appreciated.

2more comments