# Maximum product for multisets with same sum

Given a positive number N, among all multisets (containing only positive numbers) with sum N, is there a reliable method for determining the set with the maximum product? For example, for N = 5, the following are some possible sets, with their products (i.e. all their elements multiplied together).

{ 5 }              : product = 5^1 = 5
{ 1, 1, 1, 1, 1 }  : product = 1^5 = 1
{ 0.5, 1, 1.5, 2 } : product = 1.5
{ 2.5, 2.5 }       : product = 6.25


I'm looking for a reliable method, for any given N, to find the multiset which produces the maximum product.

Based on comments below, it seems you want to also maximise $f(n)=\left(\dfrac{N}n\right)^n$ among $n \in \mathbb N$. For this, you may note that the corresponding continuous function is monotone or unimodal and has a maximum at $x = N/e$. Hence, look for the maximum among $\lfloor N/e\rfloor$ and $\lceil N/e \rceil$.
• $N/n$ where $n$ is the number of elements. – Macavity Sep 27 '14 at 11:41