Question 110
Let $n$ be a natural number. A finite set of integers is called $n$-balanced if there are exactly $n$ ways to partition it into two subsets with the same sum. For example, the set $\{1,3,4,5,6,7\}$ is 2-balanced, because the only ways to partition it into two subsets with the same sum are to partition it into $\{1,3,4,5\}$ and $\{6,7\}$, or into $\{1,5,7\}$ and $\{3,4,6\}$.
Find all integers $n \geqslant 0$ for which there exists an $n$-balanced set.
Find all integers $n \geqslant 0$ for which there exists an $n$-balanced set.