You are given $n$ numbers between $-n$ and $n$, the sum of numbers is $0$. Divide the given sequence on disjoint subsequences in such a way that each subsequence has zero sum. Each element should belong to exactly one subsequence. Maximize the number of subsequences. It is not required that elements in subsequences are consecutive.
It seems the problem is NP-hard (or "Can the sequence be partitioned into $k$ subsequences which all have zero sums?" is NP-complete), but can't prove it.
Example: the maximal number of subsequences for the sequence [2, 0, 1, -1, -1, -1] is 3: , [2, -1, -1] and [1, -1].