Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within the n-gon using each of the given lengths exactly once (and no vertex twice)?

Here the length of a directed diagonal within any kind of an n-gon is the number of sides of the n-gon between the starting and the end vertex counted in a fixed direction. A directed path is a sequence $(d_1,d_2,\ldots,d_t)$ of directed diagonals such that the end vertex of $d_i$ is the starting vertex of $d_{i+1}$ for $i=1,2,…,t-1$.

In number theory, it is equivalent to: For any subset $S=\{l_1,\ldots,l_t\}$ of $\{1,2,…,n-1\}$ with $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$ and $l_1+\cdots+l_t\neq 0 \pmod n$. Is there any a permutation of the elements of the subset $S$ such that no set of consecutive elements in this permutation has sum $=0 \pmod n$. Can anyone give me a counter-example or prove it?

In Graph theory, I can rephrase this question to: Let $\mathbb{Z}_n$ be a group of integers modulo $n$ and $S=\{l_1,\dots,l_t\}$ be a set of positive integers such that $1\le l_1<l_2< \dots < l_t \le n-1$, where $l_1+\dots + l_t \not\equiv 0 \pmod n$).

Let $P_t$ be a directed path of length $t$ ($t+1$ vertices). Can you label the vertices of the directed path $\overrightarrow{P_t}$ of length $t$ with distinct elements of $\mathbb{Z}_n$ such that the label differences on edges are exactly $S$? The difference is taking by subtract the head by the tail (then take mod $n$).

I have a proof for case $t\leq 6$. However, I wonder is it true or not in general. Many thanks.

afterone reads that these "lengths" are defined as "the number of sides...". It's unfortunate that the 1st paragraph makes no sense until one has read the 2nd paragraph. $\endgroup$ – Gerry Myerson Jun 16 '15 at 2:59