being initially a little bit perplexed by the observation that the possibility of calculating vertex potentials $\lbrace\pi_1,\dots,\pi_n\rbrace$ for weighted [cycle graph](https://mathworld.wolfram.com/CycleGraph.html)s $C_n,\,2\lt n$ such that the potentials $\pi_u,\pi_v$ of the vertices $u,v$ that are adjacent to the edge $(u,v)$ sum exactly to that edge's weight $|(u,v)|$, i.e. $\pi_u+\pi_v=|(u,v)|,\ \forall (u,v)\in C_n$,  
depends on the parity of $n$, i.e. a solution exist for $n=2k+1$ but not for $n=2k$. 

Therefore I would like to ask the  
>**Question:**  
>
>what are other non-trivial examples of mathematics where the parity of an integral parameter makes a crucial difference?  

I'm especially hoping for examples that are not classical results, but don't rule them out.  
Previously unknown results from personal research would of course be the most preferable answers.

But please don't hesitate to also provide classical results as answers.