Consider the following game played on a graph $G$ where each node can hold an arbitrary number of tokens. A move consists of removing two tokens from one node (that has at least two tokens) and adding one token to some neighboring node. The LastToken problem asks whether, given a graph $G$ and an initial number of tokens $t(v) \ge 0$ for each vertex $v$, there is a sequence of moves that results in only one token being left in $G$. Prove that LastToken is NP-complete.
I'm learning how to prove NP-complete recently but having trouble to understand the concept of NP. As far as I know, to prove a problem is NP-complete, we first need to prove it's in NP and choose a NP-complete problem that can be reduced from. I'm stuck on which NP-complete problem that can reduce to my problem. As I interpreted this is sequencing problem and I'm guessing I can either reduce Ham Cycle or Traveling Sales Man to my problem, but I don't see any connection between them so far. How should I start a good approach?