Let $Y_1,Y_2,\dots$ be iid random variables with $P(Y_n=0)=1-p,\; P(Y_n=1)=p$ where $p\in(0,1)$. Define $$ X_n = 2 Y_n + Y_{n+1} $$ The question is, whether $\{X_n\}$ is a Markov chain or not.
This is a homework question. The implication is that it is not a Markov chain, since it was given as a counterexample against common intuition ("I see $n$ and $n+1$ in the definition, it's gotta be a Markov chain!")
My idea: It seems like the value of $X_n$ uniquely determines the values of $Y_n$ and $Y_{n+1}$, as each possible value ($0,1,2,$ or $3$) has a specific combination of $Y$'s that yield that combination. Thus it seems to me that the probability of different outcomes of $X_{n+1}$ is uniquely determined by $X_n$ regardless of the previous history, which points towards being a Markov chain.
Now I don't like spoiling the fun of finding the answer myself. But I've literally never encountered a problem that left me so clueless, usually I am at least aware of my lack of understanding.
So, please, I do not want a complete answer. I'd like to be reassured that I am indeed missing something elementary and perhaps just nudged in the right direction.
Thank you!