Classifying a Markov chain and its states

Question

I am learning Markov chain by myself and am very new to this concept. I need a help in the following: Consider a Markov chain with state space {1,....,100}. Suppose states 2i and 2j communicate with each other and states 2i-1 and 2j-1 communicate for all $i,j\in \{1,...,50\} $ Suppose $p_{3,3}^{(2)}>0$, $p_{4,4}^{(3)}>0$ and $p_{2,5}^{(7)}>0$. Then is the Markov chain irreducible and periodic and are the states 8 and 9 recurrent? I got the answer that every state in this chain to be recurrent and the chain to be aperiodic and reducible... Am I correct. In general how to approach a problem like this? Any help please! Also please give me some study material for Markov chain if possible.