First-passage probabilities and closed classes

Suppose $\{X_n,n\ge0\}$ is a Markov chain on $S=\{1,2,3,4\}$ with

$$P=\begin{pmatrix} \tfrac12 & 0 & \tfrac12 & 0\\ 0 & \tfrac13 & 0 & \tfrac23\\ \tfrac23 & 0 & \tfrac13 & 0\\ 0 & \tfrac14 & 0 & \tfrac34 \end{pmatrix}.$$

With $f_{ij}$ the probability of a first visit to $j$ from $i$, consider (I) $f_{12}=0$, (II) $f_{24}=1$, (III) $f_{13}=1$, (IV) $f_{33}<1$.