An individual named Claudius is located at the point 0 in the accompanying diagram.
Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves to one of the four locations $B_{1}, B_{2}, B_{3}, B_{4}$. Once at one of these locations, he uses another randomization device to decide whether he next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin.
a. Let $X=$ the number of moves that Claudius makes before first returning to 0 . What are possible values of $X$ ? Is $X$ discrete or continuous?
b. If moves are allowed also along the diagonal paths connecting 0 to $A_{1}, A_{2}, A_{3}$, and $A_{4}$, respectively, answer the questions in part (a).