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 to decider 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).