Given there is a regular fair die, roll the die and stop if the sum of all previous rolls is a multiple of 3. What is the expected number of rolls?
Let $X$ denote the number of rolls until the event {the sum of all previous rolls is a multiple of 3} happens.
For $X = 1$, the rolls must be 3 or 6.
For $X = 2$, the rolls must be one of $\{(1,2), (2,1), (1,5), (5,1), (2,4), (4,2), (4,5), (5,4) \}$
For $X = 3$, the rolls must be one of $\{(1,1,1), (1,3,2),(1,3,5),(1,4,1), \ldots\}$
From the above enumeration, the pattern was not clear to me. I suppose there must be some "patterns" to simplify the computation.