On the other hand, if OP meant Moe must see 6 different numbers in a row, then we have a problem...
At first glance it appears, that in this case May takes 6 turns while Moe takes one. But it is not so. Moe does not have a fixed position from which to start counting his 6 consecutive rolls. Let's say, the rolls 1 through 6 did not reach the objective, then it is still possibe for the rolls 2 through 7, or 3 through 8, and so on.
Prime has it. Moe sees his numbers first.