[f97tosc]

Yeah good point, I think the discrepancy boils down to exactly how the frog starts off its epic journey, and that wasn't exactly well defined in the problem statement.

The answer 1-M = 0.5+Y/2 (using your notation here) essentially assumes, as you write, that there (a.s.) exist some lowest integer below which all integers are missed.

But the problem statement says that the frog "is hopping on the integers from minus infinity". So I think that, loosely speaking, rather than thinking of the frog as starting off at some particular integer, we can say that there is no point below which the frog hasn't been when he reaches 0. That is, every point can be treated as k, again using your notation. And then Y is the answer.