# Hmm, this problem seems familiar

Suppose I have a line of squares, labelled 0, 1, 2, 3, 4, ... and so on. I place a counter on the square 0. On every turn, I roll a fair die (with the numbers 1 to 6) and move forward as follows: if I was on square $$P$$ before the roll and I get a $$x$$ then I move to square $$P+x$$. Let $$X_n$$ be the chance that I land on the square labelled $$n$$, where $$n$$ is a positive integer. What is

$\lfloor\text{max}(1000X_n)\rfloor? \quad ?$

If the maximum doesn't exist, find the infimum of the set of $$X_n$$.

Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

×