We study the expected number of rolls required for the cumulative sum of a fair six-sided die to first enter a prescribed target set $H\subset\mathbb{Z}_{\ge0}$. A one-variable dynamic-programming formulation is introduced that removes dependence on the roll count. Within this framework, the infinite process is truncated at a large cutoff $N$ and corrected by an analytically derived overshoot term that accounts for the rare event of exceeding $N$ before entering $H$. Explicit bounds on this residual yield a strict two-sided estimate of the truncation error. The method is numerically efficient, requiring constant memory and linear time in the cutoff. For the perfect-square target set $H=\{n^2:n\in\mathbb{N}\}$, all quantities are evaluated explicitly, yielding \[ \mathbb{E}[T]=7.07976423755110510389555305690818489468\ldots, \] provably correct to 1,017 decimal places. This constitutes the most precise result known to date and establishes a general framework for high-accuracy computation of discrete hitting times.