Dynamic Dice Game

The Problem

You may roll a fair six-sided die up to $n$ times. After each roll you either accept the value shown (the game ends and you collect that value) or roll again, in which case the previous value is discarded. You cannot roll more than $n$ times.

Questions

1. Write the Bellman recurrence for $V(k)$, the expected payoff when $k$ rolls remain.
2. Compute $V(3)$ and describe the optimal stopping rule.