This is based on a statistics question I had over on the rpg.stackexchange site and it was suggested that I bring it over here.
The premise is a game where you have 20 d6 dice. You roll each die sequentially and add the result to your score. Each time before rolling a new die you have the option to reset your score and re-roll all previously rolled dice along with the new die.
For example: You roll a 4 on your first die, then a 2 on your second die for a score of 6. You choose to re-roll on your third die, rolling 3d6 and getting 3. On your fourth die you again choose to re-roll (4d6) and this time you get 18. etc...
What strategy of re-rolling would result in the maximum score once you have rolled all 20 dice?
Suppose you have rolled $n$ dice. If you choose to re-roll all of these, then your expected score for these dice is $7n/2$. So you should only choose to re-roll all the die if your total score after $n$ rolls is less than $7n/2$, else you'd have nothing to gain.
Now, finding the expected value of following this strategy is much more interesting. I haven't gone through the computations yet, but I can say for sure that it should be larger than the average score of $70$ that you'd expect with no re-rolling.