In a game I was playing, you roll two six-sided dice. If either of them rolls a one, your points for the turn become zero and your turn is ended. If you roll greater than one on both dice, the sum of the rolls add to your points and you can choose to roll again or choose to end your turn and add your points from the turn to your total score.
I was wondering how many rolls per turn would give you the best possible score on average, so I programmed a simple simulation that tested rolling once, twice, three times, and four times in a turn. It turned out that the best score comes from rolling a pair three times, but I can't explain why. Can anyone explain this through a proof or some equations?