From time to time, I've played a dice game called Farkle. Maybe you've played it before. To start your turn, you roll 6 dice, and, if your roll scores (see below for scoring), you can do one of three things:
1. Add the score from all the scoring dice to your total for the turn and set those dice aside. Then, pick up the remaining dice and roll again.
2. If you score in more than one way (say you roll a 1 and a 5, or two 1s), you choosing only the scoring combinations you want to keep. You add the score from those dice to your score for the turn and set them aside. Then, pick up the remaining dice and roll again. [The difference between #1 and #2 is the rejection of some scoring combinations in favor of having more dice available for your next roll.]
3. Add the score from all the scoring dice to your total for the turn and stop rolling. These points become permanently attached to your score for the game.
The twist to the game is what happens if you don't score on a particular roll. This is called a Farkle, and you lose all the points you have accumulated in that turn. You still have the points from previous turns, but anything you might have gained in previous rolls that turn is lost. Here's an example of the effect of a Farkle:
Roll 1: Three 1s appear, adding 1000 to your score for the turn. You can stop rolling now and add 1000 to your score for the game. Or, you can roll again, but with only 3 dice this time.
Roll 2: Farkle - no scoring dice appear. You turn ends, and you add nothing to your score for the game, not even the 1000 points you earned on the previous roll.
If you manage to score with all 6 dice on one or multiple rolls, you get to roll all 6 again. A Farkle, however, will still cause you to lose all your points for the turn.
Scoring:
If you roll a 1, you get 100 points
If you roll a 5, you get 50 points
If you roll 3 of a kind, you get 1000, 200, 300, 400, 500, or 600 points (for rolls of three 1s, 2s, 3s, 4s, 5s, 6s respectively)
If you roll 4 of a kind, you get twice the value of 3 of a kind
If you roll 5 of a kind, you get twice the value of 4 of a kind
If you roll 6 of a kind, you get twice the value of 5 of a kind
If you roll 3 pair, you get 750
If you roll a straight (1 through 6), you get 1500
For the 3, 4, 5, 6 of a kind, 3 pair, or straight, you have to have all those dice appear on a single roll. For example, rolling a 1 on your first roll can't be followed by rolling 2, 3, 4, 5, 6 on your second roll to score a straight.
Obviously, the goal is to score as quickly as possible. Most times, the game is played to a certain limit (say first player to reach 10,000 wins). It's easy to score - but the trick is knowing when to stop.
Enough about the rules. Here's my questions:
1. Given a certain number of dice remaining to roll (1-6), what is your probable score for the next roll (assuming you keep all your scoring dice)?
2. Assuming you're required to keep all your scoring dice and keep rolling (ignore possibilities #2 & #3 above), what is the value of an average Farkle turn after the first roll? After the second? Third? Fourth?
3. When is it advantageous to apply option #2 above and not keep all your scoring dice?
4. Can you develop a rational method for stopping your turn (#3 above)?
