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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)?

Edited by HoustonHokie

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Expected scores for number of dice remaining is:

Dice remaining and Expected scores

1 ----- 25,0

2 ----- 50,0

3 ----- 86,8

4 ----- 143,5

5 ----- 225,8

6 ----- 381,8

Dice remaining and Probability of farke

1 ----- 67%

2 ----- 44%

3 ----- 28%

4 ----- 16%

5 ----- 8%

6 ----- 2%

Given this, and the expected score, it is best to stop if the score is more than the below scores for each number of dice remaining

Dice remaining and the rationale to stop if score in the turn thus far is more than

1 ----- 38

2 ----- 113

3 ----- 313

4 ----- 912

5 ----- 2 926

6 ----- Always Play

Edited by DeeGee
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1 25
2 50
3 86.80555556
4 143.5185185
5 206.5007716
6 305.6680813
`Dice	Avgscore`

For 5 Dice I could change two numbers in my spreadsheet from 3125 to 4125 and get deegees answer and I also used those numbers to some extent in the avg score for 6 dice

but then again after 4dice i'm not too sure about my answers

If I can get these answers confirmed i'll work on the second part

I think it would only be advantageous to not keep all your scoring dice if the points gained by keeping the extra dice is lower than the difference between the average score of the number of dice you would roll if you didn't keep (the larger number of dice) them and the average score of the number of dice you would roll if you did keep them (the smaller number of dice)

Although I can't think of many cases where this happens.

One example though is:

You roll 6 dice and the only scoring dice is a single 5.

If you keep the 5 and take the 50 points you will be rolling 5 dice and then average (about) 200 points plus the 50 from the 5 previously rolled for a total of 250 points.

But if you don't keep the 5 and roll all 6 again you will average (about) 300 points, which is 50 points more than if you would have kept the 5.

Edited by K4D

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