Each 1: 100 points
Each 5: 50 points
3 1's: 1000 points
3 2's: 200 points
...
3 6's: 600 points
4 1's: 2000 points
...
4 6's 1200 points
5 1's 3000 points
...
5 6's 1800 points
Straight 1-6 1500 points
3 Pairs (E.G. 22 33 44): 750 points
Everything else is considered to be a Farkle (0 points).
I calculated these figures under the assumption that we are trying to maximize expected value, however that is not necessarily the case. To employ a true optimal strategy, we want to maximize the probability of winning which depends on both your current score and your opponents score. However, to make the calculations easier, I ignored those two variables altogether and simply found how to maximize the expected value for any single turn.
Here is a small table that shows my findings.
Each entry in the table shows the expected value in the given situation if you were to employ this optimal strategy.
For example, lets say I have 250 points and I am rolling 3 dice, it can be seen that the best option for me is to roll again (yielding an expected value of 293.4 which is better than 250). However, if I had 500 points rolling 3, banking the 500 points is the best play because the risk of Farkling is too high to wager the 500 points.
Note that not all entries are filled in in the table. This is because they are impossible states. (I.E. you can't have 50 points rolling 3).
Now let's consider a more complex situation. Let's say I just rolled the following 6 dice:
1 2 2 2 5 4
Clearly I have a lot of options, but which one is best? Let's use the table!
50 rolling 5 (5) - EV = 363.834
100 rolling 5 (1) - EV = 399.911
150 rolling 4 (1 5) - EV = 307.673
200 rolling 3 (2 2 2) - EV = 257.584
250 rolling 2 (2 2 2 5) - EV = 250
300 rolling 2 (2 2 2 1) - EV = 300
350 rolling 1 (2 2 2 1 5) - EV = 350
So the best option is to bank the 1 and roll the other 5 dice, resulting in an average of about 400 points. Using this same methodology, you can take any situation and determine the best possible play.
Number of Dice | ||||||
Points | 1 | 2 | 3 | 4 | 5 | 6 |
0 | 548.858 | |||||
50 | 363.834 | |||||
100 | 274.776 | 399.911 | ||||
150 | 227.968 | 307.673 | ||||
200 | 221.706 | 257.584 | 343.8 | |||
250 | 274.669 | 250 | 293.4 | |||
300 | 300 | 300 | 329.228 | 802 | ||
350 | 350 | 350 | 596.22 | 846.014 | ||
400 | 400 | 400 | 400.947 | 498.2 | 638.291 | 890.036 |
450 | 450 | 450 | 450 | 540.176 | 680.368 | 934.065 |
500 | 500 | 500 | 500 | 582.157 | 722.453 | 978.099 |
550 | 550 | 550 | 550 | 624.144 | 764.545 | 1022.666 |
600 | 600 | 600 | 600 | 666.139 | 806.642 | 1067.891 |
650 | 650 | 650 | 650 | 708.139 | 849.953 | 1113.281 |
700 | 700 | 700 | 700 | 750.142 | 893.894 | 1158.674 |
750 | 750 | 750 | 750 | 792.147 | 937.847 | 1204.068 |
800 | 800 | 800 | 800 | 834.154 | 981.801 | 1249.464 |
850 | 850 | 850 | 850 | 876.162 | 1025.756 | 1295.174 |
900 | 900 | 900 | 900 | 918.17 | 1069.711 | 1341.411 |
950 | 950 | 950 | 950 | 960.178 | 1114.429 | 1387.917 |
1000 | 1000 | 1000 | 1000 | 1002.188 | 1160.198 | 1434.468 |
1050 | 1050 | 1050 | 1050 | 1050 | 1206.254 | 1481.147 |
If you're interested in seeing the extended table (up to 10000 points), or would like to understand the methods that I used to come up with this strategy, feel free to send me an email at RMcKenna21@gmail.com.