Edit: I found a bug in my code, so I've updated the simulation results.
Edit #2: I found another bug in my code, and so I've updated the results again. And, added ran a couple other strategies just to see results.
I ran an initial simulation with a simple default strategy as follows:
1) Always keep 1 and 4 if you don't already have them.
2) Always keep 6 if you already have both a 1 and a 4.
3) If you didn't keep anything in the current roll (1/4/6 as per Rules #1 and #2) , then keep a single die (the highest valued one).
| Score | Frequency | CDF | |
| Not Qualified | 4324214 | (4.32%) | (100.00%) |
| 4 | 0 | (0.00%) | (95.68%) |
| 5 | 11 | (0.00%) | (95.68%) |
| 6 | 143 | (0.00%) | (95.68%) |
| 7 | 1094 | (0.00%) | (95.68%) |
| 8 | 5670 | (0.01%) | (95.67%) |
| 9 | 21906 | (0.02%) | (95.67%) |
| 10 | 75295 | (0.08%) | (95.65%) |
| 11 | 222711 | (0.22%) | (95.57%) |
| 12 | 562945 | (0.56%) | (95.35%) |
| 13 | 1239852 | (1.24%) | (94.79%) |
| 14 | 2489387 | (2.49%) | (93.55%) |
| 15 | 4420645 | (4.42%) | (91.06%) |
| 16 | 6855657 | (6.86%) | (86.64%) |
| 17 | 9436721 | (9.44%) | (79.78%) |
| 18 | 11863564 | (11.86%) | (70.34%) |
| 19 | 14150677 | (14.15%) | (58.48%) |
| 20 | 13048294 | (13.05%) | (44.33%) |
| 21 | 11259056 | (11.26%) | (31.28%) |
| 22 | 8955723 | (8.96%) | (20.02%) |
| 23 | 6549697 | (6.55%) | (11.07%) |
| 24 | 4516738 | (4.52%) | (4.52%) |
Here are results for a different strategy. Based on these results, it seems superior in most cases -- one case where it would not be superior is if you need only to score 15 to win (the following strategy hits 15+ 85% of the time while the previously described default strategy gets you there 91% of the time).
Evaluate the dice in the following sorted order: (1+4), 6, 5, 3, 2
1) Always keep 1 and 4 if you don't already have them.
2) Always keep 6 if you already have both a 1 and a 4.
3) If you have 2 or fewer dice left to consider, then keep a 5.
4) If you have only one die left to consider, then keep a 4. (Basically, expected value of a single roll is 3.5. You are 50% to do worse than a 4, and only 33% to do better. So, keeping the 4 is best.)
5) If you didn't keep anything in the current roll (as per above rules) , then keep a single die (the highest valued one).
| Score | Frequency | CDF | |
| Not Qualified | 12031334 | (12.03%) | (100.00%) |
| 4 | 0 | (0.00%) | (87.97%) |
| 5 | 14 | (0.00%) | (87.97%) |
| 6 | 106 | (0.00%) | (87.97%) |
| 7 | 746 | (0.00%) | (87.97%) |
| 8 | 3596 | (0.00%) | (87.97%) |
| 9 | 14262 | (0.01%) | (87.96%) |
| 10 | 49197 | (0.05%) | (87.95%) |
| 11 | 147225 | (0.15%) | (87.90%) |
| 12 | 372987 | (0.37%) | (87.75%) |
| 13 | 821681 | (0.82%) | (87.38%) |
| 14 | 1633170 | (1.63%) | (86.56%) |
| 15 | 2888930 | (2.89%) | (84.93%) |
| 16 | 4484273 | (4.48%) | (82.04%) |
| 17 | 6258644 | (6.26%) | (77.55%) |
| 18 | 9710872 | (9.71%) | (71.29%) |
| 19 | 11806824 | (11.81%) | (61.58%) |
| 20 | 11650430 | (11.65%) | (49.78%) |
| 21 | 11174701 | (11.17%) | (38.13%) |
| 22 | 12093475 | (12.09%) | (26.95%) |
| 23 | 10505528 | (10.51%) | (14.86%) |
| 24 | 4352005 | (4.35%) | (4.35%) |
Here's the 'go for broke' strategy, where you are going for 24 only. This means that you are going to keep all 6's (up to 4) and keep 1 and 4 as they present themselves. This strategy is forced if someone before you had gotten 24, and so you need specifically 24 to tie.
| Score | Frequency | CDF | |
| Not Qualified | 14806947 | (14.81%) | (100.00%) |
| 4 | 1 | (0.00%) | (85.19%) |
| 5 | 9 | (0.00%) | (85.19%) |
| 6 | 117 | (0.00%) | (85.19%) |
| 7 | 728 | (0.00%) | (85.19%) |
| 8 | 3590 | (0.00%) | (85.19%) |
| 9 | 14230 | (0.01%) | (85.19%) |
| 10 | 49667 | (0.05%) | (85.17%) |
| 11 | 146853 | (0.15%) | (85.12%) |
| 12 | 374034 | (0.37%) | (84.98%) |
| 13 | 821225 | (0.82%) | (84.60%) |
| 14 | 1625200 | (1.63%) | (83.78%) |
| 15 | 2859193 | (2.86%) | (82.16%) |
| 16 | 4400751 | (4.40%) | (79.30%) |
| 17 | 6108761 | (6.11%) | (74.90%) |
| 18 | 9339354 | (9.34%) | (68.79%) |
| 19 | 11297706 | (11.30%) | (59.45%) |
| 20 | 11116484 | (11.12%) | (48.15%) |
| 21 | 10652890 | (10.65%) | (37.04%) |
| 22 | 11547947 | (11.55%) | (26.38%) |
| 23 | 10035266 | (10.04%) | (14.83%) |
| 24 | 4799047 | (4.80%) | (4.80%) |
5 comments:
We play this game almost everyday at a local bar.
Could you explain the frequency of the "not qualified"?
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I suggest another strategy. On the first roll, if you get both a 1 and a 4, don't keep the 1 (unless you get a 24 on the first roll). I suspect this will only slightly increase the chance of not qualifying while significantly increasing the expected score.
What are the chance of rolling 1/4/24 on the first roll. I play after work all the time and we contribute to a separate pot just for anyone who rolls 1/4/24 on the first roll which seems to happens about every 2-3 weeks but we play almost every day!
We also play 1/4/4 and play both versions with a 1 dice limit after each roll just to mix it up a bit.
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