Sunday, August 15, 2010

1-4-24 Simulation

So, I've written some simple simulation code for the 1-4-24 game. If you're not familiar with it, see this old post: 1-4-24 Dice Game.

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 PDF 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 PDF 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 PDF 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%)

3 comments:

The Double Down said...

We play this game almost everyday at a local bar.
Could you explain the frequency of the "not qualified"?

Elliott Starno said...

1-4-24 is now available on the iPhone/iPod!

Check it out: http://itunes.apple.com/us/app/midnight-1424/id438054437?mt=8

www.midnight1424.com

Unknown said...

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.

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