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| XGID=--AC--A--------aaaadcc----:1:-1:1:61:0:0:3:0:10 | |||||
Blue to play 61 (after adding the 15th White checker to the board) | |||||
To review the complete story, click here, but I'll also summarize in the next paragraph:
What is Blue's best play with 61 if the button (White's invisible 15th checker) is on her 2pt? Does that best play change if the button is instead on White's 1pt or on 3pt?
Scotty referred to me as "Nack Moriarty" (an anagram of Karmic Notary). This overly deserved respect (for the creation of a problem that as it happens isn't really so clever) caused him automatically to answer "yes" to the second question, which by a cruel twist turns out to be wrong (as best I can determine).
Obviously, Blue will take the 6 off, and the choice of ace is 3/2 or 2/1. These two end positions for Blue are the same (by transposition) that arose in Stick's recent thread, and that inspired me to see if I could find a subtle surprise with a different White position.
Here is XG's rollout of my button-on-2pt position:
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| XGID=--AC--A--------aaaadcc-a--:1:-1:1:61:0:0:3:0:10 | |||||
Button on 2pt | |||||
| 1. | Rollout1 | 6/Off 3/2 | eq: +1.479 | |||
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| 2. | Rollout1 | 6/Off 2/1 | eq: +1.476 (-0.003) | |||
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| 3. | Rollout1 | 6/Off | eq: +1.410 (-0.069) | |||
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| 1 5184 Games rolled with Variance Reduction. Dice Seed: 86174152 Moves and cube decisions: 4-ply | ||||||
Hmmm. I had expected 3/2 and 2/1 to converge to a 50% tie. White cannot get off the board in fewer than five rolls, and Blue's maximum (only limping with 21 every time) will be four rolls. That is, Blue is gin and it's only a matter of whether he wins a gammon. Therefore, there can be of no issue of cube efficiency (as there was in Stick's position), regardless of current cube placement.
I decided to verify not only that 3/2 and 2/1 are equal-equity plays at DMP, but also that the chances of getting off in 1, 2, 3 and 4 rolls is identical in each category.
Here's my accounting:
66 55 44 33 and 11 are identical (the first 4 doublets get off, and double 1s transposes). Also, the three-roll non-off sequence 21-21-21 (equally likely in both positions) I'll ignore for now.
65 64 63 54 53 43 favor the 2/1 play, leading to a 34- vs 26-number situation. That's a gain of 12 x 8 = 96 (in 1296).
61 51 41 31 also favor 2/1, yielding 17 vs 5. That's a gain of 8 x 12 = 96.
Total gain for 2/1 is 96 + 96 = 192 (in 1296).
Against that, 62 52 43 32 favors 3/2, yielding 25 vs 5, for a gain of 8 x 20 = 160.
21 also favors 3/2, yielding 5 vs 4, for a gain of 2 x 1 = 2.
22 also favors 3/2, yielding 36 vs 6, for a gain of 1 x 30 = 30. [i.e., 21-22 or 22-21]
Total gain for 3/2 is 160 + 2 + 30 = 192 (in 1296).
The 192 that 2/1 and 3/2 each gain over the other cancel out.
Not only is the equity dead equal, but the chances are equal in every category. Namely, here are the chances in 1296 to be off in ...
One roll ......144
Two rolls ....615
Three rolls ..537–
Four rolls .......0+
I hope someone will tell me if s/he finds a mistake in my math. If there is no mistake, then there should be no winner between the 3/2 play and the 2/1 play (my official verdict, unless proven otherwise), and all three XG 4-ply rollouts project the wrong conclusion. For reference, the rollouts that support 6/off 2/1 (for the button being on the 1pt or 3pt) are shown below.
For further reading, try The Curious Case of Benjamin's Button, Part 2
Nack
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| XGID=--AC--A--------aaaadcc--a-:1:-1:1:61:0:0:3:0:10 | |||||
Button on 1pt | |||||
| 1. | Rollout1 | 6/Off 2/1 | eq: +1.477 | |||
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| 2. | Rollout1 | 6/Off 3/2 | eq: +1.475 (-0.002) | |||
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| 3. | Rollout1 | 6/Off | eq: +1.406 (-0.071) | |||
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| 1 5184 Games rolled with Variance Reduction. Dice Seed: 86174152 Moves and cube decisions: 4-ply | ||||||
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| XGID=--AC--A--------aaaadcca---:1:-1:1:61:0:0:3:0:10 | |||||
Button on 3pt | |||||
| 1. | Rollout1 | 6/Off 2/1 | eq: +1.487 | |||
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| 2. | Rollout1 | 6/Off 3/2 | eq: +1.486 (0.000) | |||
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| 3. | Rollout1 | 6/Off | eq: +1.417 (-0.070) | |||
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| 1 5184 Games rolled with Variance Reduction. Dice Seed: 86174152 Moves and cube decisions: 4-ply | ||||||