                     |
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:0:3:0:10 | |||||
65R-54S-41K-31-53Z-32K-41-65r-64X-43H-21R-43Z-44 | |||||
In the position diagrammed above (originally posted here), bgonliners were asked:
"What would you play for Blue with double 4s, at each of the four (classically recognized) main scores?"
m (Money)...(as shown)
The best move is B (Both) = 24/16* 13/9(2), chosen by three respondents in this chronological order:
Gudgen: [Without comment]
Joe Russell: "A Ballard Challenge usually features a different play for each state. Assuming that is the case, two of the states seem easy. [Joe now stated his choices for gammon save and gammon go, to be reiterated later.] The toughies are Money and DMP. I have no clue. Normally, I would not be surprised if the same play wax right for both. I see the merit of four plays: C, B, b and R; but since two of the plays have been used, I must limit my decisions to the other two. I would not have considered either at DMP, anyway, and would not have considered the save play for money... Since I think R looks better for DMP than for for money, that must be my DMP pick and B must be my money pick. (If this were not a Ballard Challenge, I think I would have gone with B for both states.)"
Igor: "The 41S-44 second roll reference [that B is best for money -- see Igor's later notes under DMP] cannot be extended to money and GG since hitting two is correct for both, but is not an option... In the actual position. I'm [not sure] for money; [hit and] making the 4pt with C is also a very strong contender. I expect B and C to be pretty close for money, but I still chose B."
By the way, Joe and Igor nailed the best play at all four scores. Moreover, Igor nactated all four moves correctly! (Other respondent's apparent uses of Nactation were mostly my editing. B (Both) = 24/16* 13/9(2), b (both) = 24/20(2) 13/9(2), C (Cross) = 24/16* 8/4(2), and R (Run) = 24/16*(2).)
Here is the rollout:
........m (Money) 3-ply
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:0:3:0:10 | |||||
m[B C18 R25 b47 c93] "<=31*20 5 | |||||
| 1. | Rollout1 | 24/16* 13/9(2) | eq: +0.455 | |||
| ||||||
| 2. | Rollout1 | 24/16* 8/4(2) | eq: +0.437 (-0.018) | |||
| ||||||
| 3. | Rollout2 | 24/16*(2) | eq: +0.431 (-0.025) | |||
| ||||||
| 4. | Rollout3 | 24/20(2) 13/9(2) | eq: +0.409 (-0.046) | |||
| ||||||
| 5. | Rollout3 | 24/20(2) 8/4(2) | eq: +0.362 (-0.093) | |||
| ||||||
| 1 31104 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller 2 20736 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller 3 5184 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller | ||||||
d (DMP)...
The best move is R (Run) = 24/16*(2), selected by just two respondents:
Joe: [See his earlier comments for Money]
Igor: "I'm pretty sure about my DMP and GS choices since the same moves are best for 41S-44 second roll, which we know. I don't think having an extra spare on the 8pt is sufficient to change these plays. I'm surprised nobody mentioned 41S-44 reference in their analyses."
[Igor brings up a valuable technique for making early-game decisions: that of comparison to already-well-known plays in similar positions. Personally, I like comparing the feature position to 54D-44 and 64S-44 even more because they are three-score (instead of two-score) positions. See my bonus analysis "Constructing a Four-Score Position" in the final leg of this post.]
Below are rollouts, 3-ply and 4-ply:
........d (DMP) 3-ply
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:0:0:1:10 | |||||
d[R B8 b13 C29 c40] "<=23*5 | |||||
| 1. | Rollout1 | 24/16*(2) | eq: +0.207 | |||
| ||||||
| 2. | Rollout1 | 24/16* 13/9(2) | eq: +0.199 (-0.008) | |||
| ||||||
| 3. | Rollout2 | 24/20(2) 13/9(2) | eq: +0.194 (-0.013) | |||
| ||||||
| 4. | Rollout2 | 24/16* 8/4(2) | eq: +0.178 (-0.029) | |||
| ||||||
| 5. | Rollout2 | 24/20(2) 8/4(2) | eq: +0.167 (-0.040) | |||
| ||||||
| 1 23328 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller | ||||||
........d (DMP) 4-ply
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:0:0:1:10 | |||||
d[R B13] "^5 | |||||
| 1. | Rollout1 | 24/16*(2) | eq: +0.206 | |||
| ||||||
| 2. | Rollout1 | 24/16* 13/9(2) | eq: +0.193 (-0.013) | |||
| ||||||
| 1 5184 Games rolled with Variance Reduction. Dice Seed: 91208355 Moves and cube decisions: 4-ply Search interval: Large | ||||||
s (Gammon Save)...
The best move is b (both) = 24/20(2) 13/9(2), supported by Mike Clapsadle, Lenny, Gudgen (those three without comment), Joe ("seems easy" [assuming a four-score distribution]), and Igor (see his notes under DMP).
Here is the rollout:
........s (Gammon Save) 3-ply
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:1:0:1:2:10 | |||||
s[b c31 R49 B67 C88] "<= | |||||
| 1. | Rollout1 | 24/20(2) 13/9(2) | eq: +0.062 | |||
| ||||||
| 2. | Rollout1 | 24/20(2) 8/4(2) | eq: +0.032 (-0.031) | |||
| ||||||
| 3. | Rollout1 | 24/16*(2) | eq: +0.013 (-0.049) | |||
| ||||||
| 4. | Rollout1 | 24/16* 13/9(2) | eq: -0.005 (-0.067) | |||
| ||||||
| 5. | Rollout1 | 24/16* 8/4(2) | eq: -0.025 (-0.088) | |||
| ||||||
| 1 5184 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller | ||||||
g (Gammon Go)...
The best move is C (Cross) = 24/16* 8/4(2), the choice of Gudgen, Joe ("easy" [assuming a four-score position]), Mike ("seems the most obvious") and Igor (refer to his notes under Money).
Below are rollouts, 3-ply and 4-ply:
........g (Gammon Go) 3-ply
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:1:1:2:10 | |||||
g[C B9 b74 c79 O82 R89] "<=46*5 | |||||
| 1. | Rollout1 | 24/16* 8/4(2) | eq: +0.545 | |||
| ||||||
| 2. | Rollout1 | 24/16* 13/9(2) | eq: +0.537 (-0.009) | |||
| ||||||
| 3. | Rollout2 | 24/20(2) 13/9(2) | eq: +0.472 (-0.074) | |||
| ||||||
| 4. | Rollout2 | 24/20(2) 8/4(2) | eq: +0.466 (-0.079) | |||
| ||||||
| 5. | Rollout2 | 13/9(2) 8/4(2) | eq: +0.464 (-0.082) | |||
| ||||||
| 6. | Rollout2 | 24/16*(2) | eq: +0.456 (-0.089) | |||
| ||||||
| 1 46656 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves: 3-ply, cube decisions: XG Roller | ||||||
........g (Gammon Go) 4-ply
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:1:1:2:10 | |||||
g[C B9] "^10 | |||||
| 1. | Rollout1 | 24/16* 8/4(2) | eq: +0.546 | |||
| ||||||
| 2. | Rollout1 | 24/16* 13/9(2) | eq: +0.536 (-0.009) | |||
| ||||||
| 1 10368 Games rolled with Variance Reduction. Dice Seed: 64431039 Moves and cube decisions: 4-ply Search interval: Large | ||||||
The nacbracs summary below lists the best play at each score followed by successively weaker plays (with error sizes), according to the rollouts (diagrammed above). For example, for m (Money) 3-ply, B is best, then relatively C is –.018, R is –.025, b is –.047, and c is –.093.
...............3-ply.................................................4-ply
..... m[B C18 R25 b47 c93] "<=31*20 5
...... d[R B8 b13 C29 c40] "<=23*5 ........... d[R B13] "^5
...... s[b c31 R49 B67 C88] "<=5
...... g[C B9 b74 c79 O82 R89] "<=46*5 .... g[C B9] "^10
Key: B (Both) = 24/16* 13/9(2), b = 24/20(2) 13/9(2), C (Cross) = 24/16* 8/4(2), c = 24/20(2) 8/4(2), O (Outer) = 13/9(2) 8/4(2), R (Run) = 24/16*(2).
Nactation tutorial • explanation of nacbracs
In the above summary, a quick look down the left column reveals that each of the four scores (m, d, s, g) champions a different best play (B, R, b, c). Moreover, the margins of .018, .008, .031 and .009 exceed their respective "tied" error-scale cusps of .010, .0045, .007 and .007.
The power score (of this double 4s position) is [sqrt 18 + sqrt 8*20/9 + sqrt 31*10/7 + sqrt 9*10/7] = 18.70. For more information on four-score positions, error-scale cusps and power scores, click mainly here, also here and here.
Constructing a Four-Score Position
Here's how the creation of this thread's feature position came about:
Having previously found at least one R4 (real four-score position) for each of the small doublets (1s, 2s, 3s), I sought one for double 4s. (For more on four-score positions, click here.) It occurred to me that there conveniently exist a couple of second-roll double-4s positions that qualify as three-score (one T3, one R3), so I decided to see if, by adjusting one of those, I might be able to create an R4.
These two reply positions, 54D-44 and 64S-44, are diagrammed below (side by side, if you widen your window enough):
     |
| ||||
| XGID=-b----E-C---cE--ad-e----B-:0:0:1:44:0:0:3:0:10 | |||||
54D-44 | |||||
 |
| ||||
| XGID=-a----EaC---dE--ac-e----B-:0:0:1:44:0:0:3:0:10 | |||||
64S-44 | |||||
The first set of nacbracs are for 54D-44 (left-hand position), the second set for 64S-44 (right-hand position). The first column is for XGR++ evaluation, the second column for a rollout (if I have one, in green).
----------------------------------------------------------------------------------------------
... 54D-44 eval.......................... 54D-44 ro.......... 41T-54D-44 eval................ –2J-54D-44 eval
m[B b43 R60 E108 C116]e... m[B b24] "<31..... m[C c58 B73 R74 b91]e..... m[C B12 c67 b78 R86]e
.d[B R3 b24 E60 C72 c84]e... d[R B3] "<31....... d[R C7 c22 B24 b30]e........ d[B C6 R13 b34 c37]e
.s[b R45 B46 c89 e101]e..................................... s[c b1 R30 C36 B64]e......... s[b c9 B27 R37 C37]e
.g[B C41 E50 b69 R88]e..................................... g[C B73 c95 E112 R127]e... g[C B38 c93 b124 R143]e
----------------------------------------------------------------------------------------------
.... 64S-44 eval.......................... 64S-44 ro.......... 41T-64S-44 eval................ –2J-64S-44 eval
.d[R B20 E66 b83 C108]e...... d[R B22] /5.......... d[R C27 B49 E77]e............. d[R C9 B9 b90 c92]e
.s[R b18 B33 e76 E104]e....... s[b R16] "<=5........ s[R b33 c38 C43]e.............. s[R b23 C24 B24 c37]e
.g[B E24 R49 C64]e............... g[B E31] "<=15..... g[C B85 E92 R105]e........... g[C B51 R113 E151]e
----------------------------------------------------------------------------------------------
Nactation key: B (Both) = 24/16* 13/9(2), b = 24/20(2) 13/9(2), C (Cross) = 24/16* 8/4(2)), c = 24/20(2) 8/4(2), E (Each) = 24/16* 6/2(2), e = 24/20(2) 6/2(2), R (Run) = 24/16*(2). Nactation tutorial
----------------------------------------------------------------------------------------------
In blue typeface, I've marked any pair of best and second best plays for which the rollout reverses the relative evaluation. As you can see, there are three different best plays -- same for both positions. For money, DMP and gammon save, the best play, respectively, is B, R and b. In short, 54D-44 and 64S-44 are both three-score positions.
What prevents 54D-44 and 64S-44 from being four-score positions is that in each case the best play at gammon go (g) is the same as it is for money (m). Therefore, I looked to somehow adjust the position to generate a fourth play that could be best for either money or gammon go (but not both) while also keeping the best plays for DMP and gammon save intact. If successful, I could achieve at least a T4, and with enough perseverence or luck an R4.
It seemed to me that putting a fourth checker on Blue's 8pt had a chance of working. This change makes it reasonable to play C (Cross, 24/16* 8/4(2)) at gammon go, and/or conceivably open up c (cross, 24/20(2) 8/4(2)) as a best play at some score.
Suppose we try giving Blue an extra 5 down (13/8), which is represented by the pre-play of "41T" in the third column. Sure enough, in both positions, C becomes best at gammon go. Unfortunately, that also makes C best for money, because B now strips the midpoint. (We could try obtaining a different gammon go play of E (Each, 24/20* 6/2*(2) by also or instead moving some White checker to her 23pt, but in that case E dominates other scores as well -- essentially the same problem.)
To keep the midpoint at five checkers (so as not to strip it with 13/9(2)), what I actually did was to move back the fifth checker on the 6pt instead. This change (spare from 6pt to 8pt) is represented by "–2J" (a minus 2-pip Jump) in the fourth column.
We're closer now. As in the third column, in the fourth column the C play is best at gammon go. Money has still come along for the ride, but B's inferiority to C is as small as .012 in the case of –2J-54D-44. Granted, the DMP (d) eval has defected (from R) to B, but the eval/rollout swing of .006 in the first column suggests that the B/R margin in the fourth column is likely smaller than .013.
[These observations are useful, as XGR++ evaluations often go astray. A striking example is the 64S-44-s eval of [R b18], reversed in the rollout to [b R16]. The existence of such misevaluations complicates the task of finding a four-score-position, but by extrapolating from this information, one can better predict what reversals might occur when testing checker adjustments.]
The R play maintains its lead in the case of –2J-64S-44-d, which suggests that splitting White's back checkers (or putting a third one back), in conjunction with 54D, might generate an even more score-balanced division of best plays for Blue. (Indeed, that turns out to the the case.)
After quite a bit of educated trial and error, I honed in on the feature position, diagrammed at the beginning of this post and repeated below. (It can be reached by the perfectly-played 11th-roll sequence captioned, or from the modified opening position shown in the fifth diagram here and White plays 42Z.) As this post is long enough, I'll spare you the remaining details of the procedure by which (passing up at least one T4 along the way) I was ultimately fortunate enough to pinpoint an R4. For enthusiasts who may someday like to create their own four-score position, this partial account will hopefully help.
Nack
|
| ||||
| XGID=-a-a--D-D---dE--ad-d----B-:0:0:1:44:0:0:3:0:10 | |||||
65R-54S-41K-31-53Z-32K-41-65r-64X-43H-21R-43Z-44 | |||||