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Four Score Position Cluster

Posted By: Nack Ballard
Date: Tuesday, 22 August 2017, at 8:45 p.m.

Constructing an "R4" (real four-score position), even with extensive educated-trial-and-error adjustments, is a process akin to finding a needle in a haystack. Making a checker adjustment to accommodate one score almost always throws at least one of the other scores out of kilter. Not only must there be a different best play at each of the four main scores (money, DMP, gammon save, and gammon go), but none of the best plays may be "tied" with any of its rival best plays: to qualify as an R4, those margins must be cleared. For more information on R4s, T4s, power ratings, etc., click here (and there are three more R4s here, here and here).

This post features the first cluster of real four-score positions ever discovered. Fundamentally, these ten R4s are variants of each other. Blue's position remains constant, while White has different numbers of checkers on six offensive points (13pt, 10pt, 8pt, 6pt, 4pt and 3pt). Each initial diagram caption lists this distribution, along with the power rating and a perfectly played money sequence (above it) that reaches the position.

The first of these positions includes a bonus feature: a fifth score at which there is yet another best play. As that bonus best play is only technically "tied," the position is defined as an "R4+T1" (real four-score position plus a technical best play at a fifth score). A detailed rollout is shown at each of the five scores.

After that, in the interests of space, I've included only ONE diagram (instead of four) for each of the other nine positions (presented in order of power rating). Rollout results are 3-ply checker, XGR cube. For any results near a tied cusp, I've also verified with 4-ply rollouts, which are included in the nacbracs summaries. I'll be happy to post any full rollout details upon request.

Money

		White is Player 2 score: 0 pip: 146 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-bC-b-c-d--b-B-:0:0:1:44:0:0:3:0:10
43D-32Z-63P-31P-63S-31P-21U-44 White distribution 13/10/8/6/4/3 = 223402, Power rating = 23.29

1.	Rollout1	24/20(2) 7/3*(2)	eq: +0.263
	Player: Opponent: 55.02% (G:17.58% B:0.48%) 44.98% (G:8.36% B:0.33%) Conf.: ± 0.001 (+0.262...+0.264) - [100.0%] Duration: 1 day 10 hours 03 minutes
2.	Rollout1	24/20(2) 8/4(2)	eq: +0.251 (-0.012)
	Player: Opponent: 55.31% (G:15.19% B:0.44%) 44.69% (G:7.82% B:0.30%) Conf.: ± 0.001 (+0.250...+0.252) - [0.0%] Duration: 1 day 11 hours 24 minutes
3.	Rollout2	24/20(2) 13/9(2)	eq: +0.218 (-0.045)
	Player: Opponent: 55.07% (G:14.04% B:0.56%) 44.93% (G:8.75% B:0.35%) Conf.: ± 0.008 (+0.210...+0.226) - [0.0%] Duration: 48 minutes 53 seconds
4.	Rollout2	8/4(2) 7/3*(2)	eq: +0.205 (-0.058)
	Player: Opponent: 53.41% (G:23.48% B:1.00%) 46.59% (G:14.48% B:0.95%) Conf.: ± 0.008 (+0.196...+0.213) - [0.0%] Duration: 26 minutes 41 seconds
5.	Rollout2	24/16(2)	eq: +0.204 (-0.059)
	Player: Opponent: 55.95% (G:8.84% B:0.35%) 44.05% (G:10.33% B:0.34%) Conf.: ± 0.006 (+0.197...+0.210) - [0.0%] Duration: 39 minutes 40 seconds
6.	Rollout2	24/20(2) 6/2(2)	eq: +0.180 (-0.083)
	Player: Opponent: 53.46% (G:14.47% B:0.44%) 46.54% (G:8.50% B:0.33%) Conf.: ± 0.006 (+0.173...+0.186) - [0.0%] Duration: 34 minutes 26 seconds
7.	Rollout2	7/3*(2) 6/2(2)	eq: +0.149 (-0.114)
	Player: Opponent: 52.11% (G:22.19% B:1.10%) 47.89% (G:15.63% B:1.10%) Conf.: ± 0.008 (+0.141...+0.156) - [0.0%] Duration: 27 minutes 44 seconds
1 362880 Games rolled with Variance Reduction. Dice Seed: 95623456 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 95623456 Moves: 3-ply, cube decisions: XG Roller

DMP

		White is Player 2 score: 0 pip: 146 1 point match pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-bC-b-c-d--b-B-:0:0:1:44:0:0:0:1:10
Blue to play 44

1.	Rollout1	24/16(2)	eq: +0.119
	Player: Opponent: 55.95% (G:9.73% B:0.84%) 44.05% (G:10.76% B:0.96%) Conf.: ± 0.000 (+0.118...+0.119) - [100.0%] Duration: 18 hours 07 minutes
2.	Rollout1	24/20(2) 13/9(2)	eq: +0.113 (-0.006)
	Player: Opponent: 55.63% (G:14.20% B:1.30%) 44.37% (G:11.80% B:0.86%) Conf.: ± 0.001 (+0.112...+0.113) - [0.0%] Duration: 18 hours 06 minutes
3.	Rollout2	24/20(2) 8/4(2)	eq: +0.109 (-0.010)
	Player: Opponent: 55.44% (G:15.96% B:1.15%) 44.56% (G:9.70% B:0.74%) Conf.: ± 0.002 (+0.107...+0.110) - [0.0%] Duration: 1 hour 10 minutes
4.	Rollout3	24/20(2) 7/3*(2)	eq: +0.101 (-0.017)
	Player: Opponent: 55.07% (G:18.86% B:1.24%) 44.93% (G:9.89% B:0.70%) Conf.: ± 0.002 (+0.099...+0.104) - [0.0%] Duration: 32 minutes 08 seconds
5.	Rollout3	24/20(2) 6/2(2)	eq: +0.072 (-0.047)
	Player: Opponent: 53.61% (G:15.10% B:1.01%) 46.39% (G:10.47% B:0.83%) Conf.: ± 0.002 (+0.070...+0.075) - [0.0%] Duration: 33 minutes 21 seconds
6.	Rollout3	8/4(2) 7/3*(2)	eq: +0.063 (-0.056)
	Player: Opponent: 53.13% (G:25.47% B:2.94%) 46.87% (G:15.08% B:2.69%) Conf.: ± 0.003 (+0.060...+0.065) - [0.0%] Duration: 30 minutes 31 seconds
1 119232 Games rolled with Variance Reduction. Dice Seed: 41086525 Moves: 3-ply, cube decisions: XG Roller 2 10368 Games rolled with Variance Reduction. Dice Seed: 41086525 Moves: 3-ply, cube decisions: XG Roller 3 5184 Games rolled with Variance Reduction. Dice Seed: 41086525 Moves: 3-ply, cube decisions: XG Roller

Gammon Save

		White is Player 2 score: 0 pip: 146 2 point match Crawford pip: 157 score: 1 Blue is Player 1
XGID=-a-a--EBB-A-bC-b-c-d--b-B-:0:0:1:44:1:0:1:2:10
Blue to play 44

1.	Rollout1	24/20(2) 8/4(2)	eq: -0.031
	Player: Opponent: 54.66% (G:14.11% B:0.94%) 45.34% (G:6.22% B:0.34%) Conf.: ± 0.003 (-0.034...-0.028) - [100.0%] Duration: 30 minutes 56 seconds
2.	Rollout1	24/20(2) 13/9(2)	eq: -0.048 (-0.017)
	Player: Opponent: 54.17% (G:12.95% B:1.09%) 45.83% (G:6.56% B:0.37%) Conf.: ± 0.003 (-0.051...-0.045) - [0.0%] Duration: 33 minutes 20 seconds
3.	Rollout1	24/20(2) 7/3*(2)	eq: -0.055 (-0.024)
	Player: Opponent: 54.62% (G:18.26% B:1.35%) 45.38% (G:7.36% B:0.46%) Conf.: ± 0.003 (-0.058...-0.052) - [0.0%] Duration: 31 minutes 24 seconds
4.	Rollout1	24/20(2) 6/2(2)	eq: -0.082 (-0.051)
	Player: Opponent: 52.78% (G:13.99% B:0.82%) 47.22% (G:6.88% B:0.45%) Conf.: ± 0.003 (-0.085...-0.079) - [0.0%] Duration: 28 minutes 45 seconds
5.	Rollout1	24/16(2)	eq: -0.108 (-0.076)
	Player: Opponent: 55.91% (G:11.03% B:0.95%) 44.09% (G:11.29% B:0.68%) Conf.: ± 0.003 (-0.111...-0.104) - [0.0%] Duration: 31 minutes 21 seconds
1 5184 Games rolled with Variance Reduction. Dice Seed: 295083 Moves: 3-ply, cube decisions: XG Roller

Gammon Go

		White is Player 2 score: 1 pip: 146 2 point match Crawford pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-bC-b-c-d--b-B-:0:0:1:44:0:1:1:2:10
Blue to play 44

1.	Rollout1	8/4(2) 7/3*(2)	eq: +0.522
	Player: Opponent: 53.98% (G:22.13% B:2.38%) 46.02% (G:15.16% B:2.52%) Conf.: ± 0.004 (+0.518...+0.526) - [100.0%] Duration: 13 minutes 57 seconds
2.	Rollout1	7/3*(2) 6/2(2)	eq: +0.487 (-0.035)
	Player: Opponent: 52.53% (G:21.81% B:2.27%) 47.47% (G:16.72% B:3.19%) Conf.: ± 0.004 (+0.483...+0.491) - [0.0%] Duration: 13 minutes 57 seconds
3.	Rollout1	24/20(2) 7/3*(2)	eq: +0.463 (-0.060)
	Player: Opponent: 55.54% (G:17.58% B:1.03%) 44.46% (G:11.14% B:1.04%) Conf.: ± 0.003 (+0.459...+0.466) - [0.0%] Duration: 16 minutes 00 second
1 5184 Games rolled with Variance Reduction. Dice Seed: 295083 Moves: 3-ply, cube decisions: XG Roller

17c (need 17, Crawford)

		White is Player 2 score: 16 pip: 146 17 point match Crawford pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-bC-b-c-d--b-B-:0:0:1:44:0:16:1:17:10
Blue to play 44

1.	Rollout1	24/20(2) 13/9(2)	eq: +0.145
	Player: Opponent: 55.65% (G:14.04% B:0.62%) 44.35% (G:11.86% B:0.89%) Conf.: ± 0.001 (+0.145...+0.146) - [100.0%] Duration: 6 hours 31 minutes
2.	Rollout1	24/20(2) 8/4(2)	eq: +0.143 (-0.003)
	Player: Opponent: 55.45% (G:15.53% B:0.50%) 44.55% (G:9.52% B:0.65%) Conf.: ± 0.001 (+0.142...+0.143) - [0.0%] Duration: 5 hours 56 minutes
3.	Rollout1	24/16(2)	eq: +0.139 (-0.006)
	Player: Opponent: 55.90% (G:9.46% B:0.39%) 44.10% (G:10.81% B:0.97%) Conf.: ± 0.001 (+0.139...+0.140) - [0.0%] Duration: 4 hours 48 minutes
4.	Rollout1	24/20(2) 7/3*(2)	eq: +0.139 (-0.006)
	Player: Opponent: 55.04% (G:18.31% B:0.51%) 44.96% (G:9.86% B:0.76%) Conf.: ± 0.001 (+0.139...+0.140) - [0.0%] Duration: 6 hours 41 minutes
5.	Rollout2	8/4(2) 7/3*(2)	eq: +0.122 (-0.023)
	Player: Opponent: 53.41% (G:24.64% B:0.87%) 46.59% (G:15.19% B:2.64%) Conf.: ± 0.003 (+0.120...+0.125) - [0.0%] Duration: 21 minutes 15 seconds
6.	Rollout2	24/20(2) 6/2(2)	eq: +0.103 (-0.042)
	Player: Opponent: 53.54% (G:15.09% B:0.48%) 46.46% (G:10.87% B:0.79%) Conf.: ± 0.002 (+0.101...+0.105) - [0.0%] Duration: 23 minutes 46 seconds
7.	Rollout2	7/3*(2) 6/2(2)	eq: +0.097 (-0.048)
	Player: Opponent: 52.23% (G:23.38% B:0.95%) 47.77% (G:16.34% B:3.12%) Conf.: ± 0.003 (+0.095...+0.100) - [0.0%] Duration: 20 minutes 56 seconds
1 62208 Games rolled with Variance Reduction. Dice Seed: 41086525 Moves: 3-ply, cube decisions: XG Roller 2 5184 Games rolled with Variance Reduction. Dice Seed: 41086525 Moves: 3-ply, cube decisions: XG Roller

Below is a nacbracs summary of the five rollouts (detailed above):

m[C C12 B45 J58 R59 E83 A114] "<=362*5, m[C C12] "^38
d[R B6 C10 C17 E47 J56] "<119*5, d[R B5] "^72
s[C B17 C24 E51 R76] "<5
g[J A35 C60] "<5

17c[B C3 R6 C6 J23 E42 A48] "<62***5, 17c[B C3 R7 C7] "^55*20

Key: B (Both) = 24/20(2), C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

Other rollouts (not shown) suggest that B is the best play when Blue (trailing) needs any odd number of points and White is at Crawford (e.g., 3c, 5c, 7c, etc.)

___________________________________________________________________________

The Other Nine Positions

		White is Player 2 score: 0 pip: 157 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-eC---b-d--b-B-:0:0:1:44:0:0:3:0:10
53P-61P-63R-21H-21U-44 White distribution 13/10/8/6/4/3 = 502402, Power rating = 28.70

m[C J36 C37 R45 E71 A95] "<=5
d[R C31 C36 B57] "<5
s[C R14 C14 B28 E46] "<5
g[J A45 C69] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

Note: The 28.70 power rating of this R4 is currently the highest ever seen. (Until now, the range has been 17.43 to 28.52.)

___________________________________________________________________________

		White is Player 2 score: 0 pip: 156 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-dC-a-c-c--b-B-:0:0:1:44:0:0:3:0:10
61P-63R-21H-33Y_44 White distribution 13/10/8/6/4/3 = 413320, Power rating = 28.33

m[C J44 R56 C57 A100 B101 E117] "<=5
d[R C35 C43 B48] "<5
s[C C10 R14 B19 E46] "<31*15 5
g[J A42 C58] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

Note: This is the only known R4 that arises as early as the fifth roll of the game.

___________________________________________________________________________

		White is Player 2 score: 0 pip: 149 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-cC---d-d-b--B-:0:0:1:44:0:0:3:0:10
61P-42P-32Z-51S-54R-62H-31H-53Z-44 White distribution 13/10/8/6/4/3 = 304420, Power rating = 27.38

m[C C50 R84 B85 J117 E127] "<=5
d[R C8 C15 B16 E53] "<12*5
s[C C7 B20 R52 E52] "<307*5, s[C C8] "^64
g[J C26 A50] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

___________________________________________________________________________

		White is Player 2 score: 0 pip: 152 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-dC---c-d--b-B-:0:0:1:44:0:0:3:0:10
53P-61P-63R-21H-53Z-44 White distribution 13/10/8/6/4/3 = 403402, Power rating = 25.66

m[C C30 J52 R72 B77] "<=5
d[R C17 C19 B29 E55] "<5
s[C C16 B24 R41 E55] "<5
g[J A46 C60] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

___________________________________________________________________________

		White is Player 2 score: 0 pip: 152 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-dC---c-c-c--B-:0:0:1:44:0:0:3:0:10
65R-32S-42P-61U-634-53S-51H-65R-64H-41H-61U-41P-21U-44 White distribution 13/10/8/6/4/3 = 403330, Power rating = 25.53

m[C R50 C50 B99 J105] "<=5
d[R C29 C33 B44] "<5
s[C C10 R17 B26 E56] "<31*5
g[J C31 A48] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

___________________________________________________________________________

		White is Player 2 score: 0 pip: 145 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-cC---c-e--b-B-:0:0:1:44:0:0:3:0:10
43D-55P-21P-63H-53H-F-61R-21U-44 White distribution 13/10/8/6/4/3 = 303502, Power rating = 24.27

m[C J27 C43 B58 A69 E114] "<=5
d[B C9 C10 R18 J28 E46] "<12*10 5
s[C B9 C11 E51 R76] "<41**5, s[C C9 B10] "^31
g[J A34 C85] "<5

Note: At "d" (DMP), B is the best play instead of the usual R.

Key: B (Both) = 24/20(2), C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), A (Attack) = 7/3*(2) 6/2(2), E (Each) = 24/20(2) 6/2(2), R (Run) = 24/16(2). Nactation nacbracs

___________________________________________________________________________

		White is Player 2 score: 0 pip: 148 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-bC-b-c-d-b--B-:0:0:1:44:0:0:3:0:10
61P-42P-63R-53D-31Z-32Z-44 White distribution 13/10/8/6/4/3 = 223420, Power rating = 23.97

m[C C34 R41 B56 E100 J120] "<=5
d[R C29 C31 C32] "<5
s[C C15 B18 R41 E52] "<5
g[J C21 A37 C77] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

___________________________________________________________________________

		White is Player 2 score: 0 pip: 152 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-dC---b-e-b--B-:0:0:1:44:0:0:3:0:10
61P-41S-54R-54P-32Z-62H-31H-21U-44 White distribution 13/10/8/6/4/3 = 402520, Power rating = 21.51

m[C C52 R82 B104 J108 E120] "<=5
d[R C11 C18 B33] "<5
s[C C8 B27 R35 E52] "<195*5, s[C C8] "^75
g[J C25 A43 C83] "<5

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

___________________________________________________________________________

		White is Player 2 score: 0 pip: 154 Unlimited Game Jacoby Beaver pip: 157 score: 0 Blue is Player 1
XGID=-a-a--EBB-A-dC---c-d-b--B-:0:0:1:44:0:0:3:0:10
42P-61P-63R-21H-53Z-44 White distribution 13/10/8/6/4/3 = 403420, Power rating = 21.10

m[C C44 R57 B102] "<=5
d[R C24 C30 B45] "<5
s[C C10 R28 B31 E52] "<41*5
g[J C8 A44 C64] "<200*5, g[J C9] "^41

Key: C (Cross) = 24/20(2) 7/3*(2), C = 24/20(2) 8/4(2), J (Jump) = 8/4(2) 7/3*(2), R (Run) = 24/16(2), A (Attack) = 7/3*(2) 6/2(2), B (Both) = 24/20(2), E (Each) = 24/20(2) 6/2(2). Nactation nacbracs

Note: This last position (which inspired the entire R4 cluster that I found) was originally posted by Karol Szczerek here as an interesting money problem. It is surely the only R4 that has ever been posted unintentionally!

Nack