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The Curious Case of Benjamin's Button, Part 2 -- SOLUTIONS

Posted By: Nack Ballard
Date: Wednesday, 22 November 2017, at 7:21 p.m.

In Response To: The Curious Case of Benjamin's Button, Part 2 (Nack Ballard)





White is Player 2

score: 4
pip: 84
5 point match
Crawford
pip: 17
score: 2

Blue is Player 1
XGID=--AC--A------------n------:0:0:1:61:2:4:1:5:10

Where is the missing White checker?


eXtreme Gammon Version: 2.10

Here is the complete story, but I'll also summarize here:

Blue has three legal moves with 61. Each one of these moves, according to long 4-ply rollouts, is solely best, depending upon the location of Benjamin's button (White's invisible 15th checker), which is known to be somewhere in one of the home boards. What are each of the three separate locations that make this possible?

Solutions:

If the button is on White's 20pt, Blue plays 6/5*/off.
If the button is on White's 24pt, Blue plays 6/off 3/2.
If the button is on White's 21pt, Blue plays 6/off 2/1.

Below are supportive rollouts for each location.






White is Player 2

score: 4
pip: 104
5 point match
Crawford
pip: 17
score: 2

Blue is Player 1
XGID=--AC-aA------------n------:0:0:1:61:2:4:1:5:10

Button is on 20pt, Rollout favors 6/5*/off


1.Rollout16/5* 5/Offeq: +1.0606
Player:
Opponent:		98.75% (G:60.38% B:2.73%)
1.25% (G:0.00% B:0.00%)		Conf.: ±0.0013 (+1.0593...+1.0619) - [100.0%]
Duration: 3 minutes 30 seconds
2.Rollout16/Off 2/1eq: +1.0318 (-0.0288)
Player:
Opponent:		100.00% (G:64.85% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0318...+1.0318) - [0.0%]
Duration: 0.1 second
3.Rollout16/Off 3/2eq: +1.0318 (-0.0288)
Player:
Opponent:		100.00% (G:64.83% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0318...+1.0318) - [0.0%]
Duration: 0.1 second
1 5184 Games rolled with Variance Reduction.
Dice Seed: 73764214
Moves and cube decisions: 4-ply

eXtreme Gammon Version: 2.10, MET: Kazaross XG2
White is Player 2

score: 4
pip: 108
5 point match
Crawford
pip: 17
score: 2

Blue is Player 1
XGID=-aAC--A------------n------:0:0:1:61:2:4:1:5:10

Button is on 24pt, Rollout favors 6/off 3/2


1.Rollout16/Off 3/2eq: +1.1222
Player:
Opponent:		99.96% (G:79.16% B:4.11%)
0.04% (G:0.00% B:0.00%)		Conf.: ±0.0002 (+1.1221...+1.1224) - [100.0%]
Duration: 7.6 seconds
2.Rollout16/Off 2/1* eq: +1.0512 (-0.0710)
Player:
Opponent:		98.86% (G:63.87% B:2.09%)
1.14% (G:0.00% B:0.00%)		Conf.: ±0.0014 (+1.0498...+1.0526) - [0.0%]
Duration: 3 minutes 07 seconds
3.Rollout16/Offeq: +1.0342 (-0.0881)
Player:
Opponent:		98.99% (G:57.08% B:1.29%)
1.01% (G:0.00% B:0.00%)		Conf.: ±0.0014 (+1.0328...+1.0355) - [0.0%]
Duration: 2 minutes 30 seconds
1 5184 Games rolled with Variance Reduction.
Dice Seed: 73764214
Moves and cube decisions: 4-ply

eXtreme Gammon Version: 2.10, MET: Kazaross XG2
White is Player 2

score: 4
pip: 105
5 point match
Crawford
pip: 17
score: 2

Blue is Player 1
XGID=--ACa-A------------n------:0:0:1:61:2:4:1:5:10

Button is on 21pt, Rollout favors 6/off 2/1


1.Rollout16/Off 2/1eq: +1.0341
Player:
Opponent:		100.00% (G:69.57% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0341...+1.0341) - [100.0%]
Duration: 0.1 second
2.Rollout16/Off 3/2eq: +1.0341 (0.0000)
Player:
Opponent:		100.00% (G:69.47% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0341...+1.0341) - [0.0%]
Duration: 0.0 second
3.Rollout16/Offeq: +1.0321 (-0.0020)
Player:
Opponent:		100.00% (G:65.44% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0321...+1.0321) - [0.0%]
Duration: 0.1 second
1 5184 Games rolled with Variance Reduction.
Dice Seed: 73764214
Moves and cube decisions: 4-ply

eXtreme Gammon Version: 2.10, MET: Kazaross XG2

The results of the first two rollouts seem clear enough.

The arbiter was stated to be long 4-ply rollouts. Therefore, regarding the third result (immediately above), button-on-the-21pt prevails as the 6/off 2/1 "solution." I believe, however, XG's rollout conclusion is wrong (see comments by Philippe Michel, Maik Stiebler and Neil Robins). If you examine my breakdown for 3332-bearoff with an ace to play (with no contact and no material cube in play), you can verify that 2/1 and 3/2 are always exactly tied.

White's 21pt is the unique location for 2/1 (albeit improperly) winning XG's rollout. With the button on any of the six points in White's inner board, XG's rollout produces the right answer of an exact tie -- example below. [The progress window reported 50.0 / 50.0%. I don't know why that breakdown is reported instead as 53.1 / 46.9 in the summary details -- seemingly another quirk, but I'll take it on faith that with enough trials it would also show 50/50.]

Nack





White is Player 2

score: 4
pip: 87
5 point match
Crawford
pip: 17
score: 2

Blue is Player 1
XGID=--AC--A------------n--a---:0:0:1:61:2:4:1:5:10
Blue to play 61

1.Rollout16/Off 3/2eq: +1.0018
Player:
Opponent:		100.00% (G:3.58% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0018...+1.0018) - [53.4%]
Duration: 2.6 seconds
2.Rollout16/Off 2/1eq: +1.0018
Player:
Opponent:		100.00% (G:3.58% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0018...+1.0018) - [46.6%]
Duration: 2.5 seconds
3.Rollout16/Offeq: +1.0017 (-0.0001)
Player:
Opponent:		100.00% (G:3.40% B:0.00%)
0.00% (G:0.00% B:0.00%)		Conf.: ±0.0000 (+1.0017...+1.0017) - [0.0%]
Duration: 2.5 seconds
1 663552 Games rolled with Variance Reduction.
Dice Seed: 73764214
Moves and cube decisions: 4-ply

eXtreme Gammon Version: 2.10, MET: Kazaross XG2

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