Bearing off: forced to leave a shot

[Tim Chow]

XGID=-CBaDBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 45 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63

---
Tim Chow

[Paul Epstein]

Post by Tim Chow
XGID=-CBaDBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 45 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63

The only candidates are 4/1 and 6/3*
Gammons are too unlikely to be worth considering
so it's all about safety. To start the analysis,
we will need to count the number of safetying rolls
if we're missed. However, 6/3* has the advantage
that it gives the opponent nine immediate losing rolls
(rolling 6 without rolling a 3)
So 6/3* might be the play even if the safetying count
favours 4/1.
Ok, so let's count the number of safetying rolls after 6/3*.
Any roll with a 6 or a 5 needs a 2 to be safe.
I think that the rolls with no 6 and no 5 are safe
with the exception of 44.
So 6/3* gives 19 rolls which don't leave a further blot.
After 4/1 any 6 that isn't 65 is safe.
As far as I can see, 4/1 leaves only 65 as a blotting roll.

So 4/1 is correct.
How confident am I? I basically have total confidence.
But it would be wrong to give a figure of 100%, particularly
in a quiz. An analysis can always be wrong for unseen reasons.
Wiles's original FLT proof circa 1993 was wrong, before Taylor got
involved.
So maybe Paul's analysis is wrong and Tim needs to correct it.
My play stands up to a QF consideration since a careless player
might think "Both plays leave a direct shot but hitting can win the game
immediately so I'll play that".
The normal threshold for "highly surprising" is 5%. Since I'll
be highly surprised if I'm wrong, I'll play 4/1 with 95% confidence.

Paul Epstein

[Peter]

Post by Paul Epstein

Post by Tim Chow
XGID=-CBaDBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 45 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63

The only candidates are 4/1 and 6/3*
Gammons are too unlikely to be worth considering
so it's all about safety. To start the analysis,
we will need to count the number of safetying rolls
if we're missed. However, 6/3* has the advantage
that it gives the opponent nine immediate losing rolls
(rolling 6 without rolling a 3)
So 6/3* might be the play even if the safetying count
favours 4/1.
Ok, so let's count the number of safetying rolls after 6/3*.
Any roll with a 6 or a 5 needs a 2 to be safe.
I think that the rolls with no 6 and no 5 are safe
with the exception of 44.
So 6/3* gives 19 rolls which don't leave a further blot.
After 4/1 any 6 that isn't 65 is safe.
As far as I can see, 4/1 leaves only 65 as a blotting roll.
So 4/1 is correct.

Yes.

[Tim Chow]

XGID=-CBaDBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 45 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63

I think that Paul hit all the key points. I've included a variant where
4/1 leaves many more blotting rolls than in the original position.

1. Rollout¹ 6/Off 4/1 eq:+0.351
Player: 71.75% (G:1.77% B:0.00%)
Opponent: 28.25% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.349..+0.353) - [100.0%]

2. Rollout¹ 6/3* 6/Off eq:+0.303 (-0.048)
Player: 69.29% (G:3.11% B:0.05%)
Opponent: 30.71% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.301..+0.304) - [0.0%]

¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.19.207.pre-release

-------
Variant
-------

XGID=-CBaCBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 41 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63

1. Rollout¹ 6/3* 6/Off eq:+0.300
Player: 69.54% (G:4.79% B:0.07%)
Opponent: 30.46% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.298..+0.302) - [100.0%]

2. Rollout¹ 6/Off 4/1 eq:+0.246 (-0.054)
Player: 67.99% (G:2.62% B:0.02%)
Opponent: 32.01% (G:0.00% B:0.00%)
Confidence: ±0.003 (+0.243..+0.249) - [0.0%]

¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

[Paul Epstein]

Post by Tim Chow
XGID=-CBaDBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 45 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63
I think that Paul hit all the key points. I've included a variant where
4/1 leaves many more blotting rolls than in the original position.
1. Rollout¹ 6/Off 4/1 eq:+0.351
Player: 71.75% (G:1.77% B:0.00%)
Opponent: 28.25% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.349..+0.353) - [100.0%]
2. Rollout¹ 6/3* 6/Off eq:+0.303 (-0.048)
Player: 69.29% (G:3.11% B:0.05%)
Opponent: 30.71% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.301..+0.304) - [0.0%]
¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
eXtreme Gammon Version: 2.19.207.pre-release
-------
Variant
-------
XGID=-CBaCBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 41 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63

Assuming O misses, let's count (or attempt to count)
the number of shotleavers in the variant.
With 6/3*, I think the shotleavers are 44 and
rolls which contain a 6 or a 5 without containing a 2.
This is just as previously. There are 17 such rolls.

With 4/1, the shotleavers are 66/65/64/63/55/54/53/44/43
There are 15 such rolls.
Although 4/1 leaves slightly fewer shortleavers, this doesn't outweigh
the advantage of 6/3* that it gives the opponent nine rolls to lose immediately.
So 6/3* is correct here, and my approach works here, too.

My approach is basically: Choose 6/3* to try to "get the opponent out of
your hair" unless 4/1 is much safer. 5.6% safer isn't enough.

At lower levels, people don't count shots enough.

Paul

[Paul Epstein]

Post by Tim Chow
XGID=-CBaDBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 45 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63
I think that Paul hit all the key points. I've included a variant where
4/1 leaves many more blotting rolls than in the original position.
1. Rollout¹ 6/Off 4/1 eq:+0.351
Player: 71.75% (G:1.77% B:0.00%)
Opponent: 28.25% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.349..+0.353) - [100.0%]
2. Rollout¹ 6/3* 6/Off eq:+0.303 (-0.048)
Player: 69.29% (G:3.11% B:0.05%)
Opponent: 30.71% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.301..+0.304) - [0.0%]
¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
eXtreme Gammon Version: 2.19.207.pre-release
-------
Variant
-------
XGID=-CBaCBB-----------adbcbb--:1:-1:1:63:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
[phone]------19-20-21-22-23-24-+
| O | | O O O O O | +---+
| | | O O O O O | | 2 |
| | | O O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X |
| | | X X X X X |
| | | X X X O X X |
[phone]-------6--5--4--3--2--1-+
Pip count X: 41 O: 85 X-O: 0-0
Cube: 2, O own cube
X to play 63
1. Rollout¹ 6/3* 6/Off eq:+0.300
Player: 69.54% (G:4.79% B:0.07%)
Opponent: 30.46% (G:0.00% B:0.00%)
Confidence: ±0.002 (+0.298..+0.302) - [100.0%]
2. Rollout¹ 6/Off 4/1 eq:+0.246 (-0.054)
Player: 67.99% (G:2.62% B:0.02%)
Opponent: 32.01% (G:0.00% B:0.00%)
Confidence: ±0.003 (+0.243..+0.249) - [0.0%]
¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
eXtreme Gammon Version: 2.19.207.pre-release

Re the variant. Removing a checker from X gives X more gammons
which incentivises 6/3* more than in the original position.

In other words, your variantizing makes 6/3* more attractive for two
reasons:
1) The safety gap between the two plays is narrower than the original.
2) Removing a checker makes gammonish plays more attractive.

Paul