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OLM 20141229A The Prime Factors

Posted By: Taper_Mike
Date: Monday, 29 December 2014, at 11:34 p.m.

In Response To: OLM 20141229A The Prime Factors (Casper Van der Tak)

Mehhh you don't play the second 21 after 7/6 4/2 with the first. [If you play] 11/9 2/1 then 2s except 22 miss, [or if you play 11/8 then 1s except 11 miss], but [if you play] 11/10 3/1, [then] you miss then on the next roll with 11, 21, 31 and 32.

Nice catch, Casper! I made the same mistake in my original calculations. The update is below. If my math is right, all three candidates are within 0.47% of each other. At DMP, you can double that to get an equity difference of 0.0094 (or 0.0093 if you carry more decimal places).

Although the second- and third-best plays have swapped positions (compared to what happens when the second 21 is played 11/8), the top play remains:

7/6, 3/1


On the assumption that the Dilly Builders roll a doublet that rips four checkers off, letís count the ways we can bear a checker off after three candidate plays.

Candidate 1 Ė 7/6 4/2
If we play 7/6 4/2, then we will have a gap on the 1pt and a checker on the 11pt. 66, 65, 64, 63, 62, 55, 54, 53, 52, 44, 33, and 22 will bear off a checker on the following turn. Thatís 19 rolls. Of the remaining rolls, all except 21 would bring the outside checker at least as far as the 7pt. From there, we would be guaranteed to beat the gammon on the subesequent turn. Were we to roll 21, however, Casper points out that we should play 11/10 3/1. Thereafter, 7 rolls (32, 31, 21, and 11) would fail to bear off on the second turn.

Candidate 2 Ė 7/6 3/1
Can we eliminate the 21 sequence by putting a checker on the ace point now? Take a look at 7/6 3/1. After this move, 66, 65, 64, 63, 61, 55, 54, 53, 51, 44, and 33 rip off a checker on the following turn. Thatís only 18 rolls compared the 19 rolls we had before. All the remaining rolls advance the rear checker at least as far as the 8pt. From there, we can always bear off on the second turn.

Is is better to sacrifice 1 immediate number in order to avoid the 21 sequence? The only time it matters is when the Dillies roll a doublet that rips off four checkers on either of the next two turns. They could do that with 66, 55, or 44 on the next turn. If they roll 65, 64, 54, or 22 on the first turn, they would add 33 to that list for the second turn. If they roll 33 on the first turn, they would add both 33 and 22 to the list.

Candidate 3 Ė 11/9 7/6
If, instead, we play O = 11/9 7/6 on this turn, then we will have gaps on the 1pt and 2pt, with our outside checker on the 9pt. 66, 65, 64, 63, 55, 54, 53, 44, 43, 33, and 22 will bear off a checker on the subsequent turn. Thatís 17 rolls. Any other roll will bring the outside checker home, so we are guaranteed to bear a checker off in two rolls.

Dilly Buildersí
Roll on First Turn
Odds That Dilly Builders
Bear Off Four Checkers
on Second Turn
33 5/36
65, 64, 54, 22 4/36
Any other roll except
66, 55, 44
3/36

P1 = P( Dillies bear off 4 checkers next turn ) = 3/36
P2 = P( Dillies bear off 4 checkers on subsequent turn ) = (1/36)*(5/36) + (7/36)*(4/36) + (25/36)*(3/36) = 3/36
P( Dillies win in 2 rolls ) = P1 + P2 = 1/6

Candidate Move Odds of Bearing Off
on Next Roll
Odds of Bearing Off
Our First Checker on Subsequent Roll
Odds of Beating the Gammon
7/6 4/2 19/36 (15/36)*(36/36) + (2/36)*(29/36) = 598/1296 (1/6)*(19/36) + (5/6)*(19/36 + 598/1296) = 91.23%
7/6 3/1 18/36 (18/36) * (36/36) = 648/1296 (1/6)*(18/36) + (5/6)*(18/36 + 648/1296) = 91.67%
11/9 7/6 17/36 (19/36) * (36/36) = 684/1296 (1/6)*(17/36) + (5/6)*(17/36 + 684/1296) = 91.20%

Although the second- and third-best plays have swapped positions (compared to what happens when the second 21 is plays 11/8), the top play remains:

7/6, 3/1

Mike

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