This Article Is Brought To You By The Numbers 324 & 1296

by Gerry Tansey

Imagine this bearoff scenario.  You own a 2-cube in a money game and have a 2-roll position (3 checkers on the ace-point, for instance).  Your opponent has two checkers left somewhere in his home board.  Under what conditions should you redouble?  Under what conditions should your opponent pass a redouble?

Let’s answer the second question first.  Your opponent must pass a redouble if his winning chances are less than 25%.  Your opponent’s winning requires a parlay:  you must fail to roll doubles, and he must then bear off both of his checkers.  You fail to roll doubles with probability 30/36, and your opponent bears off with probability x/36, where x is the number of rolls that bear off.  So your opponent’s chance of winning is the product of these two fractions, or 30x/1296.

The denominator of this fraction is a good one to know: 1296.  An even better number to know is 324, which is 25% of 1296.  So if 30x is less than 324, your opponent should pass.  If your opponent has 10 rolls that bear off, then 30x = 300, which is less than 324, so your opponent should pass.  With 11 good rolls, 30x = 330, which is bigger than 324, so your opponent has a take.

What does a pass look like?  Here’s one example.

Analyzed in 4-ply Player Winning Chances:   76.85% (G:0.00% B:0.00%) Opponent Winning Chances: 23.15% (G:0.00% B:0.00%) Cubeless Equities: No Double=+0.537, Double=+1.074 Cubeful Equities:        No redouble:     +0.537 (-0.463)        Redouble/Take:   +1.074 (+0.074)        Redouble/Pass:   +1.000 Best Cube action: Redouble / Pass eXtreme Gammon Version: 2.10

Analyzed in 4-ply
Player Winning Chances: 76.85% (G:0.00% B:0.00%)
Opponent Winning Chances: 23.15% (G:0.00% B:0.00%)
Cubeless Equities: No Double=+0.537, Double=+1.074
Cubeful Equities:
No redouble: +0.537 (-0.463)
Redouble/Take: +1.074 (+0.074)
Redouble/Pass: +1.000
Best Cube action: Redouble / Pass
eXtreme Gammon Version: 2.10

With checkers on the 5 and the 4 point, the rolls 65, 64, 54, 33, 44, 55, and 66 bear off, which is 10 numbers out of 36, so your opponent must pass.  But if we improve the position by one pip, by placing both checkers on the 4-point, your opponent now has one additional roll that bears off, namely 22.  This 11th number gives your opponent a close take.  (A much better one-pip improvement is to place the checkers on the 5- and 3-points, which gives your opponent 14 winning rolls).

Analyzed in 4-ply Player Winning Chances:   74.54% (G:0.00% B:0.00%) Opponent Winning Chances: 25.46% (G:0.00% B:0.00%) Cubeless Equities: No Double=+0.491, Double=+0.981 Cubeful Equities:        No redouble:     +0.491 (-0.491)        Redouble/Take:   +0.981        Redouble/Pass:   +1.000 (+0.019) Best Cube action: Redouble / Take eXtreme Gammon Version: 2.10

Analyzed in 4-ply
Player Winning Chances: 74.54% (G:0.00% B:0.00%)
Opponent Winning Chances: 25.46% (G:0.00% B:0.00%)
Cubeless Equities: No Double=+0.491, Double=+0.981
Cubeful Equities:
No redouble: +0.491 (-0.491)
Redouble/Take: +0.981
Redouble/Pass: +1.000 (+0.019)
Best Cube action: Redouble / Take
eXtreme Gammon Version: 2.10

Now, when can you redouble your opponent?  You should definitely redouble your opponent if your opponent is not a favorite to bear off when you fail to roll doubles.  You are a favorite in a last-roll position, and your opponent cannot profitably redouble you back.

But what if your opponent is only a very tiny favorite to bear off when you do not roll doubles?  You gain from a redouble when you roll a set, but your opponent will redouble to 8 when you don’t roll a set, which is most of the time!  Can your gains possibly outweigh your losses?  Well take a look at this:

Analyzed in 4-ply Player Winning Chances:   56.02% (G:0.00% B:0.00%) Opponent Winning Chances: 43.98% (G:0.00% B:0.00%) Cubeless Equities: No Double=+0.120, Double=+0.241 Cubeful Equities:        No redouble:     +0.120 (-0.028)        Redouble/Take:   +0.148        Redouble/Pass:   +1.000 (+0.852) Best Cube action: Redouble / Take eXtreme Gammon Version: 2.10

Analyzed in 4-ply
Player Winning Chances: 56.02% (G:0.00% B:0.00%)
Opponent Winning Chances: 43.98% (G:0.00% B:0.00%)
Cubeless Equities: No Double=+0.120, Double=+0.241
Cubeful Equities:
No redouble: +0.120 (-0.028)
Redouble/Take: +0.148
Redouble/Pass: +1.000 (+0.852)
Best Cube action: Redouble / Take
eXtreme Gammon Version: 2.10

You have a small redouble in this position, even though your opponent has 19 rolls to bear off.  Most of the games will end up with your holding an 8-cube as a small underdog after you fail to roll doubles.  We won’t go through the math in detail here, but it turns out that the gain you get from redoubling before rolling your 6 immediate winning numbers just barely outweighs the fact that most of the time you will be playing as a 19-to-17 dog holding an 8-cube.  Of course, I’ve heard examples of opponents beavering the redouble to 4 here, which is a *huge* mistake.  Just comfort yourself with that knowledge the 44% of the time you are paying your opponent 16 points!

 

 

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