Cube Handling

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Endgame close out: Michael's 432 rule

From:   Michael Bo Hansen
Date:   27 February 1998
Subject:   Michael's 432 rule
Google:   6d72ln$sv$

Michael's 432-Rule
I have analysed some positions when bearing off.
The positions were about being hit while you are bearing off,
and your opponent has a closed board. An example is shown below.
O is bearing off, and sometime in the bear-off, he was hit.
After that X managed to closed his board with O on the bar.
This sequence is quite common in BG today.

   |                  |   |                 O |
   |                  |   |                 O |
   |                  |   |                 O |
   |                  |   |                 O |
   |                  |   |                 O |
   |                  | O |                   |
   |                  |   |                   |
   |                  |   |  X  X  X          |
   |                  |   |  X  X  X  X  X  X |
   |                  |   |  X  X  X  X  X  X |
   +------------------------------------------+   X on roll
    12 11 10  9  8  7        6  5  4  3  2  1

The question is now:
What is the probability of X winning the game?
I have found out a simple rule that can give you the necessary information,
and I've named the rule "Michael's 432 rule"

The rule is as follows: When O have 4 men left on his ace point, the
probability of X winning is between 30 and 20% (4,3,2), dependent of where
his extra builders are placed (cube less). For an optimum distribution of
spares on the 6-,5- and 4 point (see figure) will give X 30% of winning
chances, while having all the spares on the ace-point, which is the worst
condition, will give X 20% probability of winning the game. You just have
to remember the 4,3,2 sequence: When the opponent has N builders left ,
your chances of winning are between 10*(N-1)% and 10*(N-2)%. The formula
can be extended up to O having 9 men on the ace-point. Then X's probability
of winning is between 80% and 70% (9,8,7). The formula is accurate within
2-3%, which is accurate enough for human players. When O has below 4 men
and beyond 9 men, the formula isn't accurate enough. The formula also work
"in reverse". This means, that if you are hit while bearing off, you have a
take (in MG) when you have a maximum number of 7 men on the ace point. Then
the opponent's winning chances are between 60 and 70%. The 7 men is also
what Bill Robertie consider to be the turning point. I've used this formula
a lot, and I found it quiet easy to use. I hope it can help other players
around the world.

Hi from
Michael Bo (snog at FIBS)

Michael Bo Hansen  writes:

It seems that my 432-rule has been greatly commentated here in the r.g.b.,
even though there is a small error in the last part of the text. When YOU
have 7 men on the ace point, the OPPONENT has between 50% and 60% of
winning, meaning YOU have between 40% and 50%. If using the formula YOU
have a take (in moneygame) having upto 8 men on your own ace-point, and one
on the bar.

Michael Bo

Michael Bo Hansen  writes:

A optimised version of the rule can be found on the following address:
Did you find the information in this article useful?          

Do you have any comments you'd like to add?     


Cube Handling

Against a weaker opponent  (Kit Woolsey, July 1994) 
Closed board cube decisions  (Dan Pelton+, Jan 2009) 
Cube concepts  (Peter Bell, Aug 1995)  [Long message]
Early game blitzes  (kruidenbuiltje, Jan 2011) 
Early-late ratio  (Tom Keith, Sept 2003) 
Endgame close out: Michael's 432 rule  (Michael Bo Hansen+, Feb 1998)  [Recommended reading]
Endgame close out: Spleischft formula  (Simon Larsen, Sept 1999) 
Endgame closeout: win percentages  (David Rubin+, Oct 2010) 
Evaluating the position  (Daniel Murphy, Feb 2001) 
Evaluating the position  (Daniel Murphy, Mar 2000) 
How does rake affect cube actions?  (Paul Epstein+, Sept 2005) 
How to use the doubling cube  (Michael J. Zehr, Nov 1993) 
Liveliness of the cube  (Kit Woolsey, Apr 1997) 
PRAT--Position, Race, and Threats  (Alan Webb, Feb 2001) 
Playing your opponent  (Morris Pearl+, Jan 2002)  [GammOnLine forum]
References  (Chuck Bower, Nov 1997) 
Robertie's rule  (Chuck Bower, Sept 2006)  [GammOnLine forum]
Rough guidelines  (Michael J. Zehr, Dec 1993) 
Tells  (Tad Bright+, Nov 2003)  [GammOnLine forum]
The take/pass decision  (Otis+, Aug 2007) 
Too good to double  (Michael J. Zehr, May 1997) 
Too good to double--Janowski's formula  (Chuck Bower, Jan 1997) 
Value of an ace-point game  (Raccoon+, June 2006)  [GammOnLine forum]
Value of an ace-point game  (Øystein Johansen, Aug 2000) 
Volatility  (Chuck Bower, Oct 1998)  [Long message]
Volatility  (Kit Woolsey, Sept 1996) 
When to accept a double  (Daniel Murphy+, Feb 2001) 
When to beaver  (Walter Trice, Aug 1999) 
When to double  (Kit Woolsey, Nov 1994) 
With the Jacoby rule  (KL Gerber+, Nov 2002) 
With the Jacoby rule  (Gary Wong, Dec 1997) 
Woolsey's law  (PersianLord+, Mar 2008) 
Woolsey's law  (Kit Woolsey, Sept 1996) 
Words of wisdom  (Chris C., Dec 2003) 

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