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Cube Handling
Endgame close out: Michael's 432 rule
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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)
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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
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Cube Handling
- Against a weaker opponent (Kit Woolsey, July 1994)
- Closed board cube decisions (Dan Pelton+, Jan 2009)
- Cube concepts (Peter Bell, Aug 1995)
- 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)
- 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)
- References (Chuck Bower, Nov 1997)
- Robertie's rule (Chuck Bower, Sept 2006)
- Rough guidelines (Michael J. Zehr, Dec 1993)
- Tells (Tad Bright+, Nov 2003)
- 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)
- Value of an ace-point game (Øystein Johansen, Aug 2000)
- Volatility (Chuck Bower, Oct 1998)
- 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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