Move 104 - 2 checker bear off

[don]

I saw this during a FIBS match where O made the wrong move and reconstructed the position here... see next post for spoilers.

gnu position ID: DwAAAAoAAAAAAA
   gnu match ID: cIkkAAAAAAAA

[don]

This is basic backgammon and shows a situation where it's valuable to actually count the number of rolls that will bear off on O's next roll.  In this case:

Spoiler

• 6/2 leaves 19
• 6/3 5/4 leaves 17
• 5/1 leaves 15

so 6/2 is correct!  The actual player moved 6/3 5/4.  Almost a 10% difference in game winning chance!

[close]

Here's a table of 2-man bearoff odds sorted in order of best/worst bear off:

Spoiler

 points   OFF     FAIL     %

   1-1   36/36    0/36   1.00
   1-2   36/36    0/36   1.00
   1-3   34/36    2/36   0.94
   1-4   29/36    7/36   0.81
   2-2   26/36   10/36   0.72
   2-3   25/36   11/36   0.69
   1-5   23/36   13/36   0.64
   2-4   23/36   13/36   0.64
   2-5   19/36   17/36   0.53
   3-3   17/36   19/36   0.47
   3-4   17/36   19/36   0.47
   1-6   15/36   21/36   0.42
   3-5   14/36   22/36   0.39
   2-6   13/36   23/36   0.36
   4-4   11/36   25/36   0.31
   3-6   10/36   26/36   0.28
   4-5   10/36   26/36   0.28
   4-6    8/36   28/36   0.22
   5-5    6/36   30/36   0.17
   5-6    6/36   30/36   0.17
   6-6    4/36   32/36   0.11

[close]

--
don

[Hardy_whv]

I stopped counting moves in such situations. When you have to place two checkers to maximise your chances to bear them all off in the next roll, there is a principle that makes counting or moves obsolete.

Spoiler

Thomas Koch calles it the 2.5 rule. When you choose your play in such way, that the thwo remaining checkers are 2.5 pips apart, the chance to bear them both of is maximised.

Okay, two checkers can never be exactly 2.5 pips apart, but try to get as close as possible to the 2.5.

In this example you can move:

6/2 --> 3 pips apart
6/3 5/4 --> 1 pip apart
5/1 --> 5 pips apart

6/2 comes closest to the number 2.5, so thats the best move here.

Note that this only helps if there are two checkers remaining. If you have 3 or 4 checkers, you have to search manually for a move that optimizes for doubles.

[close]

Hardy 

[spielberg]

Hardy's right but I think his number is very slightly too low.

Spoiler

I think it's actually 2.7 and believe it was Kleinman who first spotted this shotcut.

[close]

If I'm right I shall be extremely proud - correcting Hardy is an event of some magnitude!

[don]

Excellent comments from Hardy and ET (at least I learned something)!

After a Google or two, I can tell you that Thomas Koch (wintom) got the idea for the rule from a discussion at http://groups.google.com/group/rec.games.backgammon/browse_thread/thread/8a2b7015ac3ba4a2/9740ef2c7d4851be?lnk=gst&q=wintom+rule#  and it was called the rule of 2 or 3:

• Rule of 2 or 3..." Play to have your last two pieces two or three pips apart and if that is not possible have them as close to 2 or 3 as possible"
• Why didn't I ask the pros in the first place ;-)

Here's my previous chart, sorted by win then pip count to show why this rule works:

 points   OFF     FAIL     %    pip  separation

   1-1   36/36    0/36   1.00    2       0
   1-2   36/36    0/36   1.00    3       1

   1-3   34/36    2/36   0.94    4       2
   2-2   26/36   10/36   0.72    4       0

   1-4   29/36    7/36   0.81    5       3
   2-3   25/36   11/36   0.69    5       1

   1-5   23/36   13/36   0.64    6       4
   2-4   23/36   13/36   0.64    6       2
   3-3   17/36   19/36   0.47    6       0
   3-4   17/36   19/36   0.47    6       1
   1-6   15/36   21/36   0.42    6       5

   2-5   19/36   17/36   0.53    7       3

   3-5   14/36   22/36   0.39    8       2
   2-6   13/36   23/36   0.36    8       4
   4-4   11/36   25/36   0.31    8       0

   3-6   10/36   26/36   0.28    9       3
   4-5   10/36   26/36   0.28    9       1

   4-6    8/36   28/36   0.22   10       2
   5-5    6/36   30/36   0.17   10       0

   5-6    6/36   30/36   0.17   11       1
   6-6    4/36   32/36   0.11   12       0

It looks to me like the rule of 2.5 works just fine.

--
don

[Hardy_whv]

Yes, I got to know this principle from Thomas Koch's Backgammon Blog.

And for all who understand German, that's the link to the 2.5 article: http://graswurzelbackgammon.wordpress.com/2007/12/10/ungefahr-genau-25/

Hardy 

[don]

For those (like me) who don't read German, Google/translate does a frequently amusing but useful job on Koch's excellent graswurzel (grass roots?) site:
http://translate.google.com/translate?u=http%3A%2F%2Fgraswurzelbackgammon.wordpress.com%2F&langpair=de%7Cen&hl=en&ie=UTF-8

Yes, I got to know this principle from Thomas Koch's Backgammon Blog.

And for all who understand German, that's the link to the 2.5 article: http://graswurzelbackgammon.wordpress.com/2007/12/10/ungefahr-genau-25/

Hardy 

The apple is to meet such a thing (Den Apfel zu treffen ist so eine Sache.)!!!  You will find a discussion of the rule of 2.5 on the bottom of the page which extends this conversation nicely.

--
don

[wintom]

Thanks for the promotion Hardy and thanks Don for waking me up and pointing out this discussion! In fact Don already posted the rec.games.backgammon link where Neil Kazaross explains the 2 or 3 rule wich I converted into the 2.5 rule. Positions like this don't come up too often and the equity loss is not too big, but why sacrifice peanuts when you are hungry? Life rarely is as simple as remembering 2.5...

Thomas

[spielberg]

I apologise for my error and will now be using 2.5. Amusingly don initially agreed with me and has now edited his error out.

PS I should have known - Hardy's very, very good and his site http://www.hardys-backgammon-pages.com/ an excellent resource.

[don]

Ya caught me in the act, spielberg!

Amusingly don initially agreed with me and has now edited his error out.

Guilty as charged.

--
don