Forum Archive : Puzzles

Lowest probability of winning

From:   masque de Z
Address:   (none)
Date:   12 April 2012
Subject:   Smallest backgammon nonzero game equity ever possible
Forum:   2+2 Backgammon Forum

What is the smallest win probability in backgammon over 0 of course. 1
point game say. Just for fun consider boards that say white has
astronomically small probability to win. Do you think you can solve this?
Does it have a trivial answer or not so fast ...?

Bill Robertie  writes:

A pretty clear answer, I think. We can eliminate all contact positions,
since they're 'easy' to win, relatively speaking. For non-contacts, how
about this:

     13  14  15  16  17  18      19  20  21  22  23  24     O's home
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |         X   O         |
    |                       |   |                       |
    |                       |   |                       |
    |                       |   |                       |   +---+
    |                       |   |                       |   | 2 |
    |                       |   |                       |   +---+
     12  11  10   9   8   7       6   5   4   3   2   1     X's home

O has 15 men on his 3-point. Black has 15 men on his 21-point. Black on
roll wins with 15 consectutive big doubles, getting 60 crossovers, while
White responds with 15 consecutive 2-1s. It's not unique since Black
doesn't have to throw 6-6 every roll, just most of the time.

uberkuber  writes:

Nice one Bill! After that, you start tossing white checkers on 1-pt and 2-
pt until it's a lock and you rollback 1 pip.

TomCowley  writes:

Yeah, I thought about this while we were down and you can actually improve
by moving one checker to the 1 point. This position is basically a parlay

1) Black must get off in 15 rolls
2) White must not get off in 14 rolls (he always gets off in 15)

You can't make 1 harder without contact (or it being impossible for white
not to win), but you can make 2 harder, because as is, he can roll 6-1 the
first 2 times, or 6-2 once, and still not get off in 14. If you move a
checker up to the 1, then any 31-61 or 32-62 will let you to bear off 2 by
the 14th at the latest. I think moving 2 men to the 2 is also equally
difficult (both requiring exactly 14 2-1s in a row to avoid being able to
win in 14).
Did you find the information in this article useful?          

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



13 blots  (Timothy Chow+, Aug 2009) 
Alice, who is not on the bar, discovers that however she plays she ends up with 13 blots. What is her position and roll?
All-time best roll  (Kit Woolsey+, Dec 1997) 
What position and roll give the greatest gain in equity?
All-time worst roll  (Tim Chow+, Feb 2009) 
Find a position that goes from White being too good to double to Black being too good to double.
All-time worst roll  (Michael J. Zehr, Jan 1998) 
What position and roll give the greatest loss in equity?
Back to Nack  (Zorba+, Oct 2005) 
How can you go from the backgammon starting position to Nackgammon?
Cube ownership determines correct play  (Kit Woolsey, Jan 1995) 
Find a position and roll where the correct play depends on who owns the cube.
Highest possible gammon rate  (Robert-Jan Veldhuizen+, May 2004)  [GammOnLine forum]
What is the highest possible gammon rate in an undecided game?
Infinite loops  (Timothy Chow, Mar 2013) 
Is this position reachable?  (Timothy Chow+, Feb 2013) 
Janowski Paradox  (Robert-Jan Veldhuizen+, Nov 2000) 
Position that's a redouble but not a double?
Least shots on a blot within direct range  (Raymond Kershaw, Dec 1998) 
Find a position with no men on bar that has the least number of shots out of 36 to hit a blot within direct range.
Legal but not likely  (David desJardins, July 2000) 
Find a position that can be legally reached but never through optimum play.
Lowest probability of winning  (masque de Z+, Apr 2012) 
What is the smallest win probability in backgammon, greater than zero.
Mirror puzzle  (Nack Ballard, Apr 2010) 
Go from the starting position to the mirror position (colors reversed)
Most checkers on the bar  (Tommy K., May 1997) 
What is the maximum total possible checkers on the bar?
Most possible plays  (Kees van den Doel+, May 2002) 
Find the position and dice roll which have the most possible plays.
Not-so-greedy bearoff  (Kit Woolsey, Mar 1997) 
Find a no-contact position where it is better to move a checker than bear one off.
Not-so-greedy bearoff  (Walter Trice, Dec 1994) 
Find a no-contact position where it is better to move a checker than bear one off.
Priming puzzle  (Gregg Cattanach+, May 2005)  [GammOnLine forum]
From the starting position, form a full 6-prime in three rolls.
Pruce's paradox  (Alan Pruce+, Dec 2012) 
Quiz  (Martin Krainer, Oct 2003) 
Replace the missing checkers  (Gary Wong+, Oct 1998) 
Returning to the start  (Nack Ballard, May 2010) 
What is the least number of rolls that can return a game to the starting position?
Returning to the start  (Tom Keith+, Nov 1996) 
What is the least number of rolls that can return a game to the starting position?
Shortest game  (Stephen Turner+, Jan 1996) 
What is the shortest (cubeless) game in which both players play reasonably?
Small chance of ending in doubles  (Walter Trice, Dec 1999) 
Find a position where the probability of the game ending in doubles is less than 1/6.
Three-cube position  (Timothy Chow+, Sept 2011) 
Find a position and roll for which three different checker plays are best, depending on the location of the cube.
Trivia question  (Walter Trice, Dec 1998) 
What is the symmetric bearoff with the smallest pip count that is not an initial double?
Worst possible checker play  (Gregg Cattanach+, June 2004) 
What position and roll have the largest difference between best and worst play?
Worst possible opening move  (Gregg Cattanach, June 2004) 
What is the worst possible first move given any choice of dice?
Worst symmetric bearoff of 8 checkers  (Gregg Cattanach+, Jan 2004)  [GammOnLine forum]
What symmetric arrangement of 8 checkers in each player's home board gives roller least chance to win?
Worst takable position  (Christopher Yep, Jan 1994) 
What position has lowest chance of winning but is a correct take if doubled?
Zero equity positions  (Kit Woolsey, Apr 1995) 
Find a position with exactly zero equity in (1) money play or (2) cubeless.

[GammOnLine forum]  From GammOnLine       [Long message]  Long message       [Recommended reading]  Recommended reading       [Recent addition]  Recent addition

  Book Suggestions
Computer Dice
Cube Handling
Cube Handling in Races
Extreme Gammon
Fun and frustration
GNU Backgammon
Luck versus Skill
Magazines & E-zines
Match Archives
Match Equities
Match Play
Match Play at 2-away/2-away
Opening Rolls
Pip Counting
Play Sites
Probability and Statistics
Source Code
Strategy--Bearing Off
Strategy--Checker play


Return to:  Backgammon Galore : Forum Archive Main Page