Puzzles

Forum Archive : Puzzles

 Zero equity positions

 From: Kit Woolsey Address: kwoolsey@netcom.com Date: 17 April 1995 Subject: Re: doubles, redoubles, re-redoubles, etc. Forum: rec.games.backgammon Google: kwoolseyD760MF.35t@netcom.com

```Christopher Yep wrote:
> This leaves open questions which I have wondered about for awhile:
> Are there any backgammon positions (other than the initial position,
> before either player has rolled) where:
>
> 1. The equity is 0? [money play]
> 2. The player on roll has a 50% chance to win the game? [1 pt. match
>    (no cube)]

1) I know of a couple.  One which is very easy to see is as follows:

X has one checker on 9 point, O has one checker on 6 point, X owns cube,
X is on roll.  X wins outright on 12 of his rolls.  Since all of his
rolls get him to at least the 6 point (where he has a claim or break-even
on O's decision whether or not to take a redouble), in essence X wins 1/4
of the 24 rolls he fails to get off immediately (since O gets off on 3/4
of his rolls), which comes to the desired 18 out of 36.

Another example which is more complex but my trusty computer says is so is:

X has two checkers on 4 point and one checker on 6 point, O has three (or
four) checkers on ace point, X owns cube, and X is on roll.  Check it out
if you want to spend some pencil and paper time.

An example of a contact position might be as follows:

X has a closed board -- his other three checkers are two on O's 3 point
and one on O's 7 point.  O's position:  3 on 1 point, 3 on 2 point, 4 on 4
point, 4 on 5 point, 1 on 12 point.  X is on roll, cube is in center
(money game, Jacoby rule in use).  Note that X has exactly 18 hitting
numbers, and if he misses O certainly has a claim with the cube.  If we
assume that after hitting X's gammon chances are less than half his losing
chances (which seems like a proper assumption to me) then it is not
correct for X to double now.  Therefore X wins when he hits, loses when he
misses, for equity of 0.

2) With no cube in play, I don't know any zero equity positions.  If there
were one in a race I'm sure our regurgitating computer programs would
have spit it out by now and I would have heard of it.  I guess in theory
there could be one in a contact position, but that would be awfully hard
to prove -- keep in mind that "gin" positions just aren't 100% gin.

Kit
```

### Puzzles

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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)
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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)
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Infinite loops  (Timothy Chow, Mar 2013)
Is this position reachable?  (Timothy Chow+, Feb 2013)
Janowski Paradox  (Robert-Jan Veldhuizen+, Nov 2000)
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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.
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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)
From the starting position, form a full 6-prime in three rolls.
Pruce's paradox  (Alan Pruce+, Dec 2012)
Quiz  (Martin Krainer, Oct 2003)
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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.
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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)
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.