Puzzles

Forum Archive : Puzzles

 From: Alan Pruce Address: prucea@gmail.com Date: 21 December 2012 Subject: Crawford Rule Brainteaser Forum: rec.games.backgammon

```You are playing an equal opponent in a match for \$100, and are trailing 3-
away, 2-away. How much would you be willing to pay to turn off the Crawford
Rule for the duration of the match?
```

 Sam Pottle  writes: ```Nothing. I'm better off with it on. ```

 Joe Russell  writes: ```The trailer should pay up to \$5.40 to keep the Crawford rule. The reason the Crawford rule favors the trailer is the amount of increase in winning chances after winning 2-points, his most common result. His winning chances are 16.46% greater with the Crawford rule in effect. ```

 Daniel Murphy  writes: ```It seems to me that Joe and Sam are right, that the Crawford rule favors the trailer at 3-away 2-away -- by quite a lot -- so I would pay nothing to turn the rule off. I make these simplifying assumptions: * 2/3 of initial doubles should be taken * Players are equally likely to make the initial double at this score * 2/3 of initial doubles should be taken at this score * The doubler wins 70% of initial doubles at this score. * Gammon rate = 20% * Ignore backgammons and undoubled gammons Then: Freq Result 1/6 I double, pass. Score is -2-2. Crawford rule doesn't matter. 2/6 He doubles, I take. The game decides the match (ignore last roll doubles with no redouble possible). Crawford rule doesn't matter. 1/6 He doubles, I drop. Trailing 3-away 2-away, with Crawford rule, I have about 25% MWC, without the Crawford rule I have about 30% MWC. 30%-25% = 5%. 1/6 * 5% = 0.83% disadvantage with Crawford rule. 2/6 I double, he takes. If I win a gammon, or lose, Crawford rule doesn't matter. The only games that matter are my single wins which make the score 1-away 2-away (as Joe wrote, "The reason the Crawford rule favors the trailer is the amount of increase in winning chances after winning 2-points, his most common result"). Single wins = 2/6 * .7 * .8 = 18.7% (that's 2/6 times the percentage of games I win times the percentage of games I win that aren't gammons). Leading 1-away 2-away, with Crawford rule, my MWC is 70%, without Crawford rule, about 50%. 70% - 50% = 20%. 20% * 18.7% = 3.7% advantage with Crawford rule. ```

 Alan Pruce  writes: ```Wow a lot of great responses already, so I guess I will post my analysis now. Clearly, the Crawford Rule helps the Leader when he reaches the Crawford Game, so it seemed intuitive to me that the leader should always benefit from having the Crawford Rule in effect. However, like many of the posters have figured out, it seems the trailer actually benefits from having the Crawford Rule in effect at the {-2,-3} score, so he is actually willing to pay to keep the Crawford Rule on. Based on my calculations, the trailer is willing to pay up to ~\$3.60 to keep the rule in effect. Also, I want to point out that I did not just find this effect at {-2,-3}. The Crawford Rule at {-4,-5} hurts the leader even more than at {-2,-3}, and there were several other scores where the Crawford Rule decreased the leader's MWC. Now if I could only get myself to spend 10% of the time studying actual bg as opposed to doing this type of mental buffoonery, maybe I could finally get my PR in the single digits ;) ```

### Puzzles

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)
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.
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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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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)
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.