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Forum Archive :
Puzzles
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Some us might have heard of the Kauder paradox; it's also in Robertie's
AB Vol.II. It exists because of the Jacoby Rule: (back)gammons only
count after a double. There are bg positions that are proper initial
doubles (activating (back)gammons), yet also proper beavers (after
taking, the opponent is the favorite with his cube ownership).
In Janowski's paper, I read about another paradox (I forgot the name),
also existing because of the Jacoby rule: bg positions that are proper
redoubles, but not initial doubles!
Normally, initial doubles can be offered earlier than redoubles, because
with a redouble you give your opponent cube access that he didn't have
before. So, a position that is an initial double might not be a
redouble; a redouble generally needs a bit more.
With the Jacoby rule however, an initial double activates gammons, which
a redouble does not. Simply put, if activating gammons in itself is
negative for the doubler, he might not want to give an initial double;
with gammons already activated he might want to redouble though.
In practice, it doesn't happen too often that you would like to double
when the gammon games are quite clearly *not* in your favour. Still it
doesn't seem at all impossible and it looks more natural to occur than
proper double/beaver positions.
So here's a nice puzzle, especially for the people that can rollout some
play with a bot: construct a position in a money game, Jacoby that is a
proper redouble yet not an initial double! :-)
Robert-Jan/Zorba
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Mary Hickey writes:
Sorry it took me so long to get around to reading your post, Zor-bie, but
I thought it might be better to respond better now than never, which is
what it would have been if I'd waited any longer, as your post would have
expired and I'd have never read it at all :0(
X on roll. Jacoby in use. Cube action?
| +24-23-22-21-20-19-------18-17-16-15-14-13-+
| | X O O O O | | |
| | X O O | | |
| | O O | | |
| | | X | |
| | | | |
| | |BAR| |
| | | | |
| | | | |
| | | | |
| | X X X X X X | | |
| | X X X X X X | | |
| +-1--2--3--4--5--6--------7--8--9-10-11-12-+
|
| CubeValue: A. 1 centered, or B. 2 with X owning it
I think this position qualifies as a solution to your "Paradox". Here is
a 1440 game JellyFish 3.0 Level 6 full-rollout, which it says its variance
reduction makes equal to 32,000 games, and who are we mere humanoids to
question this:
X wins = 67.1%
O wins 32.9%, of which 13.8% are singles, 15.2% are gammons, and 3.9% are
backgammons.
First, the center-double case:
If he waits, X will hit 25/36 times and win with the cube; 11/36 times he
will miss and be doubled out, since O has no reason to play on with
gammons not activated:
25/36 x 1 + 11/36 x -1
= 69.44 - 30.56 = + 38.88 points in 100 games
If he doubles now, I'll use the win % JF gets even though in reality
there'd be some small adjustment for O being able to turn the cube
sometimes after being hit if X has an accident bringing this home
after hitting.
67.1 x +2 + 13.8 x -2 + 15.2 x -4 + 3.9 x -6
= 134.2 - 27.6 - 60.8 - 23.4 = + 22.4 points in 100 games
It is apparent that if the cube is centered, X is better off to wait.
Now, about that redouble:
If X holds the cube at 2, his wins will be the 69.44% we saw in the
wait from the center case. This is higher by 2.34 % than his wins if
the game is played out, and it should come entirely at the expense of
O's single wins, not his gammons or backgammons, so I adjusted the
numbers for the double-from-center by this amount:
2.34 x 2 = swing on this case in X's favor = 4.68 x 2 cube = 9.36 points
in 100 games
22.4 + 9.36 = 31.76 points in 100 games
If instead X redoubles, he ends up with the same result as if he'd
doubled from the center, except now he's on a 4 cube:
22.4 x 2 = 44.8 points in 100 games
So it would appear X should redouble here to get the higher equity,
despite the gammon risk, even though because of Jacoby he could not
double from the center with this position.
Mary Hickey
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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)
![[GammOnLine forum]](pics/GammOnLine.gif)
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)
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)
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.
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