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Forum Archive :
Puzzles
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Don Woods wrote:
> I know that if there's still contact, I need to be careful about how
> I bear off to reduce the chance of leaving blots, etc. But once
> contact is broken, is it ever wrong to bear off the most men possible
> on every roll? E.g., are there cases where I could bear off 4 men on
> a doublet or 2 men on a non-doublet, but shouldn't? Failing that, are
> there times when I do better to bear off fewer men than I could, e.g.
> bearing off 2 instead of 3 on a doublet or zero instead of 1, etc.?
Yes, there are rare situations where toggle greedy will not lead you to
the best play. The best known one (from Magriel's book) is:
5 on ace, 1 on two, 5 on three, 2 on four -- 6-2 to play. Here 4/2 4/off
is superior to 4/off, 2/off. The reason is that if you roll a non-two
next turn and then roll a two the following two turns you will miss twice
if you had taken two men off, costing you a roll.
There are others. The three point provides a similar sort of situation.
Consider: 3 on ace, 3 on two, 1 on three, 4 on four, 2 on six. 6-3 to
play. 6/3, 6/off is superior to 6/off, 3/off because of the danger of
rolling subsequent threes.
Even with a four there are possibilities. For example, consider:
5 on ace, 4 on three, 1 on four, 2 on five, 1 on six. 6-4 to play. 6/2,
5/off is superior to 6/off, 4/off due the the great value of filling in
the gaps -- taking two men off leaves you missing on twos and fours for
quite a while.
I'm sure there are plenty of other examples. Perhaps someone with a
bearoff database and the programming tools to get inside it could rattle
off a list of such positions if it isn't too large.
In practice, I always toggle greedy when there is no contact anyway. The
chance of running across one of these positions combined with the cost of
making the wrong play is small indeed, so it is worth taking this
risk in order to save time.
Kit
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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)
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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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