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Forum Archive :
Puzzles
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Bill Daly wrote:
"On occasion, when someone publishes a chess or bridge problem, it turns
out to be impossible, in the sense that the problem position cannot
possibly be created by any legal sequence of plays. This would appear to
be less of an issue with backgammon, since there are fewer constraints
on the kinds of positions that can be created. Nevertheless, there are
certainly positions which cannot legally arise, for example all 30
checkers on the bar. I have been ruminating about this somewhat
inconclusively, but my focus is currently on two questions. 1) Assuming
that both players cooperate and are free to choose the rolls of the
dice, how can they play so as to put all 15 of X's checkers on the bar?
Can they do this in such a way that O also has at least one checker on
the bar? And 2) what is the maximum number of checkers (on both sides)
which can be on the bar at the same time?
Perhaps these questions would be better suited to one of the puzzles
newsgroups, but I'm not sure that backgammon is well enough understood
elsewhere to make this interesting.
Regards,
Bill"
Tommy K. Replies:
It is a simple matter for a player [X] to spread out his 15 checkers on
different points and get them all hit. The opposing player [O] cannot
simultaneously have a checker on the bar, however. Once [X] has 1-4 men on
the bar (a necessary transitional stage) [O] can only hit at most 6 more
men before getting all of his men in to hit [X]'s checkers outside the home
board. If [X] hits a man when most of his guys are already up, [O] will
need to get all of his guys in in order to his the entering [X] checker
back, so it is impossible for one player to have 15 men up and the other to
have at least one up.
As for the total number of checkers that can be on the bar, that is also
15, by similar arguments. Once [X] has even one man on the bar, he can
only hit [O] by bringing a checker in at the same time, so the total number
of checkers on the bar doesn't increase.
The most checkers that the lesser of the two players can have up is 6,
which can occur with 7 opposing checkers up. This can occur by [X] leaving
13 blots exposed, with one innerboard point closed. [O] then picks up the
13 blots, and slots his own inner board. The final slot consists of
breaking [O]'s last inner board point and leaving one blot on each point.
[X] then rolls to hit all six blots in the next three rolls, bringing in 6
guys of his own. [O] meanwhile can bounce because [X] had one inner board
point closed. After the third roll [X] has 7 men on the bar, and [O] has
6.
Tom Keisler
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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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