Puzzles

Forum Archive : Puzzles

 
Quiz

From:   Martin Krainer
Address:   martin.krainer@aon.at.NOSPAM
Date:   16 October 2003
Subject:   A new quizz - Correct Solution
Forum:   rec.games.backgammon
Google:   3f8e8d70$0$19908$91cee783@newsreader01.highway.telekom.at

Hi Backgammon Friends,

Today I've found a quiz and have sent my solution per e-mail.
Even when I've sent it already, I would like to ask you, if I have
overseen anything and what do you say about the solution?  I needed
more than 1 hour to solve it.

I have forgotten the straight way to calculate it.  (Still I hope the
numbers are correct). Could anyone please tell me me the easier (and
common) way to calculate such positions?

Here is the quiz and my solution:

  How in the world could it possibly be
  that he was favored more than me?

  A bear-off position, with two men each,
  but a lesson to me, it surely did teach.

  He doubled me, it could not be believed.
  I scooped up the cube and beavered with greed.

  I said, "I'm a favorite to get off in one roll.
  You sir are not. Now you'll pay the toll."

  He cast his dice and rolled a failing four pips.
  I smiled and said, "Now you're in quite a fix."

  He said, "We were tied in the pip count;
  it was right to double."

  I said, "Now I'll send back the recube
  for all of your trouble."

  It surprised me to see his calm disposition--
  to take an 8-cube in the resulting position.

  With one mighty shake, as hard as I could,
  out came a double, but it was no good.

  "I was favored when you cubed me!"
  My temperature was hot.

  He looked at me, then shook his head
  and countered, "You were not."

  Almost two-thirds of the time,
  he would win, it was shown,

  because often my dice
  wouldn't even be thrown.

  It was then and there, I was forced to relent
  that my wins totaled well under 40 percent."

----

Solution:

He has 2 checkers on his 3-point. You have 1 checker on your 5-point
and 1 checker on your 1-point, or you have 1 checker on your 4-point
and 1 checker on your 2-point.

Lets examine it:

After he failed to bear off, you said: "With one mighty shake, as hard
as I could, out came a double, but it was no good."  That means you must
have at least 5 pips left. There are 2 possible positions with 5 pips
left: 1 ckecker on 4-point and 1 ckecker on 1-point (29/36 probability
to bear off in 1 roll), and 1 ckecker on 3-point and 1 checker on 2-point
(25/36 probaility to bear off in 1 roll).

He said, "We were tied in the pip count; it was right to double."
Then, after you said, "I'm a favorite to get off in one roll," he said
"You sir are not. Now you'll pay the toll."

In both positions where you are 5 pips away, you are favorite to bear off
in 1 roll, so you must be more than 5 pips away.

How about the positions where both are 6 pips away?  There is one position
where you are not favorite to bear off in one roll:  when both checkers are
on the 3-point (17/36 probability).  The other two 6-pip positions are
equal:  when one checker is on the 5-point and one checker on the 1-point,
or when one checker is on the 4-point and one checker is on the 2-point.
Both have a 23/36 probability to to bear off in one roll and all three
positions have a 100% chance to bear off in two rolls.

What about the other positions when both are 7 pips or more away?
In this case there is just one possible position where you are favorite to
bear off in one roll:  one checker on the 5-point and one checker on the
2-point (19/36 probability).  His checkers must be on the 6- and 1-points
(15/36) or on the 4- and 3-points (17/36).  (There are no positions where
both are 8 pips or more away and he is favorite to bear off in 1 roll.)

That leads to following.  You said: "Almost two-thirds of the time, he
would win, it was shown, because often my dice wouldn't even be thrown."
The point here is that in both cases where you both are 7 pips away,
he is more than 2/3 favorite, not "almost two-thirds of the time".

Conclusion:

The take was correct, but the beaver was too much.  After he doubled and
you owned the cube and he failed to bear off, it was correct to redouble,
but of course it still was a take (he had still 36.11% (13/36) to win the
game).  In both correct solutions (see above) he was 66.3% favorite, as
you said:  "almost two-thirds of the time".

Cheers
Martin Krainer
 
Did you find the information in this article useful?          

Do you have any comments you'd like to add?     

 

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]
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)  [GammOnLine forum]
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)  [GammOnLine forum]
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.

[GammOnLine forum]  From GammOnLine       [Long message]  Long message       [Recommended reading]  Recommended reading       [Recent addition]  Recent addition
 

  Book Suggestions
Books
Cheating
Chouettes
Computer Dice
Cube Handling
Cube Handling in Races
Equipment
Etiquette
Extreme Gammon
Fun and frustration
GNU Backgammon
History
Jellyfish
Learning
Luck versus Skill
Magazines & E-zines
Match Archives
Match Equities
Match Play
Match Play at 2-away/2-away
Miscellaneous
Opening Rolls
Pip Counting
Play Sites
Probability and Statistics
Programming
Propositions
Puzzles
Ratings
Rollouts
Rules
Rulings
Snowie
Software
Source Code
Strategy--Backgames
Strategy--Bearing Off
Strategy--Checker play
Terminology
Theory
Tournaments
Uncategorized
Variations

 

Return to:  Backgammon Galore : Forum Archive Main Page