Puzzles

Forum Archive : Puzzles

 
Priming puzzle

From:   Gregg Cattanach
Address:   gcattanach@prodigy.net
Date:   9 May 2005
Subject:   Priming puzzle
Forum:   GammOnLine

From the opening position, with X moving three times consecutively,
form a full 6 prime in front of O's anchor. (O doesn't move, doubles
are OK on the first roll.)

In other words, reach this position after three rolls:

     24  23  22  21  20  19      18  17  16  15  14  13
    +---+---+---+---+---+---+---+---+---+---+---+---+---+
    | X                   O |   |     O               X |
    | X                   O |   |     O                 |
    |                     O |   |     O                 |
    |                     O |   |                       |
    |                     O |   |                       |
    |                       |   |                       |
    |                       |   |                     O |
    |                       |   |                     O |
    |                       |   |                     O |
    | O   X   X   X   X   X |   | X                   O |
    | O   X   X   X   X   X |   | X                   O |
    +---+---+---+---+---+---+---+---+---+---+---+---+---+
      1   2   3   4   5   6       7   8   9  10  11  12

How do you do it?

Řystein Johansen  writes:

This is the Bill Davis "Great Prime" puzzle. It was published in an
online backgammon magazine in December 1999.

Semi-Spoiler:
How to start the thinking progress. Trying and failing won't take you
anywhere. The first thing to think about is: How many pips do you need?
Which dice combinations of 3 rolls can give me this number?

-Řystein

TarHeelFan  writes:

There are two "tricks" to solving this sort of problem. The first is to
determine which dice rolls could possibly get you to the position. For
this one, we need 52 pips in 3 rolls. It's very unlikely that using the
same set of doubles twice will be usefull, so I'll ignore sets like 55,
55, 33 and 55, 44, 44. That leaves 66, 55, 22 and 66, 44, 33 as the only
combos I see that give the correct number of pips. Since 5's don't allow
us to diversify our checkers, we'll start with 66, 44, 33 as the most
likely rolls.

The second trick is to forget about how you would normally play
checkers. (I learned this trick from a similar chess problem where you
had to put your queen in a position to be captured to solve it)

Solution:

  - We have to more 3 checkers from the 6, and we have to use 2 rolls to
    do it. That means either 2 fours and a three or 2 threes and a four.

  - We have the same situalion with the 3 checkers on the 8 point (we
    can't play 8/5(2), 8/2 with 33 since that wouldn't leave a 3 for the
    6 point checkers)

  - Obviously, any solution would include 13/7(2), but there is no
    solution involving 8/2(2), since that would leave no 4 to play from
    the 6 point

So far, we have 13/7(2), 8/5, 8/4, 6/3, 6/2 as our "forced" plays,
leaving 2 6's, 2 3's and 2 4's:

     24  23  22  21  20  19      18  17  16  15  14  13
    +---+---+---+---+---+---+---+---+---+---+---+---+---+
    | X                   O |   |     O               X |
    | X                   O |   |     O               X |
    |                     O |   |     O               X |
    |                     O |   |                       |
    |                     O |   |                       |
    |                       |   |                       |
    |                       |   |                     O |
    |                       |   |                     O |
    |                     X |   |                     O |
    | O                   X |   | X                   O |
    | O   X   X   X   X   X |   | X   X               O |
    +---+---+---+---+---+---+---+---+---+---+---+---+---+
      1   2   3   4   5   6       7   8   9  10  11  12

  - We still have 2 checkers on the mid, which cannot play 6's. We can
    quickly see that playing both of these checkers with the same number
    does not work, so we need to play 13/10, 13/9

  - The remaining plays are trivial: 10/4, 9/3, 8/5, 6/2

So, the plays are (the order of the 44 and 33 doesn't matter):

44: 13/9, 8/4, 6/2(2)
33: 13/10, 8/5(2), 6/3
66: 13/7(2), 10/4, 9/3

I haven't looked for a solution using 66, 55, 22 or repeating rolls, but
I find it highly unlikely that such a solution exists.

Gregg Cattanach  writes:

It appears there are at least two different ways to do this, considering
transpositions as not being a separate solution:

Here's the way I've always done it.

66: 13/7(3), 8/2
44: 13/5, 8/4, 6/2
33: 8/5, 7/4, 6/3(2)

Cool! I didn't know there was a 2nd way to do it!

Gregg

TarHeelFan  writes:

This is something of a transposition, but:

33: 13/10, 8/5(2), 6/3
66: 13/7(3), 10/4
44: 8/4, 7/3, 6/2(2)

My error was when I said "We still have 2 checkers on the mid, which
cannot play 6's..." I had no logical backing for that, and the solution
branches at this point.
 
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