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Puzzles
Is this position reachable?
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Here's a puzzle I created while helping to design the 2013 MIT Mystery
Hunt. This puzzle ended up not being used but I thought some readers here
might enjoy it.
Can the position below be reached in a legal game of backgammon? If so, can
you deduce who's on roll?
12 11 10 9 8 7 6 5 4 3 2 1
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | X X X | O's home
| | X | |
| | X | |
| | X | |
| | | |
| | | |
| | | |
| | | O |
| | | O X X | X's home
+---+---+---+---+---+---+---+---+---+---+---+---+---+
12 11 10 9 8 7 6 5 4 3 2 1
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Benf writes:
Cool puzzle.
Can the position below be reached in a legal game of backgammon? Yes
If so, can you deduce who's on roll? X must be on roll otherwise there
would be a fourth checker in O's home board from X being able to bring in
two checkers on the last two rolls.
The sequence that makes this possible is below in both text and diagrams.
From the starting position O is on roll:
(O) 5-1: 5/off, 5/4 (X) 4-2: bar/21*, bar/23
(O) 4-1: bar/21, 5/4* (X) 4-3: bar/21*, bar/22
(O) 3-1: bar/24, 24/21
Starting position:
12 11 10 9 8 7 6 5 4 3 2 1
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | X | O |
| | X | O |
| | X | O |
| | X | |
| | X | |
| | X | |
| | | |
| | | |
| | | X X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
13 14 15 16 17 18 19 20 21 22 23 24
(O) 5-1: 5/off, 5/4
13 14 15 16 17 18 19 20 21 22 23 24
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | X | O O |
| | X | |
| | X | |
| | X | |
| | X | |
| | X | |
| | | |
| | | |
| | | X X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
12 11 10 9 8 7 6 5 4 3 2 1
(X) 4-2: bar/21*, bar/23
12 11 10 9 8 7 6 5 4 3 2 1
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | O X X |
| | X | |
| | X | |
| | X | |
| | X | |
| | | |
| | O | |
| | | |
| | | X X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
13 14 15 16 17 18 19 20 21 22 23 24
(O) 4-1: bar/21, 5/4*
13 14 15 16 17 18 19 20 21 22 23 24
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | X | O X |
| | X | |
| | X | |
| | X | |
| | X | |
| | | |
| | | |
| | | |
| | | O X X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
12 11 10 9 8 7 6 5 4 3 2 1
(X) 4-3: bar/21*, bar/22
12 11 10 9 8 7 6 5 4 3 2 1
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | X X X |
| | X | |
| | X | |
| | X | |
| | | |
| | | |
| | O | |
| | | |
| | | O X X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
13 14 15 16 17 18 19 20 21 22 23 24
(O) 3-1: bar/24/21
13 14 15 16 17 18 19 20 21 22 23 24
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | X X X |
| | X | |
| | X | |
| | X | |
| | | |
| | | |
| | | |
| | | O |
| | | O X X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
12 11 10 9 8 7 6 5 4 3 2 1
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Puzzles
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Find a position that can be legally reached but never through optimum play.
- Lowest probability of winning (masque de Z+, Apr 2012)
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- Mirror puzzle (Nack Ballard, Apr 2010)
Go from the starting position to the mirror position (colors reversed)
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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.
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- Pruce's paradox (Alan Pruce+, Dec 2012)
- Quiz (Martin Krainer, Oct 2003)
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- 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)
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- Trivia question (Walter Trice, Dec 1998)
What is the symmetric bearoff with the smallest pip count that is not an initial double?
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- Worst takable position (Christopher Yep, Jan 1994)
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- 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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