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Backgammon Problem of the Week The Discussion
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Black to play 3-1
Black can’t believe the roll he has made! The 3-1 roll does not allow him to bear off even one man!! Black looks at the board and knows at best White is off in 3 more rolls, maybe just 2 more rolls if he should roll a double so Black must now make the best move with the 3-1 roll that gives himself the highest probability of bearing off both men on his next roll.
Black has a choice of three moves. He can move one man from his 5 point to his 1 point; he can move one man from his 6 point to his 2 point; or he can move two men – one from the 6 point to the 3 point and one from the 5 point to the 4 point. The correct move in this case will enable Black to offer White the doubling cube before his next roll. What position does Black require in order to make a reasonable double?
Discussion
With exactly two men left on your board, the answer is based upon the
probability of bearing both men off with a single roll of the dice. Detailed below
are the exact number of rolls that will bear off the last two men.
1, 3 34/36 (94.4%) 2/36 (5.6%)
1, 4 29/36 (80.5%) 7/36 (19.5%)
1, 5 23/36 (63.8%) 13/36 (36.2%)
1, 6 15/36 (41.6%) 21/36 (58.2%)
2, 2 26/36 (72.2%) 10/36 (27.8%)
2, 3 25/36 (69.4%) 11/36 (30.6%)
2, 4 23/36 (63.8%) 13/36 (36.2%)
2, 5 19/36 (52.7%) 17/36 (47.3%)
2, 6 13/36 (36.2%) 23/36 (63.8%)
3, 3 17/36 (47.3%) 19/36 (52.7%)
3, 4 17/36 (47.3%) 19/36 (52.7%)
3, 5 14/36 (38.9%) 22/36 (61.1%)
3, 6 10/36 (27.8%) 26/36 (72.2%)
4, 4 11/36 (30.6%) 25/36 (69.4%)
4, 5 10/36 (27.8%) 26/36 (72.2%)
4, 6 8/36 (22.2%) 28/36 (77.8%)
5, 5 6/36 (16.7%) 30/36 (83.2%)
5, 6 6/36 (16.7%) 30/36 (83.2%)
6, 6 4/36 (11.1%) 32/36 (88.9%)
Now the answer to our problem of what two points to leave our two men on is
Crystal Clear. Leaving two men on 6 and 1 points gives us a 41.6% chance of
bearing them both off on our next roll. Leaving two men on our 4 and 3 points
gives us a 47.3% chance of bearing them both off on our next roll. Finally, leaving
two men on 5 and 2 points gives us a 52.7% chance of bearing them both off on
our next roll. If you answered “move 6 point to 5 point with the “1” and then
move 5 point to 2 point with the “3”, congratulations – you are playing winning
backgammon.
One last point, should you double before rolling the dice after White rolled a
double 4 and took off four men from his 1 point, leaving him just one man
remaining to bear off on his next turn? The answer is pure math. Over 100 games
at $1/point, you will win $2.00 x 52.7 of the time = $105.40. You will lose $2.00 x
47.3 of the time =$94.60. Over 100 games, you will realize a Profit of $10.80. Yes,
you will hate it after you double the cube and roll a 1 and fail to get both men off
but over the long run, the best play is to double the cube every time when you
are better than 50% probability to bear both men off on your last roll.
The number of the
points that men are on
Rolls of dice
they get off
Rolls of dice
they don't get off
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