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Hi Backgammon Friends,
Today I've found a quiz and have sent my solution per email.
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 bearoff 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 8cube 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 twothirds 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 3point. You have 1 checker on your 5point
and 1 checker on your 1point, or you have 1 checker on your 4point
and 1 checker on your 2point.
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 4point and 1 ckecker on 1point (29/36 probability
to bear off in 1 roll), and 1 ckecker on 3point and 1 checker on 2point
(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 3point (17/36 probability). The other two 6pip positions are
equal: when one checker is on the 5point and one checker on the 1point,
or when one checker is on the 4point and one checker is on the 2point.
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 5point and one checker on the
2point (19/36 probability). His checkers must be on the 6 and 1points
(15/36) or on the 4 and 3points (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 twothirds 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 twothirds 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 twothirds of the time".
Cheers
Martin Krainer




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