Forum Archive :
Puzzles
Lowest probability of winning
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From: |
masque de Z |
Address: |
(none) |
Date: |
12 April 2012 |
Subject: |
Smallest backgammon nonzero game equity ever possible |
Forum: |
2+2 Backgammon Forum |
What is the smallest win probability in backgammon over 0 of course. 1
point game say. Just for fun consider boards that say white has
astronomically small probability to win. Do you think you can solve this?
Does it have a trivial answer or not so fast ...?
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Bill Robertie writes:
A pretty clear answer, I think. We can eliminate all contact positions,
since they're 'easy' to win, relatively speaking. For non-contacts, how
about this:
13 14 15 16 17 18 19 20 21 22 23 24 O's home
+---+---+---+---+---+---+---+---+---+---+---+---+---+
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | X O |
| | | |
| | | |
| | | | +---+
| | | | | 2 |
| | | | +---+
+---+---+---+---+---+---+---+---+---+---+---+---+---+
12 11 10 9 8 7 6 5 4 3 2 1 X's home
O has 15 men on his 3-point. Black has 15 men on his 21-point. Black on
roll wins with 15 consectutive big doubles, getting 60 crossovers, while
White responds with 15 consecutive 2-1s. It's not unique since Black
doesn't have to throw 6-6 every roll, just most of the time.
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uberkuber writes:
Nice one Bill! After that, you start tossing white checkers on 1-pt and 2-
pt until it's a lock and you rollback 1 pip.
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TomCowley writes:
Yeah, I thought about this while we were down and you can actually improve
by moving one checker to the 1 point. This position is basically a parlay
of
1) Black must get off in 15 rolls
2) White must not get off in 14 rolls (he always gets off in 15)
You can't make 1 harder without contact (or it being impossible for white
not to win), but you can make 2 harder, because as is, he can roll 6-1 the
first 2 times, or 6-2 once, and still not get off in 14. If you move a
checker up to the 1, then any 31-61 or 32-62 will let you to bear off 2 by
the 14th at the latest. I think moving 2 men to the 2 is also equally
difficult (both requiring exactly 14 2-1s in a row to avoid being able to
win in 14).
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