Puzzles

Forum Archive : Puzzles

 Lowest probability of winning

 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 ...?
```

 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. ```

 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. ```

 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). ```

### Puzzles

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What position and roll give the greatest loss in equity?
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Find a position and roll where the correct play depends on who owns the cube.
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Find a position that can be legally reached but never through optimum play.
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What is the smallest win probability in backgammon, greater than zero.
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Go from the starting position to the mirror position (colors reversed)
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Find a no-contact position where it is better to move a checker than bear one off.
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Find a no-contact position where it is better to move a checker than bear one off.
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Quiz  (Martin Krainer, Oct 2003)
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