Came across a wonderful backgammon position today (on DailyGammon). At first glance, it is totally trival. All the complications are gone. In three rolls the game will be decided.
Still I managed to squeeze a blunder of nearly 0.2 equity out of it. After 10 minutes of thinking. With a piece of paper. And a calculator.
That’s how mysteriously complicated backgammon can be, even in its most simple positions.
Here we go:
This is a match to 15, score is 2:2.Should white double? Should black take?
I have still not figured out how to solve this one.
What’s your suggestion?
Spoiler starts here.
The trivial part is:
- if white rolls 44, 55, 66, white wins (1/12 probability)
- if white rolls something else, and black rolls a double, black wins (11/12*1/6 probability)
- if white rolls something else, black rolls no double, then black has only a single checker on 1 and will win for sure. Unless white can bear off the three stones in two rolls.
But how likely is this?
I was black, my opponent (correctly) doubled.
I arrived at the conclusion that I need 25% wins to take, the second dot gave me only 15%. So I need to win about 10% of all games in case 3. Since even with a 21 roll, white still has many combinations to win, I concluded that nearly all 2-roll combinations will bring the three checkers home and passed.
Tuns out, the correct cube action is double/take. In this position, black will win almost 30% of games (28,97%, actually; I even did a rollout).
I still have no clue how somebody might solve this OTB. And I’m still amazed that such a simple position is so difficult to solve. But that’s the beauty of backgammon.
White is a favorite (has better than a 50 % chance to win after his/her second roll, unless white’s first roll is 2-1), since having one checker on the 2 point and a second checker on the 5 point is a predetermined (known) reference point. All other first rolls (not 2-1) can be moved to arrive at a position greater than 50% chance to win on white’s second roll.
Thanks, seems I have to learn a few more reference points… Didn’t know of the 52 one.