By Valentin Updated Puzzle Bear-off
One position, one roll, one best play. Study the board, choose your move, then open the answer below. The position is drawn from your side: your checkers are the light ones and your home board is at the bottom right.
The position
- You are light and it is your turn. Both sides are bearing off and the checkers can no longer meet.
- Your opponent finishes on their next turn only with a double.
- The cube is not in play. No gammon is possible: both sides have borne off checkers.
- Pip counts: you 11, your opponent 3. Your roll: 4-3.
What is the best play?
You cannot get both checkers off with 4-3. How do you play it to give yourself the best chance of finishing next turn?
Show the answer
The answer: Play 5/1 6/3, leaving one checker on the 3 point and one on the 1 point. Next turn every roll except 2-1 bears them both off.
Why it is right
You need eleven pips to bear off both checkers and you rolled seven, so this turn cannot win the game. It is about the next one. Your opponent has three checkers on their ace point and is off next turn only with a double, 6 rolls in 36. Five times in six you get one more roll, and the whole question is which two checkers give that roll the best chance.
There are only two different plays. 5/2 6/2 puts both checkers on the 2 point. It looks tidy and it has the lowest pip count, but any roll with a 1 in it fails: a 1 cannot bear a checker off the 2 point, so the other die takes off only one. Ten of the 36 rolls contain a 1 and are not 1-1 (which plays four times and does get both off), so from the 2 point you finish with 26 rolls in 36.
5/1 6/3 leaves one checker on the 3 point and one on the 1 point. Now any die of 3 or more takes the checker off the 3 point and the other die, whatever it is, takes the last one off the 1 point. The small doubles work too: 1-1 plays 3/2/1/off and 1/off, and 2-2 plays 3/1/off and 1/off. Only 2-1 fails, so you finish with 34 rolls in 36.
Multiply each by the five in six chance that you get to roll at all, and 5/1 6/3 wins 78.7 percent of the time against 60.2 percent for 5/2 6/2. The lesson travels well beyond this position: with your last checkers, spread them so that different numbers do the work, and never stack two on a low point where a single small number strands one.
The numbers
| Play | Winning chances |
|---|---|
| 5/1 6/3best | 78.7% |
| 5/2 6/2 | 60.2% |
The gap between the best play and the next one is 18.5 percentage points of winning chances, or 0.370 in cubeless equity.
Checked by exact calculation. With no contact left, every sequence of rolls from this position to the end of the game can be enumerated, so the winning chances on this page are exact, not estimates. We computed them twice, with two programs written independently, and they agree to every digit shown. The cube is not in play, and no gammon is possible because both sides have already borne off checkers. This one is short enough to check by hand: 34 rolls in 36 against 26, each times five in six.
More puzzles
A new puzzle comes out every Monday. The next one is Puzzle 2: 6-1 with one checker still outside. All of them are on the backgammon puzzles page, and if a rule in this one surprised you, the bearing off rules and how to play backgammon cover it from the start.