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. The checkers can no longer meet: this is a pure race.
- Your opponent's last four checkers are on their 6, 6, 2 and 2 points, 16 pips.
- The cube is not in play. No gammon is possible: both sides have borne off checkers.
- Pip counts: you 17, your opponent 16. Your roll: 5-2.
What is the best play?
You can bear a checker off with 5-2, or you can keep all four on the board. Which play wins more often?
Show the answer
The answer: Play 6/1 4/2 and bear nothing off. One checker on each of the 1, 2, 3 and 4 points finishes in two turns more often than three checkers on the 4, 4 and 3.
Why it is right
The natural play is 6/4 4/off. The 2 clears the 6 point, and the 5 then takes a checker off the 4 point, which is legal because nothing is left higher. Three checkers remain instead of four, and fewer checkers usually means fewer turns. Not here.
Look at what each play leaves. After 6/4 4/off you have checkers on the 4, 4 and 3 points, and they only come off with a 3 or more. Apart from 2-2, any roll with a 1 or a 2 in it takes off one checker at most, and that is 19 rolls in 36. After 6/1 4/2 you have one checker on each of the 1, 2, 3 and 4 points, and every number has a checker waiting for it: every roll that is not a double bears off two. From there you finish within two turns 87.0 percent of the time, against 79.6 percent from the 4, 4 and 3.
That gap is what decides the game. Your opponent, with their 6, 6, 2 and 2 points, needs two turns at least and finishes within two only 21.2 percent of the time, and you roll first, so the play that most often finishes in two turns is the play that wins. 6/1 4/2 wins 72.0 percent, 6/4 4/off 67.3 percent, and 6/1 3/1, which stacks two checkers on the 1 point, only 50.8 percent.
Bearing off the most checkers you can is a habit, not a rule. In the last turns of a race, count how many turns each play is likely to need, and prefer the shape in which every number has a job. Gaps and stacks waste rolls; one checker per point wastes almost none.
The numbers
| Play | Winning chances |
|---|---|
| 6/1 4/2best | 72.0% |
| 6/4 4/off | 67.3% |
| 6/1 3/1 | 50.8% |
The gap between the best play and the next one is 4.7 percentage points of winning chances, or 0.094 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.
More puzzles
A new puzzle comes out every Monday. The one before this is Puzzle 3: 3-2 with two checkers on the 6-point. 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.