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 three checkers are on their 6, 6 and 4 points, 16 pips.
- The cube is not in play. No gammon is possible: both sides have borne off checkers.
- Pip counts: you 14, your opponent 16. Your roll: 6-1.
What is the best play?
One of your checkers has not reached home yet. What is the best way to play 6-1?
Show the answer
The answer: Play 8/2 1/off. The 6 brings the last checker home, and once all your checkers are home the 1 may bear a checker off in the same turn.
Why it is right
Two rules meet here. You may not bear off while any of your checkers is outside your home board, and you choose the order in which you play your dice. Play the 6 first, 8/2, and every checker is home; from that moment the 1 is allowed to take a checker off the 1 point. Players who think of the turn as bring it in now, bear off next time miss this play, and it is the whole puzzle.
The quiet alternatives, 8/2 2/1 or 8/2 3/2, look neat but leave five checkers on the board. Each ordinary roll bears off at most two, so five checkers need three more turns unless you roll a double. After 8/2 1/off you have four checkers on the 3, 2, 1 and 1 points, and from there you finish within two more turns 99.7 percent of the time.
Your opponent needs two turns as well: from their 6, 6 and 4 points they finish in one only with 6-6, and within two turns 37.3 percent of the time. So after the right play the race is your two turns against their two, and you roll first. After a quiet play it is your three turns against their two or three, and they rarely need the third.
That is why the gap is so large: 8/2 1/off wins 66.4 percent of the time, both quiet plays only 24.1 percent. Near the end of a race, count checkers before pips. Pips tell you who is ahead over a long journey; in the last few turns the number of checkers left decides how many turns you need, and a checker borne off now is a whole turn saved.
The numbers
| Play | Winning chances |
|---|---|
| 8/2 1/offbest | 66.4% |
| 8/2 2/1 | 24.1% |
| 8/2 3/2 | 24.1% |
The gap between the best play and the next one is 42.3 percentage points of winning chances, or 0.846 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 1: 4-3 with your last two checkers. The next one 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.