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, 5 and 1 points, 18 pips.
- The cube is not in play. No gammon is possible: both sides have borne off checkers.
- Pip counts: you 14, your opponent 18. Your roll: 3-2.
What is the best play?
Nothing can come off with 3-2, so both dice must move. Which checkers do you move?
Show the answer
The answer: Play 6/3 6/4 and clear the 6 point. Moving one checker twice, 6/3/1, leaves a lone checker on the 6 point, which needs a 6, or a roll of six pips or more, to come off.
Why it is right
Nothing can be borne off this turn. Your 3 and 2 points are empty, and with checkers still on the 6 point neither die may take a checker off a lower point, so both dice must be played inside. There are only two different results: move both checkers off the 6 point, 6/3 6/4, or move one checker twice, 6/3/1, which leaves three checkers on the 1 point and one on the 6.
6/3/1 looks efficient. It sets up three easy bear-offs from the 1 point, and the pip count is the same either way. The trouble is the checker it leaves behind. A lone checker on the 6 point comes off directly only with a 6; otherwise it needs a roll of six pips or more, and that spends most of a roll on one checker. With 6, 1, 1 and 1 left you finish within two turns only 53.0 percent of the time.
6/3 6/4 clears the 6 point instead. The four checkers on the 4, 3, 1 and 1 points let almost every number do useful work: any 3 or 4 takes a checker off directly, a 5 or 6 takes one off the highest point, and a 1 takes one off the 1 point. You finish within two turns 74.2 percent of the time.
Your opponent, with their 6, 6, 5 and 1 points, finishes within two turns only 17.6 percent of the time, so most games come down to whether you need two turns or three. That is where the 21 point gap in two turn finishes goes: 6/3 6/4 wins 67.2 percent, 6/3/1 only 53.0 percent. Clear the highest point first. In a bear-off, a straggler on the 6 point is the most expensive checker you have.
The numbers
| Play | Winning chances |
|---|---|
| 6/3 6/4best | 67.2% |
| 6/3/1 | 53.0% |
The gap between the best play and the next one is 14.3 percentage points of winning chances, or 0.285 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 2: 6-1 with one checker still outside. The next one is Puzzle 4: 5-2, take a checker off or not?. 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.